Finite Group Databasebeta

← #33057246

Subgroups — #33057246

Order 18688.

592 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620009C211yesmaximal, characteristic
2#22619086C211yesmaximal, characteristic
3#22619019C211yesmaximal, characteristic, complemented
4#22608841C211yesmaximal, characteristic, complemented
5#22608830C211yesmaximal, characteristic, complemented
6#22608781C211yesmaximal, characteristic, complemented
7#22608780C211yesmaximal, characteristic, complemented
8#22608775C211yesmaximal, characteristic, complemented
9#22608767C211yesmaximal, characteristic, complemented
10#22608747C211yesmaximal, characteristic, complemented
11#22608715C211yesmaximal, characteristic, complemented
12#22608388C211yesmaximal, characteristic
13Q8⋊C8×D73C211yesmaximal, characteristic, complemented
14#22608222C211yesmaximal, characteristic, complemented
15C73×SD16⋊C8C211yesmaximal, characteristic, complemented
16#17989119C2211yescharacteristic
17C4×Q8×D73C2211yescharacteristic
18#17988919C411yescharacteristic
19#17988899C411yescharacteristic
20#17988897C411yescharacteristic
21#17988896C411yescharacteristic
22#17988808C411yescharacteristic
23#17988807C411yescharacteristic
24#17988744C411yescharacteristic
25#17988724C411yescharacteristic
26#17988717—21no—
27#17988717C2211yescharacteristic
28#17988717C2211yescharacteristic
29#17988714—21no—
30#17988714C2211yescharacteristic
31Q8×#10596831C2211yescharacteristic
32#17988711C2211yescharacteristic
33C4×Dic292C2211yescharacteristic
34C4×#11924018C2211yescharacteristic
35C4×#11924016C2211yescharacteristic
36#17988665C2211yescharacteristic
37#17988661—21no—
38C8×C22⋊D73—21no—
39#17988645—21no—
40#17988643—21no—
41#17988643C2211yescharacteristic
42#17988638—21no—
43#17988638C2211yescharacteristic
44#17988638C2211yescharacteristic
45#17988613—21no—
46C4⋊C8×D73C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988339C2211yescharacteristic
49#17988338C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988319C2211yescharacteristic
52#17988319C2211yescharacteristic
53Q8⋊#10596831C2211yescharacteristic
54#17988317C2211yescharacteristic
55#17988317C2211yescharacteristic
56D4⋊#10596831C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988309C2211yescharacteristic
59#17988308C2211yescharacteristic
60C8×Q8×C73C2211yescharacteristic
61C4×SD16×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×Q8×D73C2×C411yescharacteristic
68C2×Dic292C2×C411yescharacteristic
69C2×#11924018C2×C411yescharacteristic
70C2×#11924016C2×C411yescharacteristic
71C2×#11924012—21no—
72C2×C8×D73—21no—
73C2×C8×D73—21no—
74#14556584—21no—
75#14556582C2×C411yescharacteristic
76#14556578C2×C411yescharacteristic
77#14556573—21no—
78#14556571C2×C411yescharacteristic
79#14556562C2×C411yescharacteristic
80Q8×Dic73C2×C411yescharacteristic
81C4×Dic146—21no—
82C4×Dic146C2311yescharacteristic
83C4×Dic146C2311yescharacteristic
84#14556556—21no—
85#14556556C2×C411yescharacteristic
86C4×C22⋊D73—21no—
87#14556550—21no—
88C4⋊C4×D73—21no—
89C4⋊C4×D73C2×C411yescharacteristic
90C42×D73—21no—
91C42×D73C2311yescharacteristic
92#14556525C2×C411yescharacteristic
93#14556524C2×C411yescharacteristic
94#14556522C2×C411yescharacteristic
95C4⋊#10596831—21no—
96C4⋊#10596831C2311yescharacteristic
97C4⋊#10596831C2311yescharacteristic
98#14556520—21no—
99#14556520—21no—
100#14556520—21no—
101#14556520—21no—
