Finite Group Databasebeta

← #11790785

Subgroups — #11790785

Order 1152.

805 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#10595029C211yesmaximal, characteristic
2C2×#63652C211yesmaximal, characteristic, complemented
3C2×C3×#825C211yesmaximal, characteristic, complemented
4C3×#5118C211yesmaximal, characteristic, complemented
5C4×Dic32C211yesmaximal, characteristic, complemented
6C2×#63570C211yesmaximal, characteristic, complemented
7C2×#63568C211yesmaximal, characteristic, complemented
8#70512—31nomaximal
9#70356—31nomaximal
10C2×#3828C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C4×Dic3C2211yescharacteristic
15C2×C3×C4×Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17#63978—22no—
18C2×Dic32—22no—
19#63911C411yescharacteristic, complemented
20C4×#626—22no—
21#63866C411yescharacteristic, complemented
22#63864C411yescharacteristic, complemented
23#63860C411yescharacteristic, complemented
24#63652—22no—
25C3×#825—22no—
26C3×Dic3⋊C8—22no—
27#63570—22no—
28#63568—22no—
29#5832—31no—
30#5477—31no—
31C3×#423—31no—
32C2×C3×#104—31no—
33C2×#837—31no—
34C2×#825—31no—
35C2×#825—31no—
36C2×#825—31no—
37C2×#825—31no—
38C2×#825—61no—
39C2×#825D311yescharacteristic, complemented
40#5118D311yescharacteristic, complemented
41C42×Dic3—31no—
42C42×Dic3—31no—
43C2×#805—31no—
44C22×C32×C4C2311yescharacteristic
45C2×#638—21no—
46C22×C3⋊Dic3—21no—
47#3828—21no—
48#3828—21no—
49#3828—21no—
50#3828—21no—
51C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
52C2×C4×C3⋊D3—21no—
53C2×#634C2×C411yescharacteristic
54C2×C3×#67—21no—
55C22×C3×Dic3—21no—
56C22×C3×Dic3D412yes—
57C3×#235—21no—
58C3×#235—21no—
59C3×#235—21no—
60C3×#235—21no—
61C2×C3×D12C2×C411yescharacteristic
62C2×C3×C4×D3—21no—
63C2×C3×Dic6C2×C411yescharacteristic
64C2×#626—22no—
65C32×M16—21no—
66C32×M16—21no—
67C2×C32×C8—21no—
68C4×C3⋊Dic3—21no—
69C4×C3⋊Dic3D412yescomplemented
70C3×C4×Dic3—21no—
71C3×C4×Dic3C2×C411yescharacteristic
72C3×C4×Dic3C2×C411yescharacteristic
73C3×C4×Dic3D412yes—
74C3×C4×Dic3D412yescomplemented
75C3×#208—21no—
76C3×#208—21no—
77C2×C3×C3⋊C8—21no—
78Dic3⋊Dic3—22no—
79#3722—22no—
80(C2×D3)⋊Dic3—22no—
81Dic32—28no—
82#1962—91noSylow
83C2×C3×#41—31no—
84C2×#235—31no—
85C2×#235—31no—
86C2×#235—61no—
87C2×#235C2×D311yescharacteristic
88C2×C3×M16—21no—
89C2×C3×M16—31no—
90C2×C3×M16C2×D311yescharacteristic
91C2×C3×M16C2×D311yescharacteristic
92C2×C3×C42—31no—
93C2×C3×C42—31no—
94#947—31no—
95#944—31no—
96#940—31no—
97#939—62no—
98#936—31no—
99#930—62no—
100C4×#67—62no—
101C2×C4×Dic3—31no—
102C2×C4×Dic3—31no—
103C2×C4×Dic3—61no—
104C2×C4×Dic3—62no—
105C2×C4×Dic3—62no—
106C2×C4×Dic3C2×D311yescharacteristic
107C2×C4×Dic3C2×D311yescharacteristic
108C2×#208C2×D311yescharacteristic
109#875—31no—
110#875—31no—
111#874—31no—
112C4×D12—62no—
113#869—31no—
114C3×C4⋊C8—62no—
115C3×#104—62no—
116#837—62no—
117#825—22no—
118#825—62no—
119#825—62no—
120#825—62no—
121#825—64no—
122#825—62no—
123Dic3⋊C8—22no—
124#805—62no—
125C23×C32C2×D411yescharacteristic
126C22×C3⋊D3—21no—
127C22×C3×D3—21no—
