Finite Group Databasebeta

← #33055865

Subgroups — #33055865

Order 18688.

706 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619937C211yesmaximal, characteristic
2#22618860C211yesmaximal, characteristic, complemented
3C8×D4×D73C211yesmaximal, characteristic
4#22608770C211yesmaximal, characteristic, complemented
5#22608768C211yesmaximal, characteristic, complemented
6#22608766C211yesmaximal, characteristic, complemented
7#22608764C211yesmaximal, characteristic, complemented
8C8×#11924017C211yesmaximal, characteristic, complemented
9C8×#11924015C211yesmaximal, characteristic, complemented
10C8×#11924016C211yesmaximal, characteristic, complemented
11#22608700C211yesmaximal, characteristic, complemented
12#22608354C211yesmaximal, characteristic
13#22608332C211yesmaximal, characteristic, complemented
14#22608213C211yesmaximal, characteristic, complemented
15C8×SD16×C73C211yesmaximal, characteristic, complemented
16C22×C8×D73—21no—
17#17989141C2211yescharacteristic
18C4×D4×D73C2211yescharacteristic
19#17988919C411yescharacteristic
20#17988916C411yescharacteristic
21#17988876C411yescharacteristic
22#17988875C411yescharacteristic
23#17988791C411yescharacteristic
24#17988790C411yescharacteristic
25#17988739C411yescharacteristic
26#17988736C411yescharacteristic
27#17988718C2211yescharacteristic, complemented
28#17988716—21no—
29#17988714C2211yescharacteristic, complemented
30C8×Dic146—21no—
31C8×Dic146C2211yescharacteristic, complemented
32C4×#11924017C2211yescharacteristic
33C4×#11924015C2211yescharacteristic
34C4×#11924016C2211yescharacteristic
35C8×D292C2211yescharacteristic
36D4×#10596831C2211yescharacteristic
37#17988652C2211yescharacteristic
38C8×C22⋊D73—21no—
39#17988650—21no—
40#17988638C2211yescharacteristic
41#17988638C2211yescharacteristic
42#17988612—21no—
43#17988609C2211yescharacteristic
44C4⋊C8×D73C2211yescharacteristic
45C22⋊C8×D73—21no—
46#17988541—21no—
47C4×C8×D73C2211yescharacteristic
48#17988338C2211yescharacteristic, complemented
49#17988337C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51#17988319C2211yescharacteristic, complemented
52Q8⋊#10596831C2211yescharacteristic, complemented
53#17988317C2211yescharacteristic, complemented
54#17988316C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic, complemented
58#17988309C2211yescharacteristic, complemented
59C8×#10596831C2211yescharacteristic, complemented
60C8×Q8×C73C2211yescharacteristic, complemented
61C4×SD16×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic
63#334×C73C2211yescharacteristic, complemented
64Q8⋊C8×C73C2211yescharacteristic, complemented
65D4⋊C8×C73C2211yescharacteristic
66C82×C73C2211yescharacteristic, complemented
67C2×D4×D73C2×C411yescharacteristic
68C22×C4×D73—21no—
69C2×#11924017C2×C411yescharacteristic
70C2×#11924016C2×C411yescharacteristic
71C2×#11924015C2×C411yescharacteristic
72C2×#11924011—21no—
73C22×#10596831—21no—
74C2×C8×D73—21no—
75C2×C8×D73—21no—
76C2×C8×D73—21no—
77C2×C8×D73—44no—
78C2×C8×D73D411yescharacteristic
79C2×C8×D73D411yescharacteristic
