Finite Group Databasebeta

← #33137568

Subgroups — #33137568

Order 18688.

778 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988991C211yesmaximal, characteristic, complemented
2C2×#17988875C211yesmaximal, characteristic
3#22615102C211yesmaximal, characteristic
4#22613018C211yesmaximal, characteristic, complemented
5#22611415C211yesmaximal, characteristic, complemented
6#22611414C211yesmaximal, characteristic, complemented
7#22611319C211yesmaximal, characteristic
8#22611315C211yesmaximal, characteristic
9#22610877C211yesmaximal, characteristic, complemented
10#22610699C211yesmaximal, characteristic, complemented
11#22610698C211yesmaximal, characteristic, complemented
12#22610648C211yesmaximal, characteristic, complemented
13#22610553C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C73×#1925C211yesmaximal, characteristic
16C2×#14556590C2211yescharacteristic
17C2×#14556590C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19#17988991—24no—
20#17988875—24no—
21C2×#14556525C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×C4⋊#10596831C2211yescharacteristic
27C2×#14556520C2211yescharacteristic
28C2×#14556520C2211yescharacteristic
29C2×#14556512C2211yescharacteristic
30C2×#14556512C2211yescharacteristic
31C2×C8×Dic73C2211yescharacteristic
32#17988499—21no—
33#17988499—21no—
34#17988493—21no—
35#17988491—21no—
36#17988490C2211yescharacteristic
37#17988486C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988484—21no—
40#17988453C2211yescharacteristic
41#17988452C2211yescharacteristic
42#17988445—21no—
43#17988445—21no—
44C4⋊C4×Dic73C2211yescharacteristic
45C4⋊C4×Dic73C2211yescharacteristic
46C4×#11924007—21no—
47C4×#11924007C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988341C2211yescharacteristic
54C2×#107×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57#390×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60#336×C73C2211yescharacteristic
61C22×Dic146—21no—
62C22×Dic146—21no—
63C23×Dic73—21no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556590—22no—
67#14556590—22no—
68#14556590—22no—
69#14556590D414yes—
70D4×Dic73—22no—
71D4×Dic73—22no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007D411yescharacteristic
81C2×#11924007D411yescharacteristic
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C22⋊Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73C2311yescharacteristic
88C2×C4×Dic73C2311yescharacteristic
89#14556525—22no—
90#14556524—22no—
91#14556524—22no—
92#14556523—22no—
93#14556522—22no—
94C4⋊#10596831—22no—
95#14556520—22no—
96#14556520—22no—
97#14556512—22no—
98#14556512—22no—
99C8×Dic73—22no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498D411yescharacteristic
109#14556498D411yescharacteristic
110C22×D4×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C22⋊C4×C73—21no—
117C2×C22⋊C4×C73—21no—
118C2×C42×C73—21no—
119C2×C42×C73C2311yescharacteristic
120#107×C73—22no—
121#102×C73—22no—
122#102×C73Q814yescomplemented
123#95×C73—21no—
124C2×Dic146—24no—
125C2×Dic146—42no—
126C2×Dic146—81no—
127C2×Dic146—81no—
128C2×Dic146—81no—
129C2×Dic146—81no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—42no—
135C22×Dic73—42no—
136C22×Dic73—42no—
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C22×Dic73C2×D411yescharacteristic
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831#4112yes—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—81no—
155C4⋊Dic73C2×D412yes—
156#11924007—24no—
157#11924007—22no—
158#11924007—22no—
159#11924007—42no—
160#11924007—81no—
161C22⋊Dic73—42no—
162C22⋊Dic73—81no—
163C22⋊Dic73—81no—
164C4×Dic73—21no—
165C4×Dic73—21no—
166C4×Dic73—42no—
167C4×Dic73—81no—
168C4×Dic73C2×D412yes—
169C4×Dic73#4112yes—
170C24×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73—21no—
174C2×D4×C73—21no—
175C2×D4×C73—21no—
176C2×D4×C73C2×Q812yes—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2411yescharacteristic
