Finite Group Databasebeta

← #16796055

Subgroups — #16796055

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#6122C211yesmaximal, characteristic
2#13131631C211yesmaximal, characteristic
3#13130476C211yesmaximal, characteristic, complemented
4#13130474C211yesmaximal, characteristic, complemented
5#13130450C211yesmaximal, characteristic, complemented
6#13130449C211yesmaximal, characteristic, complemented
7#13130449C211yesmaximal, characteristic, complemented
8#13130447C211yesmaximal, characteristic, complemented
9#13130423C211yesmaximal, characteristic, complemented
10Q16×#1057C211yesmaximal, characteristic, complemented
11#13130405C211yesmaximal, characteristic
12#13130184C211yesmaximal, characteristic
13C2×Q8⋊#1057C211yesmaximal, characteristic
14#13130037C211yesmaximal, characteristic
15#13124870C211yesmaximal, characteristic
16C2×Q8×Dic13C2211yescharacteristic
17#11719327C2211yescharacteristic
18#11719158—21no—
19#11719158—21no—
20#11719158C2211yescharacteristic
21#11719158C2211yescharacteristic
22#11719158C2211yescharacteristic
23Q8×#1057C2211yescharacteristic
24Q8×#1057C2211yescharacteristic
25Q8×#1057C2211yescharacteristic
26#11719129—21no—
27#11719129C2211yescharacteristic
28C4×#6122—21no—
29C4×#6122C2211yescharacteristic
30C2×C4⋊#1057C2211yescharacteristic
31C2×C4×#1057C2211yescharacteristic
32#11719080—21no—
33#11719080C2211yescharacteristic
34#11719078—21no—
35#11719074C2211yescharacteristic
36#11719067—21no—
37#11719062—21no—
38C13×#451C411yescharacteristic
39#11718889C411yescharacteristic
40Q16×Dic13C2211yescharacteristic
41Q16×Dic13C2211yescharacteristic
42#11718843C411yescharacteristic
43#11718843C411yescharacteristic
44#11718705C411yescharacteristic
45#11718455C411yescharacteristic
46#11718455C411yescharacteristic
47#11718451C2211yescharacteristic
48#11718451C2211yescharacteristic
49#11718383C411yescharacteristic
50#11718379C2211yescharacteristic
51Q8⋊#1057C2211yescharacteristic
52Q8⋊#1057C2211yescharacteristic
53Q8⋊#1057C2211yescharacteristic
54Q8⋊#1057C2211yescharacteristic
55Q8⋊#1057C2211yescharacteristic
56Q8⋊#1057C2211yescharacteristic
57#11718256C2211yescharacteristic
58#11718256C2211yescharacteristic
59#11718254C2211yescharacteristic
60#11718254C2211yescharacteristic
61#1057⋊C8C2211yescharacteristic
62#1057⋊C8C2211yescharacteristic
63C8×#1057C2211yescharacteristic
64C8×#1057C2211yescharacteristic
65#11718244C2211yescharacteristic
66#11718244C2211yescharacteristic
67C22×Q8×C13C2×C411yescharacteristic
68C2×#6122—21no—
69C2×#6122D411yescharacteristic
70C2×#6122D411yescharacteristic
71C22×#1057—21no—
72C22×#1057—21no—
73C2×C13×Q16C2×C411yescharacteristic
74C2×C13×Q16C2×C411yescharacteristic
75C13×C22⋊Q8C2×C411yescharacteristic
76Q8×Dic13—21no—
77Q8×Dic13—21no—
78Q8×Dic13C2×C411yescharacteristic
79Q8×Dic13C2×C411yescharacteristic
80Q8×Dic13C2311yescharacteristic
81Q8×Dic13C2311yescharacteristic
82Q8×Dic13C2311yescharacteristic
83#88198C2×C411yescharacteristic
84C2×#6107C2×C411yescharacteristic
85C2×#6107C2×C411yescharacteristic
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13C2×C411yescharacteristic
88#88178C2×C411yescharacteristic
89C2×C4×Dic13—21no—
90C2×C4×Dic13C2311yescharacteristic
91#88143—21no—
92#88142C2×C411yescharacteristic
93#88141—21no—
94#88141C2311yescharacteristic