80#14556590C2×C411yescharacteristic
81#14556589C2×C411yescharacteristic
82#14556581—21no—
83#14556571C2×C411yescharacteristic
84#14556569—21no—
85#14556567C2×C411yescharacteristic
86C4×Dic146—21no—
87C4×Dic146C2311yescharacteristic
88C4×D292C2311yescharacteristic
89D4×Dic73C2×C411yescharacteristic
90#14556553C2×C411yescharacteristic
91C4×C22⋊D73—21no—
92#14556551—21no—
93#14556548—21no—
94C4⋊C4×D73C2×C411yescharacteristic
95C22⋊C4×D73—21no—
96C42×D73C2311yescharacteristic
97#14556523C2×C411yescharacteristic
98#14556522C2×C411yescharacteristic
99#14556522C2×C411yescharacteristic
100C4⋊#10596831—21no—
101C4⋊#10596831C2311yescharacteristic
102C4⋊#10596831C2311yescharacteristic
103#14556520—21no—
104#14556520—21no—
105#14556520—21no—
106#14556520—21no—
107#14556520—21no—
108#14556520C2×C411yescharacteristic
109#14556520C2×C411yescharacteristic
110#14556516C2×C411yescharacteristic
111#14556515C2×C411yescharacteristic
112#14556514C2×C411yescharacteristic
113#14556513C2×C411yescharacteristic
114#14556512C2×C411yescharacteristic
115#14556511C2×C411yescharacteristic
116C22⋊#10596831—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503—21no—
120#14556503C2×C411yescharacteristic
121#14556503C2×C411yescharacteristic
122#14556502—21no—
123#14556501—21no—
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C8×Dic73—21no—
128C8×Dic73D411yescharacteristic
129C8×Dic73D411yescharacteristic
130C2×SD16×C73C2×C411yescharacteristic
131C22×C8×C73—21no—
132C4×Q8×C73C2311yescharacteristic
133C4×D4×C73C2311yescharacteristic
134#106×C73C2×C411yescharacteristic
135C4⋊C8×C73—21no—
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138#103×C73C2×C411yescharacteristic
139#102×C73C2×C411yescharacteristic
140C22⋊C8×C73—21no—
141C4×C8×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C23×D73—21no—
146C2×Dic146—21no—
147C2×Dic146C22×C411yescharacteristic
148C2×D292C22×C411yescharacteristic
149C2×C22⋊D73—21no—
150D4×D73—22no—
151D4×D73—22no—
152C22×Dic73—21no—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—42no—
157C2×C4×D73—42no—
158C2×C4×D73C22×C411yescharacteristic
159C2×C4×D73C2×D411yescharacteristic
160C2×C4×D73#4111yescharacteristic
161#11924017—22no—
162#11924016—22no—
163#11924015—22no—
164#11924011—42no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—42no—
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C22×C411yescharacteristic
171C2×#10596831C2×D411yescharacteristic
172C2×#10596831C2×D411yescharacteristic
173C8×D73—22no—
174C8×D73—42no—
175C8×D73—42no—
176C8×D73—42no—
177C8×D73SD1614yescomplemented
178C4⋊Dic73—21no—
179C4⋊Dic73C22×C411yescharacteristic
180C4⋊Dic73C22×C411yescharacteristic
181#11924007—21no—
182#11924007—21no—
183#11924007—21no—
184#11924007—21no—
185#11924007—21no—
186#11924007—21no—
187C22⋊Dic73—21no—
188#11924005—21no—
189#11924005—21no—
190#11924005—21no—
191#11924005—21no—
192C4×Dic73—21no—
193C4×Dic73C22×C411yescharacteristic
194C4×Dic73C2×D411yescharacteristic
195C4×Dic73#4111yescharacteristic
196C2×Q8×C73C22×C411yescharacteristic
197C2×D4×C73C22×C411yescharacteristic