102#14556520—21no—
103#14556520C2×C411yescharacteristic
104#14556520C2×C411yescharacteristic
105#14556520D411yescharacteristic
106#14556520D411yescharacteristic
107#14556516C2×C411yescharacteristic
108#14556516C2×C411yescharacteristic
109#14556515C2×C411yescharacteristic
110#14556514C2×C411yescharacteristic
111#14556514C2×C411yescharacteristic
112#14556512C2×C411yescharacteristic
113C22⋊#10596831—21no—
114#14556503—21no—
115#14556503C2×C411yescharacteristic
116#14556503C2×C411yescharacteristic
117#14556503D411yescharacteristic
118#14556503D411yescharacteristic
119#14556501—21no—
120#14556501—21no—
121#14556501—21no—
122#14556501—21no—
123C4×#10596831—21no—
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C8×Dic73—21no—
128C8×Dic73—21no—
129C2×SD16×C73C2×C411yescharacteristic
130C22×C8×C73—21no—
131C4×Q8×C73C2311yescharacteristic
132C4×D4×C73C2311yescharacteristic
133#106×C73C2×C411yescharacteristic
134C4⋊C8×C73—21no—
135C4⋊C8×C73C2311yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137#103×C73C2×C411yescharacteristic
138#102×C73C2×C411yescharacteristic
139C22⋊C8×C73—21no—
140C4×C8×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C2×Dic146—21no—
145C2×Dic146C22×C411yescharacteristic
146C2×Dic146C22×C411yescharacteristic
147Q8×D73—22no—
148Q8×D73—41no—
149C2×C22⋊D73—21no—
150C2×C4×D73—21no—
151C2×C4×D73—21no—
152C2×C4×D73C22×C411yescharacteristic
153C2×C4×D73C2×D411yescharacteristic
154C2×C4×D73#4111yescharacteristic
155Dic292—41no—
156#11924018—22no—
157#11924016—41no—
158#11924012—81no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831C22×C411yescharacteristic
163C2×#10596831C22×C411yescharacteristic
164C2×#10596831C2×D411yescharacteristic
165C2×#10596831C2×D411yescharacteristic
166C8×D73—42no—
167C8×D73—81no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73C22×C411yescharacteristic
172C4⋊Dic73C22×C411yescharacteristic
173#11924007—21no—
174#11924007—21no—
175#11924007—21no—
176#11924007—21no—
177#11924007—21no—
178C22⋊Dic73—21no—
179C22⋊Dic73—21no—
180C22⋊Dic73—21no—
181#11924005—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73C22×C411yescharacteristic
187C4×Dic73C2×D411yescharacteristic
188C4×Dic73#4111yescharacteristic
189C2×Q8×C73C22×C411yescharacteristic
190C2×D4×C73C22×C411yescharacteristic
191C22×C4×C73—21no—
192SD16×C73—22no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—81no—
197C2×C8×C73C22×C411yescharacteristic
198C2×C8×C73C22×C411yescharacteristic
199C2×C8×C73C2×D411yescharacteristic
200C2×C8×C73C2×D411yescharacteristic
201C4⋊C4×C73—21no—
202C4⋊C4×C73C22×C411yescharacteristic
203C4⋊C4×C73C22×C411yescharacteristic
204C22⋊C4×C73—21no—
205C42×C73—21no—
206C42×C73C2411yescharacteristic
207C23×C73—21no—
208C22×D73C4×D411yescharacteristic
209Q8×C73—21no—
210Q8×C73#13112yes—
211D4×C73—21no—
212D4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73C23×C411yescharacteristic
219C2×C4×C73C22×D411yescharacteristic
220C2×C4×C73C2×#4111yescharacteristic
221C22⋊D73—42no—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73C4×D411yescharacteristic
227C2×Dic73C4×D411yescharacteristic
228C2×Dic73C4×D411yescharacteristic
229C4×D73—22no—
230C4×D73—21no—
231C4×D73—21no—
232C4×D73—21no—
233C4×D73—21no—
234C4×D73—42no—
235C4×D73—42no—
236Dic146—21no—
237Dic146—21no—
238Dic146—21no—
239Dic146—21no—
240Dic146—42no—
241C8×C73—21no—
242C8×C73—21no—
243C8×C73—21no—