95#88132—21no—
96#88131—21no—
97#88112—21no—
98#88112—21no—
99#88112—21no—
100#88112—21no—
101#88112—21no—
102#88110—21no—
103#88110—21no—
104#88110—21no—
105#88110—21no—
106#88110D411yescharacteristic
107#88110D411yescharacteristic
108C4⋊#1057—21no—
109C4⋊#1057—21no—
110C4⋊#1057—21no—
111C4⋊#1057—21no—
112C4⋊#1057—21no—
113C4⋊#1057C2311yescharacteristic
114C4⋊#1057C2311yescharacteristic
115C4⋊#1057C2311yescharacteristic
116C4⋊#1057C2311yescharacteristic
117C4×#1057—21no—
118C4×#1057—21no—
119C4×#1057—21no—
120C4×#1057—21no—
121C4×#1057—21no—
122C4×#1057C2311yescharacteristic
123C4×#1057C2311yescharacteristic
124C4×#1057C2311yescharacteristic
125C4×#1057C2311yescharacteristic
126C13×#103C2×C411yescharacteristic
127C13×#103C2×C411yescharacteristic
128C13×C22⋊C8C2×C411yescharacteristic
129#88075C2×C411yescharacteristic
130#88075C2×C411yescharacteristic
131#88075C2×C411yescharacteristic
132#88075C2×C411yescharacteristic
133C22⋊#1055C2×C411yescharacteristic
134#88058C2×C411yescharacteristic
135#88058C2×C411yescharacteristic
136#88056C2×C411yescharacteristic
137#88056C2×C411yescharacteristic
138#88054C2×C411yescharacteristic
139#88054C2×C411yescharacteristic
140C8×Dic13C2311yescharacteristic
141C8×Dic13C2311yescharacteristic
142#88047C2×C411yescharacteristic
143#88047C2×C411yescharacteristic
144#17998—131nomaximal, Sylow
145C2×Q8×C13—21no—
146C2×Q8×C13—22no—
147C2×Q8×C13—41no—
148C2×Q8×C13C22×C411yescharacteristic
149C2×Q8×C13C22×C411yescharacteristic
150C2×Q8×C13C22×C411yescharacteristic
151C22×C4×C13—21no—
152C22×C4×C13C22×C411yescharacteristic
153C22×Dic13—21no—
154C22×Dic13C2×D411yescharacteristic
155C22×Dic13#4111yescharacteristic
156C2×Dic26—21no—
157C2×Dic26C22×C411yescharacteristic
158#6122—22no—
159#6122—42no—
160#6122Q1614yescomplemented
161C2×#1057—21no—
162C2×#1057—21no—
163C2×#1057—21no—
164C2×#1057—21no—
165C2×#1057—21no—
166C2×#1057—21no—
167C2×#1057—42no—
168C2×#1057—42no—
169C2×#1057C2×D411yescharacteristic
170C2×#1057C2×D411yescharacteristic
171C2×#1057C2×D411yescharacteristic
172C2×#1057C2×D411yescharacteristic
173C13×Q16—22no—
174C13×Q16—41no—
175C2×C8×C13C22×C411yescharacteristic
176C2×C8×C13C22×C411yescharacteristic
177C13×C4⋊C4—21no—
178C13×C4⋊C4C22×C411yescharacteristic
179C13×C22⋊C4—21no—
180C22⋊Dic13—21no—
181#6107—22no—
182#6107—41no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13C22×C411yescharacteristic
190C4⋊Dic13C22×C411yescharacteristic
191C4⋊Dic13C22×C411yescharacteristic
192#6101—21no—
193#6101—21no—
194#6101—21no—
195#6101—21no—
196#6101—21no—
197C4×Dic13—21no—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—21no—
201C4×Dic13—21no—
202C4×Dic13—21no—
203C4×Dic13C22×C411yescharacteristic
204C4×Dic13C2×D411yescharacteristic
205C4×Dic13#4111yescharacteristic
206C4×Dic13C2411yescharacteristic
207C2×#1055C22×C411yescharacteristic
208C2×#1055C22×C411yescharacteristic
209#3104—131no—
210Q8×M16—131no—
211#2962—131no—
212#1591—131no—
213#1591—131no—
214#1584—131no—
215#1584—131no—
216#1584—131no—
217#1584—131no—
218C8×Q16—131no—
219C8×Q16—131no—
220#1571—131no—
221#1493—131no—
222C2×Q8⋊C8—131no—
223C8×M16—131no—
224C23×C13C4×D411yescharacteristic
225Q8×C13—21no—
226Q8×C13—21no—
227Q8×C13—21no—
228Q8×C13—21no—
229Q8×C13—21no—
230Q8×C13—42no—