198C22×C4×C73—21no—
199SD16×C73—22no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—42no—
204C2×C8×C73C22×C411yescharacteristic
205C2×C8×C73C22×C411yescharacteristic
206C2×C8×C73C2×D411yescharacteristic
207C2×C8×C73C2×D411yescharacteristic
208C4⋊C4×C73—21no—
209C4⋊C4×C73C22×C411yescharacteristic
210C4⋊C4×C73C22×C411yescharacteristic
211C22⋊C4×C73—21no—
212C42×C73—21no—
213C42×C73C2411yescharacteristic
214C23×C73—21no—
215C22×D73—24no—
216C22×D73—21no—
217C22×D73—42no—
218C22×D73—42no—
219C22×D73C4×D411yescharacteristic
220Q8×C73—21no—
221Q8×C73—21no—
222D4×C73—21no—
223D4×C73#13112yes—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73C23×C411yescharacteristic
230C2×C4×C73C22×D411yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C22⋊D73—24no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—42no—
237C2×Dic73C4×D411yescharacteristic
238C2×Dic73C4×D411yescharacteristic
239C2×Dic73C4×D411yescharacteristic
240D292—21no—
241D292#13112yes—
242C4×D73—21no—
243C4×D73—21no—
244C4×D73—41no—
245C4×D73—42no—
246C4×D73—41no—
247C4×D73—41no—
248C4×D73—42no—
249C4×D73C2×SD1612yes—
250C4×D73#13512yes—
251Dic146—21no—
252Dic146—21no—
253Dic146—42no—
254C8×C73—21no—
255C8×C73—21no—
256C8×C73—21no—
257C8×C73—41no—
258C8×C73—41no—
259C8×C73—41no—
260C8×C73C2×SD1612yes—
261#10596831—21no—
262#10596831—21no—
263#10596831—21no—
264#10596831—41no—
265#10596831—41no—
266#10596831—41no—
267#10596831C2×SD1612yes—
268C22×C73—22no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73C2×C4×D411yescharacteristic
272C2×D73—21no—
273C2×D73—21no—
274C2×D73—22no—
275C2×D73—48no—
276C2×D73—41no—
277C2×D73—41no—
278C2×D73C4×SD1612yes—
279C4×C73—21no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—41no—
283C4×C73C2×#13111yescharacteristic
284C4×C73C2×#13111yescharacteristic
285C4×C73C22×SD1611yescharacteristic
286C4×C73C2×#13511yescharacteristic
287Dic73—21no—
288Dic73—21no—
289Dic73—41no—
290Dic73—41no—
291Dic73—41no—
292Dic73C4×SD1612yes—
293#17902—731nomaximal, Sylow
294C2×C73—42no—
295C2×C73#292911yescharacteristic
296C2×C73C2×C4×SD1611yescharacteristic
297C2×C73#295511yescharacteristic
298D73—22no—
299D73—42no—
300#3101—731no—
301#2960—731no—
302C2×C8×D4—731no—
303#1582—731no—
304#1582—731no—
305#1582—731no—
306#1582—731no—
307C8×SD16—731no—
308C8×SD16—731no—
309C8×SD16—731no—
310C8×SD16D7311yescharacteristic, complemented
311#1570—731no—
312#1486—731no—
313#1481—731no—
314#1450—731no—
315C73#1790211yescharacteristic, complemented, Sylow
316C23×C8—1461no—
317#533—731no—
318C2×C4×D4—731no—
319#483—731no—
320#483—731no—
321#476—731no—
322#476—731no—
323#461—731no—
324#461—731no—
325#450—731no—
326#450—731no—
327#446—731no—
328#446—731no—
329#446—1461no—
330C8×Q8—731no—
331C8×Q8—1461no—
332C8×Q8C2×D7311yescharacteristic, complemented
333C4×SD16—731no—
334C4×SD16—731no—
335C4×SD16—731no—
336C4×SD16C2×D7311yescharacteristic
337C8×D4—731no—
338C8×D4—731no—
339C8×D4—731no—
340C8×D4—1461no—
341C8×D4—1461no—
342C8×D4C2×D7311yescharacteristic
343#431—731no—
344#431—731no—
345#424—1461no—