128C2×C32×C4C22×C411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4#4111yescharacteristic
133C2×C32×C4#4111yescharacteristic
134#638—41no—
135#638—41no—
136C2×C3⋊Dic3—24no—
137C2×C3⋊Dic3—22no—
138C2×C3⋊Dic3—21no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)—21no—
141C4×C3⋊D3—41no—
142C4×C3⋊D3—41no—
143#634—21no—
144#634—21no—
145C3×#67—41no—
146C3×#67—41no—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—24no—
150C2×C3×Dic3—21no—
151C2×C3×Dic3C22⋊C412yes—
152C2×C3×Dic3C22⋊C412yes—
153C3×D12—21no—
154C3×D12—21no—
155C3×C4×D3—41no—
156C3×C4×D3—41no—
157C3×Dic6—21no—
158C3×Dic6—21no—
159#626—44no—
160C32×C8—41no—
161C3×C3⋊C8—41no—
162#532—91no—
163#423—31no—
164C2×#104—31no—
165C2×#104—91no—
166C2×#104—91no—
167C2×#104—91no—
168C43—91no—
169C3×#41—61no—
170C3×#41—61no—
171C3×#41—61no—
172C3×#41—61no—
173C2×C3×Q8—31no—
174C2×C3×D4—31no—
175C2×C3×D4—61no—
176C22×C3×C4—21no—
177C22×C3×C4—32no—
178C22×C3×C4—61no—
179C22×C3×C4—61no—
180C22×C3×C4C22×D311yescharacteristic
181C22×C3×C4C22×D311yescharacteristic
182C2×#67—21no—
183C2×#67—61no—
184C2×#67—61no—
185C2×#67—62no—
186C2×#67—121no—
187C22×Dic3—21no—
188C22×Dic3—61no—
189C22×Dic3—61no—
190C22×Dic3—121no—
191C22×Dic3D1212yes—
192#235—21no—
193#235—21no—
194#235—21no—
195#235—21no—
196#235—62no—
197#235—62no—
198#235—61no—
199#235—61no—
200#235—61no—
201#235—61no—
202#235—61no—
203#235—61no—
204#235—61no—
205#235—61no—
206#235—121no—
207#235—121no—
208C2×D12—31no—
209C2×D12—31no—
210C2×D12—61no—
211C2×D12—62no—
212C2×D12C4×D311yescharacteristic
213C2×C4×D3—21no—
214C2×C4×D3—61no—
215C2×C4×D3—61no—
216C2×C4×D3—121no—
217C2×Dic6—31no—
218C2×Dic6—31no—
219C2×Dic6—61no—
220C2×Dic6C4×D311yescharacteristic
221C3×M16—22no—
222C3×M16—21no—
223C3×M16—21no—
224C3×M16—21no—
225C3×M16—21no—
226C3×M16—41no—
227C3×M16—61no—
228C3×M16—61no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—41no—
232C2×C3×C8—61no—
233C3×C42—31no—
234C3×C42—31no—
235C3×C42—32no—
236C3×C42—32no—
237C3×C42—61no—
238#217—62no—
239(C2×D3)⋊C4—62no—
240(C2×D3)⋊C4—62no—
241(C2×D3)⋊C4—62no—
242C4⋊Dic3—62no—
243Dic3⋊C4—62no—
244C4×Dic3—21no—
245C4×Dic3—32no—
246C4×Dic3—32no—
247C4×Dic3—62no—
248C4×Dic3—68no—
249C4×Dic3—68no—
250C4×Dic3—61no—
251C4×Dic3—61no—
252C4×Dic3—121no—
253C4×Dic3C4×D311yescharacteristic
254C4×Dic3C4×D311yescharacteristic
255C4×Dic3#6712yes—
256C4×Dic3#6712yescomplemented
257#208—21no—
258#208—21no—
259C2×C3⋊C8—21no—
260C22×C32—22no—
261C22×C32C2×C22⋊C411yescharacteristic
262C22×C32#12111yescharacteristic
263C22×C32#12111yescharacteristic
264C2×C3⋊D3—41no—
265C2×C3⋊D3—42no—
266C2×C3×D3—41no—
267C2×C3×D3—42no—
268C32×C4C4×D411yescharacteristic
269C32×C4C4×D411yescharacteristic
270C32×C4#12411yescharacteristic
271C32×C4#12711yescharacteristic
272C3⋊Dic3—24no—
273C3⋊Dic3—41no—
274C3×Dic3—24no—
275C3×Dic3—22no—
276C3×Dic3—41no—
277C3×Dic3#10414yescomplemented
278C2×#41—31no—
279C2×#41—91no—
280C2×M16—31no—
281C2×M16D3211yescharacteristic
282C4⋊Q8—91no—
283#127—91no—
284#124—91no—
285#124—91no—
286#121—182no—
