Finite Group Databasebeta

← #33090358

Subgroups — #33090358

Order 18688.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988687C211yesmaximal, characteristic
2#22612577C211yesmaximal, characteristic, complemented
3#22612574C211yesmaximal, characteristic
4#22610813C211yesmaximal, characteristic
5#22610786C211yesmaximal, characteristic
6#22610668C211yesmaximal, characteristic, complemented
7#22610663C211yesmaximal, characteristic, complemented
8#22610625C211yesmaximal, characteristic
9#22610622C211yesmaximal, characteristic, complemented
10#22610611C211yesmaximal, characteristic
11#22610608C211yesmaximal, characteristic, complemented
12#22609829C211yesmaximal, characteristic, complemented
13C4×#14556516C211yesmaximal, characteristic, complemented
14#22609617C211yesmaximal, characteristic, complemented
15C4×#102×C73C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18#17988687C418yescomplemented
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556516C2211yescharacteristic
23C2×#14556516C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556502C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31#17988492C2211yescharacteristic
32#17988492C2211yescharacteristic
33#17988490C2211yescharacteristic
34#17988490C2211yescharacteristic
35#17988486C2211yescharacteristic
36#17988486C2211yescharacteristic
37#17988483C2211yescharacteristic
38#17988483C2211yescharacteristic
39#17988480—21no—
40#17988480—21no—
41#17988471—21no—
42#17988471—21no—
43#17988454—21no—
44#17988451—21no—
45#17988450—21no—
46#17988447—21no—
47C4×C4⋊Dic73C2211yescharacteristic
48#17988433C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×C4×C8×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×Dic146C2311yescharacteristic
61C2×#11924016—22no—
62C2×#11924016—22no—
63C2×#11924016C2×C414yes—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C4×Dic146—22no—
67C4×Dic146C2×C414yes—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C22⋊Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73C2311yescharacteristic
83C2×C4×Dic73C2311yescharacteristic
84#14556524—22no—
85#14556524C2×C414yes—
86#14556522—22no—
87#14556516—22no—
88#14556516—22no—
89#14556514C2×C414yes—
90#14556512C2×C414yes—
91#14556502C2×C414yes—
92#14556501—22no—
93#14556501—22no—
94C8×Dic73—22no—
95C8×Dic73—22no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498D411yescharacteristic
106#14556498D411yescharacteristic
107#14556498D411yescharacteristic
108#14556498D411yescharacteristic
109C22×D4×C73C2311yescharacteristic
110C23×C4×C73—21no—
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C4×D4×C73—22no—
114C4×D4×C73C2×C414yes—
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C22⋊C4×C73—21no—
119C2×C42×C73—21no—
120C2×C42×C73C2311yescharacteristic
121#102×C73—22no—
122#102×C73—22no—
123C4×C8×C73—22no—
124#95×C73—21no—
125C2×Dic146—21no—
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146C22×C412yes—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139#11924016—28no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831C22×C412yes—
143C2×#10596831C22×C412yes—
144C2×#10596831#4112yes—
145C2×#10596831#4112yes—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—81no—
152C4⋊Dic73—81no—
153C4⋊Dic73C22×C412yes—
154#11924007—24no—
155#11924007—81no—
156#11924007—81no—
157C22⋊Dic73—81no—
158C22⋊Dic73—81no—
159C22⋊Dic73—81no—
160C22⋊Dic73—81no—
161C4×Dic73—24no—
162C4×Dic73—21no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—21no—
166C4×Dic73—81no—
167C24×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73C22×C412yes—
174C22×C4×C73—21no—
175C22×C4×C73—24no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—81no—
179C22×C4×C73—81no—
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#4112yes—