346#423—731no—
347C2×C4⋊C8—731no—
348C2×C22⋊C8—1461no—
349#405—1461no—
350C2×C4×C8—731no—
351#334—731no—
352#334—731no—
353#334—731no—
354#334C2×D7311yescharacteristic, complemented
355Q8⋊C8—731no—
356Q8⋊C8—731no—
357Q8⋊C8—731no—
358Q8⋊C8C2×D7311yescharacteristic, complemented
359D4⋊C8—731no—
360D4⋊C8—731no—
361D4⋊C8—731no—
362D4⋊C8C2×D7311yescharacteristic
363#322—731no—
364#322—731no—
365C82—731no—
366C82C2×D7311yescharacteristic, complemented
367C22×D4—731no—
368C23×C4—1461no—
369C2×SD16—731no—
370C2×SD16—731no—
371C2×SD16—731no—
372C2×SD16C4×D7311yescharacteristic
373C2×M16—1461no—
374C22×C8—21no—
375C22×C8—2924no—
376C22×C8—731no—
377C22×C8—731no—
378C22×C8—1461no—
379C22×C8—1461no—
380C22×C8—1461no—
381C22×C8—1461no—
382C4⋊Q8—731no—
383C4⋊Q8—731no—
384#125—1461no—
385C22⋊Q8—731no—
386C22⋊Q8—731no—
387C22⋊Q8—1461no—
388C4×Q8—731no—
389C4×Q8—1461no—
390C4×Q8C22×D7311yescharacteristic
391C4×D4—731no—
392C4×D4—731no—
393C4×D4—731no—
394C4×D4—1461no—
395C4×D4—1461no—
396C4×D4C22×D7311yescharacteristic
397#117—1461no—
398C2×C4⋊C4—731no—
399C2×C22⋊C4—1461no—
400C2×C42—731no—
401#106—731no—
402#106—731no—
403#106—731no—
404#106C4×D7311yescharacteristic
405C4⋊C8—21no—
406C4⋊C8—731no—
407C4⋊C8—731no—
408C4⋊C8—731no—
409C4⋊C8—731no—
410C4⋊C8—1461no—
411C4⋊C8—1461no—
412C4⋊C8—1461no—
413C4⋊C8—1461no—
414C4⋊C8—1461no—
415C4⋊C8—1461no—
416C4⋊C8C22×D7311yescharacteristic
417C4⋊C8C22×D7311yescharacteristic
418#103—731no—
419#103—731no—
420#103—731no—
421#103C4×D7311yescharacteristic
422#102—731no—
423#102—731no—
424#102—731no—
425#102C4×D7311yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—731no—
428C22⋊C8—731no—
429C22⋊C8—1461no—
430C22⋊C8—1461no—
431C22⋊C8—1461no—
432C22⋊C8—1461no—
433#97—1461no—
434#97—1461no—
435C4×C8—21no—
436C4×C8—731no—
437C4×C8—731no—
438C4×C8—731no—
439C4×C8—731no—
440C4×C8—731no—
441C4×C8—1461no—
442C4×C8C22×D7311yescharacteristic
443C4×C8C22×D7311yescharacteristic
444C4×C8C22×D7311yescharacteristic
445C24—1461no—
446C2×Q8—731no—
447C2×Q8—1461no—
448C2×Q8C2×C4×D7311yescharacteristic
449C2×D4—731no—
450C2×D4—1461no—
451C2×D4—1462no—
452C2×D4—1462no—
453C2×D4C2×C4×D7311yescharacteristic
454C22×C4—21no—
455C22×C4—2922no—
456C22×C4—2922no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—731no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464SD16—22no—
465SD16—1462no—
466SD16—1462no—
467SD16—1462no—
468M16—2922no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—42no—
473C2×C8—2922no—
474C2×C8—2922no—
475C2×C8—2922no—
476C2×C8—2922no—
477C2×C8—731no—
478C2×C8—734no—
479C2×C8—731no—
480C2×C8—731no—
481C2×C8—731no—
482C2×C8—1462no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8C2×C4×D7311yescharacteristic
487C2×C8C2×C4×D7311yescharacteristic
488C2×C8D4×D7311yescharacteristic
489C2×C8D4×D7311yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—731no—
492C4⋊C4—731no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4C2×C4×D7311yescharacteristic
501C4⋊C4C2×C4×D7311yescharacteristic
502C22⋊C4—21no—
503C22⋊C4—1461no—
504C22⋊C4—1461no—
505C22⋊C4—1461no—