244C8×C73—21no—
245C8×C73—41no—
246C8×C73—41no—
247C8×C73—41no—
248#10596831—21no—
249#10596831—21no—
250#10596831—21no—
251#10596831—21no—
252#10596831—41no—
253#10596831—41no—
254#10596831—41no—
255C22×C73—41no—
256C22×C73—41no—
257C22×C73—41no—
258C22×C73C2×C4×D411yescharacteristic
259C2×D73—21no—
260C2×D73—21no—
261C2×D73#44412yes—
262C4×C73—22no—
263C4×C73—21no—
264C4×C73—41no—
265C4×C73—41no—
266C4×C73C2×#13111yescharacteristic
267C4×C73#56811yescharacteristic
268C4×C73#57711yescharacteristic
269C4×C73#57811yescharacteristic
270Dic73—21no—
271Dic73—21no—
272Dic73—21no—
273Dic73—41no—
274Dic73—41no—
275Dic73—41no—
276Dic73—41no—
277#18023—731nomaximal, Sylow
278C2×C73—81no—
279C2×C73#293311yescharacteristic
280C2×C73#294711yescharacteristic
281C2×C73C2×#44411yescharacteristic
282D73—22no—
283#3108—731no—
284#2994—731no—
285#2984—731no—
286#1593—731no—
287#1592—731no—
288#1584—731no—
289#1584—731no—
290#1583—731no—
291#1582—731no—
292#1579—731no—
293SD16⋊C8D7311yescharacteristic, complemented
294#1573—731no—
295#1494—731no—
296C2×Q8⋊C8—731no—
297#1452—731no—
298C73#1802311yescharacteristic, complemented, Sylow
299#529—731no—
300C2×C4×Q8—731no—
301#483—731no—
302#480—731no—
303#479—731no—
304#479—731no—
305#464—731no—
306#464—731no—
307#451—731no—
308#448—731no—
309#446—731no—
310#446—731no—
311#446—731no—
312#446—1461no—
313#446—1461no—
314C8×Q8—731no—
315C8×Q8—731no—
316C8×Q8C2×D7311yescharacteristic
317C4×Q16—731no—
318C4×Q16—731no—
319C4×SD16—731no—
320C4×SD16C2×D7311yescharacteristic
321#436—731no—
322#435—1461no—
323C8×D4—1461no—
324C8×D4C2×D7311yescharacteristic
325#433—1461no—
326#432—731no—
327#432—1461no—
328#431—731no—
329#431—731no—
330#431—1461no—
331#424—1461no—
332C2×C4⋊C8—731no—
333C2×C4×C8—731no—
334#335—731no—
335#335C2×D7311yescharacteristic
336#334—731no—
337#334—731no—
338Q8⋊C8—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8—731no—
343Q8⋊C8C2×D7311yescharacteristic
344D4⋊C8—731no—
345D4⋊C8C2×D7311yescharacteristic
346#322—731no—
347#322—731no—
348#322—731no—
349#322C2×D7311yescharacteristic
350C22×Q8—731no—
351C2×Q16—731no—
352C2×Q16—731no—
353C2×SD16—731no—
354C2×SD16C4×D7311yescharacteristic
355C2×M16—1461no—
356C22×C8—21no—
357C22×C8—1461no—
358C22×C8—1461no—
359#126—1461no—
360#125—731no—
361#124—731no—
362#123—1461no—
363C22⋊Q8—731no—
364#121—731no—
365C4×Q8—731no—
366C4×Q8—731no—
367C4×Q8—731no—
368C4×Q8—731no—
369C4×Q8—1461no—
370C4×Q8—1461no—
371C4×Q8C22×D7311yescharacteristic
372C4×D4—1461no—
373C4×D4C22×D7311yescharacteristic
374#117—1461no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—1461no—
377C2×C42—731no—
378C2×C42—1461no—
379#107—731no—
380#107—731no—
381#106—731no—
382#106C4×D7311yescharacteristic
383C4⋊C8—21no—
384C4⋊C8—731no—
385C4⋊C8—731no—
386C4⋊C8—731no—
387C4⋊C8—731no—
388C4⋊C8—731no—
389C4⋊C8—731no—
390C4⋊C8—1461no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8—1461no—
396C4⋊C8C22×D7311yescharacteristic
397C4⋊C8C22×D7311yescharacteristic
398#103—731no—
399#103—731no—
400#103—731no—
401#103—731no—
402#103—731no—
403#103C4×D7311yescharacteristic
404#102—731no—
405#102C4×D7311yescharacteristic
406C22⋊C8—21no—
407C22⋊C8—731no—
408C22⋊C8—731no—
409C22⋊C8—731no—
410C22⋊C8—731no—
411C22⋊C8—1461no—
412C22⋊C8—1461no—
413#97—1461no—
414#97—1461no—
415#97—1461no—
416#97—1461no—
417C4×C8—21no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—1461no—