287C4×D4—182no—
288C2×C42—31no—
289C2×C42—31no—
290C2×C42—91no—
291C2×C42—182no—
292C4⋊C8—62no—
293#104—62no—
294#104—182no—
295#104—182no—
296#104—182no—
297C23×C3—21no—
298C23×C3—61no—
299C23×C3C2×D1211yescharacteristic
300C23×C3D3×D411yescharacteristic
301C22×D3—21no—
302C22×D3—61no—
303C22×D3—61no—
304C22×D3—121no—
305C3×Q8—61no—
306C3×Q8—61no—
307C3×D4—61no—
308C3×D4—61no—
309C3×D4—121no—
310C3×D4—121no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—32no—
318C2×C3×C4—32no—
319C2×C3×C4—64no—
320C2×C3×C4—64no—
321C2×C3×C4—62no—
322C2×C3×C4—61no—
323C2×C3×C4—121no—
324C2×C3×C4—121no—
325C2×C3×C4C2×C4×D311yescharacteristic
326C2×C3×C4C2×C4×D311yescharacteristic
327C2×C3×C4#23511yescharacteristic
328C2×C3×C4D3×D411yescharacteristic
329C2×C3×C4D3×D411yescharacteristic
330C2×C3×C4D3×D411yescharacteristic
331C2×C3×C4D3×D411yescharacteristic
332C2×C3×C4#23711yescharacteristic
333C2×C3×C4#23911yescharacteristic
334C2×C3×C4#23911yescharacteristic
335C2×C3×C4C2×#6711yescharacteristic
336C2×C3×C4C2×#6711yescharacteristic
337#67—41no—
338#67—41no—
339#67—122no—
340#67—122no—
341#67—121no—
342#67—121no—
343#67—121no—
344#67—121no—
345#67—124no—
346C2×Dic3—24no—
347C2×Dic3—22no—
348C2×Dic3—24no—
349C2×Dic3—21no—
350C2×Dic3—64no—
351C2×Dic3—64no—
352C2×Dic3—62no—
353C2×Dic3—62no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—122no—
357C2×Dic3—121no—
358C2×Dic3—124no—
359C2×Dic3(C2×D3)⋊C412yes—
360C2×Dic3(C2×D3)⋊C412yes—
361D12—21no—
362D12—21no—
363D12—62no—
364D12—61no—
365D12—61no—
366D12—61no—
367D12—61no—
368D12—121no—
369D12—124no—
370C4×D3—41no—
371C4×D3—41no—
372C4×D3—122no—
373C4×D3—122no—
374C4×D3—121no—
375C4×D3—121no—
376C4×D3—121no—
377C4×D3—121no—
378Dic6—21no—
379Dic6—21no—
380Dic6—62no—
381Dic6—61no—
382Dic6—61no—
383Dic6—61no—
384Dic6—61no—
385Dic6—121no—
386C3×C8—42no—
387C3×C8—41no—
388C3×C8—41no—
389C3×C8—121no—
390C3⋊C8—41no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32#39011yescharacteristic
394C2×C32C2×#10412yes—
395C3⋊D3—81no—
396C3×D3—81no—
397#41—61no—
398#41—61no—
399#41—61no—
400#41—61no—
401#41—181no—
402#41—181no—
403#41—181no—
404#41—181no—
405C2×Q8—31no—
406C2×Q8—91no—
407C2×D4—31no—
408C2×D4—61no—
409C2×D4—91no—
410C2×D4—182no—
411C2×D4—181no—
412C22×C4—32no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—181no—
416C22×C4—181no—
417C22×C4C2×D3211yescharacteristic
418M16—21no—
419M16—21no—
420M16—61no—
421M16—61no—
422C2×C8—21no—
423C2×C8—61no—
424C4⋊C4—182no—
425C22⋊C4—182no—
426C22⋊C4—182no—
427C42—31no—
428C42—31no—
429C42—32no—
430C42—32no—
431C42—61no—
432C42—92no—
433C42—188no—
434C42—181no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—42no—
441C22×C3—122no—
442C22×C3—121no—
443C22×C3D3×C22⋊C411yescharacteristic
444C22×C3#88411yescharacteristic
445C22×C3#89511yescharacteristic
446C22×C3#89511yescharacteristic
447C22×C3C2×(C2×D3)⋊C411yescharacteristic
448C22×C3#93911yescharacteristic
449C2×D3—41no—
450C2×D3—42no—
451C2×D3—122no—
452C2×D3—122no—
453C2×D3—121no—
454C2×D3—121no—
455C2×D3—122no—