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73C2×Q812yes—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73C2×D412yes—
194C4⋊C4×C73C2×Q812yes—
195C22⋊C4×C73—81no—
196C22⋊C4×C73—81no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—81no—
200C23×C73—22no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73C22×D411yescharacteristic
208D4×C73—22no—
209D4×C73—22no—
210D4×C73—41no—
211D4×C73—41no—
212C2×C4×C73—22no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—22no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73C22×D411yescharacteristic
228C2×C4×C73C22×Q811yescharacteristic
229C2×C4×C73C2×#4111yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C732−1+411yescharacteristic
232C2×C4×C732−1+411yescharacteristic
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—22no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—84no—
254C2×Dic73#12112yes—
255C2×Dic73#12112yes—
256C2×Dic73#13512yes—
257C2×Dic73#13712yes—
258Dic146—84no—
259Dic146—84no—
260C8×C73—22no—
261C8×C73—41no—
262#10596831—22no—
263#10596831—41no—
264C22×C73—42no—
265C22×C73—42no—
266C22×C73—42no—
267C22×C73—81no—
268C22×C73—81no—
269C22×C73—81no—
270C22×C73—81no—
271C22×C73C2×#12111yescharacteristic
272C22×C73#54711yescharacteristic
273C22×C73D4×Q811yescharacteristic
274C22×C73C2×#13511yescharacteristic
275C22×C73C2×#13711yescharacteristic
276C22×C73#57611yescharacteristic
277C22×C73#57611yescharacteristic
278C4×C73—41no—
279C4×C73—41no—
280C4×C73—42no—
281C4×C73—42no—
282C4×C73—81no—
283C4×C73#53311yescharacteristic
284C4×C73#54111yescharacteristic
285C4×C73#54811yescharacteristic
286C4×C73D4×Q811yescharacteristic
287Dic73—22no—
288Dic73—41no—
289Dic73—41no—
290Dic73—42no—
291Dic73—81no—
292Dic73—81no—
293Dic73—81no—
294Dic73#46414yes—
295#25450—731nomaximal, Sylow
296C2×C73—82no—
297C2×C73#242911yescharacteristic
298C2×C73C2×#46411yescharacteristic
299C2×C73#304111yescharacteristic
300C2×C73#334111yescharacteristic
301C2×C73#335711yescharacteristic
302C2×C73#338011yescharacteristic
303C2×C73#338511yescharacteristic
304C2×#500—731no—
305C2×#476—731no—
306#2563—731no—
307#2369—731no—
308#2085—731no—
309#2085—731no—
310#2067—731no—
311#2067—731no—
312#1987—731no—
313#1949—731no—
314#1949—731no—
315#1939—731no—
316#1925—731no—
317#1925D7311yescharacteristic
318C4×#102—731no—
319C73#2545011yescharacteristic, complemented, Sylow
320C2×C4⋊Q8—731no—
321C2×C4⋊Q8—731no—
322C2×C4×D4—731no—
323#500—1464no—
324#476—1464no—
325C2×#107—731no—
326C2×#107—731no—
327C2×#107—731no—
328C2×#107C2×D7311yescharacteristic
329C2×#106—731no—
330C2×#106—731no—
331C2×C4⋊C8—731no—
332C2×C4⋊C8—731no—
333C2×C4⋊C8—731no—
334C2×#102—731no—
335C2×#102—731no—
336C2×#102C2×D7311yescharacteristic
337C2×#102C2×D7311yescharacteristic
338C2×C4×C8—731no—
339#393—1461no—
340#393—1461no—
341#391—1461no—
342#391—1461no—
343#390—731no—
344#390C2×D7311yescharacteristic
345#389—731no—
346#389—731no—
347#389—1461no—
348#382—731no—
349#382—731no—
350#380—1461no—
351#380—1461no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—731no—
354C4×C4⋊C4—731no—
355C4×C4⋊C4—1461no—
356C4×C4⋊C4C2×D7311yescharacteristic
357#340—731no—
358#340—731no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340C2×D7311yescharacteristic
363#336—731no—
364#336C2×D7311yescharacteristic
365C22×Q8—1461no—
366C22×Q8—1461no—
367C22×D4C22×D7311yescharacteristic
368C23×C4—1461no—
369C22×C8—731no—
370C22×C8—731no—
371C22×C8C22×D7311yescharacteristic
372C22×C8C22×D7311yescharacteristic
373C4⋊Q8—734no—
374C4⋊Q8—1462no—
375C4⋊Q8—1462no—
376C4⋊Q8—1462no—
377C4×D4—1462no—
378C4×D4—1462no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—731no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4—1461no—
389C2×C4⋊C4C22×D7311yescharacteristic
390C2×C4⋊C4C22×D7311yescharacteristic