422C4×C8—1461no—
423C4×C8—1461no—
424C4×C8C22×D7311yescharacteristic
425C4×C8C22×D7311yescharacteristic
426C4×C8C22×D7311yescharacteristic
427C2×Q8—2921no—
428C2×Q8—731no—
429C2×Q8—731no—
430C2×Q8—1461no—
431C2×Q8—1462no—
432C2×Q8C2×C4×D7311yescharacteristic
433C2×D4—1461no—
434C2×D4C2×C4×D7311yescharacteristic
435C22×C4—21no—
436C22×C4—731no—
437C22×C4—731no—
438C22×C4—731no—
439C22×C4—1461no—
440C22×C4—1461no—
441Q16—2921no—
442Q16—1462no—
443SD16—22no—
444SD16—2921no—
445M16—5841no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—81no—
450C2×C8—2922no—
451C2×C8—5841no—
452C2×C8—731no—
453C2×C8—731no—
454C2×C8—731no—
455C2×C8—731no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8C2×C4×D7311yescharacteristic
460C2×C8C2×C4×D7311yescharacteristic
461C2×C8D4×D7311yescharacteristic
462C2×C8D4×D7311yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—731no—
465C4⋊C4—731no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4C2×C4×D7311yescharacteristic
475C4⋊C4C2×C4×D7311yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—1461no—
478C22⋊C4—1461no—
479C22⋊C4—1461no—
480C22⋊C4—1461no—
481C42—21no—
482C42—731no—
483C42—731no—
484C42—731no—
485C42—1461no—
486C42—1461no—
487C42—1461no—
488C42—1461no—
489C42C23×D7311yescharacteristic
490C23—21no—
491C23—731no—
492Q8—21no—
493Q8—2922no—
494Q8—1461no—
495Q8—1461no—
496Q8—1461no—
497Q8—1461no—
498Q8#1455659712yes—
499D4—21no—
500D4—21no—
501D4—2922no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—2922no—
508C2×C4—2922no—
509C2×C4—731no—
510C2×C4—731no—
511C2×C4—731no—
512C2×C4—1462no—
513C2×C4—1461no—
514C2×C4—1461no—
515C2×C4—1461no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4—1461no—
520C2×C4—1461no—
521C2×C4C22×C4×D7311yescharacteristic
522C2×C4C2×D4×D7311yescharacteristic
523C2×C4#41×D7311yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8—2921no—
532C8—2921no—
533C8—2921no—
534C8—1461no—
535C8—1461no—
536C8—1461no—
537C8—1461no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—732no—
542C22—1461no—
543C22—1461no—
544C22C4×D4×D7311yescharacteristic
545C4—22no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—2921no—
550C4—2921no—
551C4—2921no—
552C4—2921no—
553C4—1461no—
554C4—1461no—
555C4—1461no—
556C4C2×#1455659711yescharacteristic
557C4#1798935211yescharacteristic
558C4#1798939711yescharacteristic
559C4#1798940311yescharacteristic
560C2—81no—
561C2—1462no—
562C2#2261865111yescharacteristic
563C2#2261877111yescharacteristic
564C2#444×D7311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×SD16⋊C8C2
socleC22×C73C2×C4×D4
prime radical#14556562C2×C4
cosocleC42×C73C24
hypercenterSD16⋊C8D73
hypocenterC73#18023
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
3. #22619019C2semidirect
4. #22608841C2semidirect
5. #22608830C2semidirect
6. #22608781C2semidirect
7. #22608780C2semidirect
8. #22608775C2semidirect
9. #22608767C2semidirect
10. #22608747C2semidirect
11. #22608715C2semidirect
13. Q8⋊C8×D73C2semidirect
14. #22608222C2semidirect
15. C73×SD16⋊C8C2semidirect
293. SD16⋊C8D73semidirect
298. C73#18023semidirect

Maximal subgroups

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

subgroupindex
#226200092
#226190862
#226190192
#226088412
#226088302
#226087812
#226087802
#226087752
#226087672
#226087472
#226087152
#226083882
Q8⋊C8×D732
#226082222
C73×SD16⋊C82
#1802373