188C2×C8×C73#4112yes—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73C22×C412yes—
195C22⋊C4×C73—24no—
196C42×C73—81no—
197C42×C73C22×C412yes—
198C42×C73#4112yes—
199C23×C73—22no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73C22×D411yescharacteristic
207D4×C73—22no—
208D4×C73—22no—
209D4×C73—41no—
210D4×C73—41no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—22no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—84no—
225C2×C4×C73C4×D412yes—
226C2×C4×C73C4×D412yes—
227C2×C4×C73C23×C411yescharacteristic
228C2×C4×C73C2×#4111yescharacteristic
229C2×C4×C73C2×#4111yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×Dic73—22no—
234C2×Dic73—22no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257Dic146—22no—
258Dic146—22no—
259Dic146—41no—
260Dic146—41no—
261C8×C73—22no—
262C8×C73—22no—
263#10596831—22no—
264#10596831#11714yes—
265C22×C73—42no—
266C22×C73—42no—
267C22×C73—42no—
268C22×C73—81no—
269C22×C73—81no—
270C22×C73—81no—
271C22×C73—81no—
272C22×C73C2×C4×D411yescharacteristic
273C22×C73#53211yescharacteristic
274C22×C73#53211yescharacteristic
275C22×C73#57611yescharacteristic
276C22×C73#57711yescharacteristic
277C22×C73#57711yescharacteristic
278C22×C73#57811yescharacteristic
279C4×C73—22no—
280C4×C73—22no—
281C4×C73—42no—
282C4×C73—81no—
283C4×C73—81no—
284C4×C73C2×#11711yescharacteristic
285C4×C73C4×#4111yescharacteristic
286C4×C73#52911yescharacteristic
287C4×C73#52911yescharacteristic
288Dic73—41no—
289Dic73—41no—
290Dic73—41no—
291Dic73—41no—
292Dic73—42no—
293Dic73—42no—
294Dic73—81no—
295Dic73—81no—
296#21291—731nomaximal, Sylow
297C2×C73—82no—
298C2×C73#229311yescharacteristic
299C2×C73#294611yescharacteristic
300C2×C73#294711yescharacteristic
301C2×C73#317411yescharacteristic
302C2×C73#317511yescharacteristic
303C2×C73#317811yescharacteristic
304C2×C73#317911yescharacteristic
305C2×#440—731no—
306#2320—731no—
307#2319—731no—
308#1973—731no—
309#1968—731no—
310#1943—731no—
311#1942—731no—
312#1936—731no—
313#1935—731no—
314#1934—731no—
315#1933—731no—
316#1776—731no—
317C4×#103—731no—
318C4×#102D7311yescharacteristic
319#1726—731no—
320C73#2129111yescharacteristic, complemented, Sylow
321C22×SD16—731no—
322C2×C4×Q8—731no—
323C2×C4×D4C2×D7311yescharacteristic
324#440—738no—
325C2×#107—731no—
326C2×#107—731no—
327C2×#106—731no—
328C2×#103—731no—
329C2×#103—731no—
330C2×#103—731no—
331C2×#102—731no—
332C2×#102C2×D7311yescharacteristic
333C2×#102C2×D7311yescharacteristic
334C2×#97—731no—
335C2×#97—731no—
336C2×#97—731no—
337C2×C4×C8—731no—
338C2×C4×C8—731no—
339C2×C4×C8C2×D7311yescharacteristic
340#391—731no—
341#391—731no—
342#390—731no—
343#390—731no—
344#389—731no—
345#389—731no—
346#389—731no—
347#389—731no—
348#388—1461no—
349#388—1461no—
350#387—1461no—
351#387—1461no—
352#383—1461no—
353#382—1461no—
354#381—1461no—
355#380—1461no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4C2×D7311yescharacteristic
358#376—731no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340C2×D7311yescharacteristic
364#340C2×D7311yescharacteristic
365C22×Q8—731no—
366C22×D4C22×D7311yescharacteristic
367C23×C4—21no—
368C2×SD16—734no—
369C2×SD16—1462no—
370C2×SD16—1462no—
371C22×C8—731no—
372C22×C8—731no—
373C22×C8C22×D7311yescharacteristic
374C22×C8C22×D7311yescharacteristic
375C4×Q8—734no—
376C4×Q8—1462no—
377C4×D4—22no—
378C4×D4C4×D7314yes—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4C22×D7311yescharacteristic
388C2×C4⋊C4C22×D7311yescharacteristic
389C2×C4⋊C4C22×D7311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1461no—
392C2×C22⋊C4—1461no—
393C2×C22⋊C4—1461no—
394C2×C22⋊C4—1461no—
395C2×C42—21no—
396C2×C42—731no—
397C2×C42—731no—
398C2×C42—1461no—
399C2×C42—1461no—