456C2×D3—242no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—34no—
462C3×C4—64no—
463C3×C4—62no—
464C3×C4—121no—
465C3×C4#67⋊C411yescharacteristic
466C3×C4#88511yescharacteristic
467C3×C4#89311yescharacteristic
468C3×C4#89311yescharacteristic
469C3×C4C4×#6711yescharacteristic
470C3×C4#94011yescharacteristic
471C3×C4#94011yescharacteristic
472C3×C4#94711yescharacteristic
473Dic3—24no—
474Dic3—22no—
475Dic3—41no—
476Dic3—64no—
477Dic3—64no—
478Dic3—124no—
479Dic3—122no—
480Dic3—121no—
481Dic3—121no—
482Dic3#82514yescomplemented
483C32#196211yescharacteristic, complemented, Sylow
484C23—61no—
485C23—181no—
486C23D3×D1211yescharacteristic
487Q8—61no—
488Q8—61no—
489Q8—181no—
490Q8—181no—
491D4—61no—
492D4—61no—
493D4—121no—
494D4—121no—
495D4—181no—
496D4—181no—
497D4—364no—
498D4—361no—
499D4—361no—
500C2×C4—32no—
501C2×C4—32no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—62no—
505C2×C4—61no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—184no—
509C2×C4—182no—
510C2×C4—181no—
511C2×C4—361no—
512C2×C4—361no—
513C2×C4#379611yescharacteristic
514C2×C4#379911yescharacteristic
515C2×C4C4×D3211yescharacteristic
516C2×C4D3×#6711yescharacteristic
517C2×C4D3×#6711yescharacteristic
518C2×C4#381111yescharacteristic
519C8—41no—
520C8—121no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—22no—
526C2×C3—21no—
527C2×C3—41no—
528C2×C3—41no—
529C2×C3—241no—
530C2×C3D3×#10412yes—
531C2×C3#495711yescharacteristic
532C2×C3#498811yescharacteristic
533C2×C3C2×#82512yes—
534D3—81no—
535D3—242no—
536D3—241no—
537D3—241no—
538C22—22no—
539C22—122no—
540C22—121no—
541C22—362no—
542C22—361no—
543C22#6388611yescharacteristic
544C22#6391011yescharacteristic
545C22D3×(C2×D3)⋊C411yescharacteristic
546C4—34no—
547C4—64no—
548C4—62no—
549C4—121no—
550C4—184no—
551C4—361no—
552C4#6388111yescharacteristic
553C4#6390411yescharacteristic
554C4#6390711yescharacteristic
555C4#6397911yescharacteristic
556C3—21no—
557C3#7035611yescharacteristic, complemented
558C3#7051211yescharacteristic, complemented
559C2—21no—
560C2—21no—
561C2—241no—
562C2—721no—
563C2D3×#82512yes—
564C2#1059138411yescharacteristic
subgroupG/H
centerC2×C4D3×#67
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32C2×C22⋊C4
prime radical#10595029C2
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1962
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×#63652C2semidirect
3. C2×C3×#825C2semidirect
4. C3×#5118C2semidirect
5. C4×Dic32C2semidirect
6. C2×#63570C2semidirect
7. C2×#63568C2semidirect
19. #63911C4semidirect
21. #63866C4semidirect
22. #63864C4semidirect
23. #63860C4semidirect
39. C2×#825D3semidirect
40. #5118D3semidirect
69. C4×C3⋊Dic3D4semidirect
74. C3×C4×Dic3D4semidirect
256. C4×Dic3#67semidirect
277. C3×Dic3#104semidirect
482. Dic3#825semidirect
483. C32#1962semidirect
557. C3#70356semidirect
558. C3#70512semidirect

Maximal subgroups

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

subgroupindex
#105950292
C2×#636522
C2×C3×#8252
C3×#51182
C4×Dic322
C2×#635702
C2×#635682
#705123
#703563