391C2×C4⋊C4C22×D7311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1461no—
395C2×C22⋊C4—1461no—
396C2×C22⋊C4—1461no—
397C2×C42—21no—
398C2×C42—731no—
399C2×C42—731no—
400C2×C42—1461no—
401C2×C42—1461no—
402C2×C42C22×D7311yescharacteristic
403#107—22no—
404#107—1462no—
405#107—1462no—
406#107—1462no—
407#106—1462no—
408#106—1462no—
409C4⋊C8—1462no—
410C4⋊C8—1462no—
411C4⋊C8—1462no—
412#102—22no—
413#102—1462no—
414#102—1462no—
415#102Dic14614yescomplemented
416C4×C8—1462no—
417#95—21no—
418#95—731no—
419#95—731no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428C24—21no—
429C2×Q8—2922no—
430C2×Q8—5841no—
431C2×Q8—5841no—
432C2×Q8—5841no—
433C2×Q8—5841no—
434C2×Q8—1464no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4C2×Dic14612yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—2922no—
445C22×C4—2922no—
446C22×C4—2922no—
447C22×C4—731no—
448C22×C4—731no—
449C22×C4—731no—
450C22×C4—731no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4D4×D7311yescharacteristic
456C22×C4D4×D7311yescharacteristic
457C22×C4C23×D7311yescharacteristic
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—732no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8C2×Dic14612yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—2922no—
475C4⋊C4—5841no—
476C4⋊C4—5841no—
477C4⋊C4—732no—
478C4⋊C4—1462no—
479C4⋊C4—1462no—
480C4⋊C4—1464no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4D4×D7312yes—
489C4⋊C4C2×Dic14612yes—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—2922no—
493C22⋊C4—5841no—
494C22⋊C4—5841no—
495C42—21no—
496C42—21no—
497C42—81no—
498C42—2922no—
499C42—5841no—
500C42—732no—
501C42—732no—
502C42—1461no—
503C42—1461no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D7311yescharacteristic
512Q8—5844no—
513Q8—5844no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—5844no—
545C2×C4—732no—
546C2×C4—732no—
547C2×C4—732no—
548C2×C4—732no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1462no—
555C2×C4—1461no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4C2×D4×D7311yescharacteristic
559C2×C4C22×Dic14611yescharacteristic
560C2×C4#41×D7311yescharacteristic
561C2×C4#41×D7311yescharacteristic
562C2×C4#1455663811yescharacteristic
563C2×C4#1455663811yescharacteristic
564C8—22no—
565C8—41no—
566C8—2921no—
567C8—1462no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22#121×D7311yescharacteristic
576C22#1798923511yescharacteristic
577C22D4×Dic14611yescharacteristic
578C22#135×D7311yescharacteristic
579C22#137×D7311yescharacteristic
580C22#1798938811yescharacteristic
581C22#1798939111yescharacteristic
582C4—42no—
583C4—42no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—2922no—
588C4—2921no—
589C4—2921no—
590C4—5841no—
591C4—5841no—
592C4—5841no—
593C4—734no—
594C4—1462no—
595C4#1798914011yescharacteristic
596C4#1798919411yescharacteristic
597C4#1798924511yescharacteristic
598C4D4×Dic14611yescharacteristic
599C2—82no—
600C2#2261352711yescharacteristic
601C2#464×D7311yescharacteristic
602C2#2261942911yescharacteristic
603C2#2262234811yescharacteristic
604C2#2262255011yescharacteristic
605C2#2262279811yescharacteristic
606C2#2262284211yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1925C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#1925D73
hypocenterC73#25450
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
1. C2×#17988991C2semidirect
4. #22613018C2semidirect
5. #22611415C2semidirect
6. #22611414C2semidirect
9. #22610877C2semidirect
10. #22610699C2semidirect
11. #22610698C2semidirect
12. #22610648C2semidirect
122. #102×C73Q8semidirect
319. C73#25450semidirect
415. #102Dic146semidirect

Maximal subgroups

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

subgroupindex
C2×#179889912
C2×#179888752
#226151022
#226130182
#226114152
#226114142
#226113192
#226113152
#226108772
#226106992
#226106982
#226106482
#226105532
#102×Dic732
C73×#19252
#2545073