506C22⋊C4—1461no—
507C22⋊C4—1461no—
508C42—21no—
509C42—731no—
510C42—731no—
511C42—731no—
512C42—1461no—
513C42C23×D7311yescharacteristic
514C23—21no—
515C23—2922no—
516C23—2922no—
517C23—731no—
518C23—1464no—
519C23—1461no—
520Q8—21no—
521Q8—21no—
522Q8—2922no—
523Q8—1461no—
524Q8—1461no—
525D4—21no—
526D4—732no—
527D4—1464no—
528D4—1461no—
529D4#1455659712yes—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—2921no—
536C2×C4—2922no—
537C2×C4—2922no—
538C2×C4—2922no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—731no—
542C2×C4—732no—
543C2×C4—732no—
544C2×C4—731no—
545C2×C4—731no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4C22×C4×D7311yescharacteristic
552C2×C4C2×D4×D7311yescharacteristic
553C2×C4#41×D7311yescharacteristic
554C8—21no—
555C8—21no—
556C8—21no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—2921no—
561C8—2921no—
562C8—2921no—
563C8—732no—
564C8—1461no—
565C8—1461no—
566C8—1461no—
567C8SD16×D7312yes—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—2928no—
572C22—2921no—
573C22—2921no—
574C22—732no—
575C22—1462no—
576C22—1461no—
577C22—1461no—
578C22C4×D4×D7311yescharacteristic
579C4—21no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—2921no—
584C4—2921no—
585C4—2921no—
586C4—732no—
587C4—1461no—
588C4—1461no—
589C4#131×D7311yescharacteristic
590C4C2×#1455659711yescharacteristic
591C4C2×SD16×D7311yescharacteristic
592C4#135×D7311yescharacteristic
593C2—42no—
594C2—2922no—
595C2—1462no—
596C2#2261862011yescharacteristic
597C2C4×SD16×D7311yescharacteristic
598C2#2261882911yescharacteristic
subgroupG/H
centerC2×C8D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC8×SD16×C73C2
socleC22×C73C2×C4×D4
prime radicalC2×D4×D73C2×C4
cosocleC42×C73C24
hypercenterC8×SD16D73
hypocenterC73#17902
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×SD16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22618860C2semidirect
4. #22608770C2semidirect
5. #22608768C2semidirect
6. #22608766C2semidirect
7. #22608764C2semidirect
8. C8×#11924017C2semidirect
9. C8×#11924015C2semidirect
10. C8×#11924016C2semidirect
11. #22608700C2semidirect
13. #22608332C2semidirect
14. #22608213C2semidirect
15. C8×SD16×C73C2semidirect
27. #17988718C22semidirect
29. #17988714C22semidirect
31. C8×Dic146C22semidirect
48. #17988338C22semidirect
49. #17988337C22semidirect
50. #17988337C22semidirect
51. #17988319C22semidirect
52. Q8⋊#10596831C22semidirect
53. #17988317C22semidirect
57. #17988309C22semidirect
58. #17988309C22semidirect
59. C8×#10596831C22semidirect
60. C8×Q8×C73C22semidirect
63. #334×C73C22semidirect
64. Q8⋊C8×C73C22semidirect
66. C82×C73C22semidirect
177. C8×D73SD16semidirect
310. C8×SD16D73semidirect
315. C73#17902semidirect
332. C8×Q8C2×D73semidirect
354. #334C2×D73semidirect
358. Q8⋊C8C2×D73semidirect
366. C82C2×D73semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226199372
#226188602
C8×D4×D732
#226087702
#226087682
#226087662
#226087642
C8×#119240172
C8×#119240152
C8×#119240162
#226087002
#226083542
#226083322
#226082132
C8×SD16×C732
#1790273