400C2×C42C22×D7311yescharacteristic
401#107—734no—
402#107—1462no—
403#106—1462no—
404#103—734no—
405#103—1462no—
406#103—1462no—
407#102—22no—
408#102—22no—
409#102—734no—
410#97—734no—
411#97—1462no—
412#97—1462no—
413C4×C8—22no—
414C4×C8—1462no—
415C4×C8—1462no—
416#95—21no—
417#95—731no—
418#95—731no—
419#95—731no—
420#95—731no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430C24—21no—
431C2×Q8—732no—
432C2×Q8—1461no—
433C2×Q8—1461no—
434C2×Q8—1461no—
435C2×Q8—1461no—
436C2×Q8—1461no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D7312yes—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—731no—
450C22×C4—731no—
451C22×C4—731no—
452C22×C4—731no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4D4×D7311yescharacteristic
458C22×C4D4×D7311yescharacteristic
459C22×C4C23×D7311yescharacteristic
460SD16—1468no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—732no—
466C2×C8—732no—
467C2×C8—732no—
468C2×C8—732no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8#1192402812yes—
472C2×C8#1192402812yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—5841no—
479C4⋊C4—5841no—
480C4⋊C4—5841no—
481C4⋊C4—5841no—
482C4⋊C4—732no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1464no—
488C4⋊C4—1461no—
489C4⋊C4C2×C4×D7312yes—
490C22⋊C4—24no—
491C22⋊C4—5841no—
492C22⋊C4—5841no—
493C22⋊C4—5841no—
494C22⋊C4—5841no—
495C42—81no—
496C42—5841no—
497C42—1464no—
498C42—1461no—
499C42—1461no—
500C42—1461no—
501C42—1461no—
502C42C2×C4×D7312yes—
503C42#1192402712yes—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D7311yescharacteristic
512Q8—2921no—
513Q8—2921no—
514Q8—1462no—
515Q8—1462no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—84no—
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—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1462no—
557C2×C4—1462no—
558C2×C4#1455655112yes—
559C2×C4#1455655112yes—
560C2×C4C22×C4×D7311yescharacteristic
561C2×C4#41×D7311yescharacteristic
562C2×C4#41×D7311yescharacteristic
563C2×C4C2×#1192402711yescharacteristic
564C2×C4C2×#1192402811yescharacteristic
565C2×C4C2×#1192402811yescharacteristic
566C8—22no—
567C8—22no—
568C8—734no—
569C8—1462no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C2×#1455655111yescharacteristic
578C22#1798913411yescharacteristic
579C22#1798913411yescharacteristic
580C22#1798938811yescharacteristic
581C22#1798939311yescharacteristic
582C22#1798939411yescharacteristic
583C22#1798940111yescharacteristic
584C4—22no—
585C4—22no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—2922no—
590C4—2921no—
591C4—2921no—
592C4—2922no—
593C4—2921no—
594C4—2921no—
595C4—5841no—
596C4—5841no—
597C4#117×D7311yescharacteristic
598C4C4×#1192402811yescharacteristic
599C4#1798911711yescharacteristic
600C4#1798911711yescharacteristic
601C2—82no—
602C2#2261239411yescharacteristic
603C2#2261875711yescharacteristic
604C2#2261877711yescharacteristic
605C2#2262055511yescharacteristic
606C2#2262056611yescharacteristic
607C2#2262059511yescharacteristic
608C2#2262061811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC4×#102×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenterC4×#102D73
hypocenterC73#21291
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #22612577C2semidirect
6. #22610668C2semidirect
7. #22610663C2semidirect
9. #22610622C2semidirect
11. #22610608C2semidirect
12. #22609829C2semidirect
13. C4×#14556516C2semidirect
14. #22609617C2semidirect
18. #17988687C4semidirect
320. C73#21291semidirect

Maximal subgroups

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

subgroupindex
C2×#179886872
#226125772
#226125742
#226108132
#226107862
#226106682
#226106632
#226106252
#226106222
#226106112
#226106082
#226098292
C4×#145565162
#226096172
C4×#102×C732
#2129173