231Q8×C13#13112yes—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—42no—
239C2×C4×C13C4×D411yescharacteristic
240C2×C4×C13C23×C411yescharacteristic
241C2×Dic13—21no—
242C2×Dic13—21no—
243C2×Dic13—21no—
244C2×Dic13—21no—
245C2×Dic13—41no—
246C2×Dic13—81no—
247C2×Dic13C4×D411yescharacteristic
248C2×Dic13C4×D411yescharacteristic
249C2×Dic13C2×Q1612yes—
250C2×Dic13#13512yes—
251C2×Dic13C22×D411yescharacteristic
252C2×Dic13C2×#4111yescharacteristic
253Dic26—21no—
254Dic26—21no—
255Dic26—81no—
256#1057—22no—
257#1057—21no—
258#1057—21no—
259#1057—41no—
260#1057—41no—
261#1057—41no—
262#1057—41no—
263#1057—41no—
264#1057C2×Q1612yes—
265#1057C2×Q1612yes—
266C8×C13—21no—
267C8×C13—21no—
268#1055—21no—
269#1055—21no—
270#533—131no—
271C2×C4×Q8—131no—
272#484—131no—
273#484—131no—
274#477—131no—
275#477—131no—
276#462—131no—
277#462—131no—
278#451—131no—
279#451#26111yescharacteristic
280#446—131no—
281#446—131no—
282#446—131no—
283#446—131no—
284#446—261no—
285#446—261no—
286#446—261no—
287#446—261no—
288C8×Q8—131no—
289C8×Q8—131no—
290C8×Q8—131no—
291C8×Q8—131no—
292C4×Q16—131no—
293C4×Q16—131no—
294C4×Q16—131no—
295C4×Q16—131no—
296#431—131no—
297#424—261no—
298#423—131no—
299#423—261no—
300C2×C4⋊C8—131no—
301#405—261no—
302C4×M16—131no—
303C4×M16—261no—
304C2×C4×C8—131no—
305#335—131no—
306#335—131no—
307#335—131no—
308#335—131no—
309Q8⋊C8—131no—
310Q8⋊C8—131no—
311Q8⋊C8—131no—
312Q8⋊C8—131no—
313Q8⋊C8—131no—
314Q8⋊C8—131no—
315Q8⋊C8—131no—
316Q8⋊C8—131no—
317#322—131no—
318#322—131no—
319C82—131no—
320C82—131no—
321C22×C13—21no—
322C22×C13—21no—
323C22×C13C4×Q1612yes—
324C22×C13C2×C4×D411yescharacteristic
325C4×C13—21no—
326C4×C13—22no—
327C4×C13—41no—
328C4×C13—41no—
329C4×C13—41no—
330C4×C13C2×#13111yescharacteristic
331C4×C13#56811yescharacteristic
332Dic13—21no—
333Dic13—21no—
334Dic13—41no—
335Dic13—41no—
336Dic13—41no—
337Dic13—41no—
338Dic13C4×Q1612yes—
339Dic13C22×Q1611yescharacteristic
340Dic13C2×#13511yescharacteristic
341C22×Q8C2×#26111yescharacteristic
342C2×Q16—131no—
343C2×Q16—131no—
344C2×Q16C2×#26111yescharacteristic
345C2×Q16C2×#26111yescharacteristic
346C2×M16—131no—
347C2×M16—131no—
348C2×M16—261no—
349C22×C8—261no—
350C22×C8—261no—
351C4⋊Q8—131no—
352C4⋊Q8—131no—
353#125—261no—
354C22⋊Q8—131no—
355C22⋊Q8—261no—
356C22⋊Q8C2×#26111yescharacteristic
357C4×Q8—131no—
358C4×Q8—131no—
359C4×Q8—131no—
360C4×Q8—131no—
361C4×Q8—131no—
362C4×Q8—131no—
363C4×Q8—261no—
364C4×Q8—261no—
365C4×Q8—261no—
366#117—261no—
367C2×C4⋊C4—131no—
368C2×C4⋊C4—261no—
369C2×C42—131no—
370C2×C42—261no—
371#107—131no—
372#107—131no—
373#107—131no—
374#107—131no—
375C4⋊C8—131no—
376C4⋊C8—131no—
377C4⋊C8—131no—
378C4⋊C8—131no—
379C4⋊C8—131no—
380C4⋊C8—131no—
381C4⋊C8—261no—
382C4⋊C8—261no—
383C4⋊C8—261no—
384C4⋊C8—261no—
385C4⋊C8—261no—
386C4⋊C8—261no—
387C4⋊C8—261no—
388C4⋊C8—261no—
389C4⋊C8—261no—
390C4⋊C8—261no—
391#103—131no—
392#103—131no—
393#103—131no—
394#103—131no—
395#103—131no—
396#103—131no—
397#103C2×#26111yescharacteristic
398#103C2×#26111yescharacteristic
399C22⋊C8—131no—
400C22⋊C8C2×#26111yescharacteristic
401#97—131no—
402#97—131no—
403#97—261no—
404#97—261no—
405#97—261no—
406#97—261no—
407C4×C8—131no—
408C4×C8—131no—
409C4×C8—131no—
410C4×C8—131no—
411C4×C8—131no—
412C4×C8—131no—
413C4×C8—261no—
414C4×C8—261no—
415C4×C8—261no—
416C4×C8—261no—
417C4×C8—261no—
418C2×C13—22no—
419C2×C13#293311yescharacteristic
420C2×C13C2×C4×Q1611yescharacteristic
421C2×C13#295611yescharacteristic
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8—41no—
425C2×Q8—131no—
426C2×Q8—261no—
427C2×Q8C22×#26111yescharacteristic
428C2×Q8C22×#26111yescharacteristic
429C2×Q8C22×#26111yescharacteristic
430C22×C4—21no—
431C22×C4—131no—
432C22×C4—131no—
433C22×C4—261no—
434C22×C4C22×#26111yescharacteristic
435Q16—22no—
436Q16—41no—
437Q16—262no—
438Q16—521no—
439M16—134no—
440M16—262no—
441M16—522no—
442C2×C8—131no—
443C2×C8—131no—
444C2×C8—131no—
445C2×C8—131no—
446C2×C8—131no—
447C2×C8—131no—
448C2×C8—261no—
449C2×C8—261no—
450C2×C8—261no—
451C2×C8—261no—
452C2×C8—261no—
453C2×C8—261no—
454C2×C8—522no—
455C2×C8—522no—
456C2×C8C22×#26111yescharacteristic
457C2×C8C22×#26111yescharacteristic
458C4⋊C4—21no—
459C4⋊C4—131no—
460C4⋊C4—131no—
461C4⋊C4—131no—
462C4⋊C4—261no—
463C4⋊C4—261no—
464C4⋊C4—261no—
465C4⋊C4—261no—
466C4⋊C4—261no—
467C4⋊C4—261no—
468C4⋊C4—261no—
469C4⋊C4—261no—
470C4⋊C4—261no—
471C4⋊C4—261no—
472C4⋊C4—261no—
473C4⋊C4C22×#26111yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—261no—
476C42—131no—
477C42—131no—
478C42—131no—
479C42—131no—
480C42—261no—
481C42—261no—
482C42—261no—
483C42—261no—
484C42—261no—
485C42—261no—
486C13#1799811yescharacteristic, complemented, Sylow
487C23D4×#26111yescharacteristic
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—42no—
494Q8—261no—
495Q8—261no—
496Q8—1041no—
497Q8#8823812yes—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—42no—
505C2×C4—132no—
506C2×C4—132no—
507C2×C4—131no—
508C2×C4—131no—
509C2×C4—131no—
510C2×C4—131no—
511C2×C4—261no—
512C2×C4—261no—
513C2×C4—261no—
514C2×C4—261no—
515C2×C4—521no—
516C2×C4—1041no—
517C2×C4D4×#26111yescharacteristic
518C2×C4C23×#26111yescharacteristic
519C8—21no—
520C8—21no—
521C8—132no—
522C8—132no—
523C8—261no—
524C8—261no—
525C8—262no—
526C8—261no—
527C8—261no—
528C8—521no—
529C8—521no—
530C8—521no—
531C8—521no—
532C8—521no—
533C22—21no—
534C22—21no—
535C22Q16×#26112yes—
536C22C2×D4×#26111yescharacteristic
537C4—21no—
538C4—22no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—131no—
543C4—131no—
544C4—132no—
545C4—261no—
546C4—261no—
547C4—521no—
548C4—521no—
549C4—521no—
550C4—521no—
551C4C2×#8823811yescharacteristic
552C4#1171962111yescharacteristic
553C2—22no—
554C2#1313156811yescharacteristic
555C2C2×Q16×#26111yescharacteristic
556C2#1313161111yescharacteristic
subgroupG/H
centerC22C2×D4×#261
commutatorC2×C4×C13C23×C4
FrattiniC2×C4C23×#261
FittingC13×#451C4
socleC22×C13C2×C4×D4
prime radicalC23×C13C4×D4
cosocleC4×Dic13C24
hypercenter#451#261
hypocenterC13#17998
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #13130476C2semidirect
4. #13130474C2semidirect
5. #13130450C2semidirect
6. #13130449C2semidirect
7. #13130449C2semidirect
8. #13130447C2semidirect
9. #13130423C2semidirect
10. Q16×#1057C2semidirect
160. #6122Q16semidirect
486. C13#17998semidirect

Maximal subgroups

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

subgroupindex
Q8×#61222
#131316312
#131304762
#131304742
#131304502
#131304492
#131304472
#131304232
Q16×#10572
#131304052
#131301842
C2×Q8⋊#10572
#131300372
#131248702
#1799813