Finite Group Databasebeta

← #10828651

Subgroups — #10828651

Order 768.

1030 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85718C211yesmaximal, characteristic
2C3×#2317C211yesmaximal, characteristic, complemented
3#82216C212yesmaximal, complemented
4#71591C212yesmaximal, complemented
5#71562C211yesmaximal, characteristic, complemented
6#69593C211yesmaximal, characteristic, complemented
7#69592C212yesmaximal, complemented
8#69573C211yesmaximal, characteristic, complemented
9#69563C211yesmaximal, characteristic, complemented
10#69562C212yesmaximal, complemented
11#69543C211yesmaximal, characteristic, complemented
12#24079—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16#5791—22no—
17#5791C2212yes—
18#5515—22no—
19#5514—22no—
20#5513—22no—
21#5511—22no—
22#5510C2211yescharacteristic
23#5507—22no—
24#5504C2212yes—
25#5500C2211yescharacteristic
26C3×#389—22no—
27C3×#385—22no—
28C3×#384C2211yescharacteristic
29C3×#384C2211yescharacteristic
30C3×#382C2211yescharacteristic
31C3×#382C2211yescharacteristic
32#5207C2212yes—
33#5003—22no—
34#5003—22no—
35#4996—22no—
36#4996—22no—
37#4966C2212yes—
38#4966C2212yes—
39#4964C2212yes—
40#4964C2212yes—
41C2×#805C2212yes—
42C2×#805C2212yes—
43#4530C2211yescharacteristic
44#4530C2212yes—
45#4530C2211yescharacteristic
46#4523C2212yes—
47#4523C2212yes—
48#3492—31no—
49#3093—32no—
50#2317D311yescharacteristic, complemented
51#2061—31no—
52#2060—31no—
53#2040—31no—
54#2039—31no—
55#1897—32no—
56#1860—32no—
57#1859—32no—
58#1821—31no—
59C23×C3×C4D412yes—
60C23×C3×C4C2311yescharacteristic
61C2×#235—22no—
62C2×#235C2312yes—
63C3×#117—22no—
64C3×#117—22no—
65C3×#117—22no—
66C3×#117—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4D412yes—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C42—21no—
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82#953—22no—
83#930—22no—
84#930—22no—
85#930D412yes—
86#930D412yes—
87#929—24no—
88#929—22no—
89C4×#67—24no—
90C4×#67—24no—
91C4×#67—22no—
92C4×#67—22no—
93#926—22no—
94#926C2312yes—
95#924—22no—
96#924—22no—
97#924D412yes—
98#924D412yes—
99C2×#208C2312yes—
100C2×#208C2312yes—
101#898—24no—
102#897—22no—
103#896—24no—
104#895—22no—
105#894—24no—
106#893—22no—
107#892—22no—
108D3×C4⋊C4—22no—
109#890—22no—
110#889—24no—
111#888—22no—
112#887—22no—
113C3×#95—22no—
114C3×#95—22no—
115#830—22no—
116#830—22no—
117#810—22no—
118#810—22no—
119#809—22no—
120#809—22no—
121#808—22no—
122#808—22no—
123#807—22no—
124#807—22no—
125#805—24no—
126#805—24no—
127#554—62no—
128#553—62no—
129#552—62no—
130#547—62no—
131#544—31no—
132#539—62no—
133#537—31no—
134#525—32no—
135#525—62no—
136#520—32no—
137C2×#117C2×D311yescharacteristic
138C2×#117C2×D311yescharacteristic
139C22×C4⋊C4C2×D311yescharacteristic
140#485—62no—
141#485—62no—
142#482—62no—
143#482—62no—
144#428—32no—
145#428—32no—
146C2×#104—32no—
147C2×#104—32no—
148#417—32no—
149#417—32no—
150#407—32no—
151#389—22no—
152#385—22no—
153#384C2×D311yescharacteristic
154#384C2×D311yescharacteristic
155#382C2×D311yescharacteristic
156#382C2×D311yescharacteristic
157#343—31no—
158#343—31no—
159#343—32no—
160#339—32no—
161#339—32no—
162C24×C3C2×D411yescharacteristic
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—82no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2411yescharacteristic
179C2×#67—22no—
180C2×#67—22no—
181C2×#67—22no—
182#235—22no—
183#235—22no—
184#235—22no—
185#235—22no—
186#235—84no—
187C2×D12—22no—
188C2×D12C2×D412yes—
189C2×C4×D3—22no—
190C2×C4×D3—24no—
191C2×Dic6—22no—
192C2×Dic6C2×D412yes—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—82no—
200C3×C4⋊C4—82no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—41no—
203C3×C22⋊C4—41no—
204C3×C42—21no—
205C3×C42—21no—
206C3×C42—42no—
207C3×C42#4112yescomplemented
208C3×C42#4112yescomplemented
209#217—22no—
210#217—22no—
211#217—22no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—22no—
215(C2×D3)⋊C4—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3C2×D412yes—
219C4⋊Dic3C2×D412yes—
220Dic3⋊C4—24no—
221Dic3⋊C4—24no—
222Dic3⋊C4—22no—
223Dic3⋊C4—22no—
224C4×Dic3—24no—
225C4×Dic3—22no—
226#208—22no—
227#208—22no—
228#208—22no—
229#208—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×#41—32no—
233C2×#41—62no—
234C23×C4#6712yes—
235C23×C4C22×D311yescharacteristic
236C2×M16—32no—
237C2×M16—32no—
238C4⋊Q8—62no—
239#126—64no—
240#125—62no—
241#125—64no—
242#123—64no—
243#123—64no—
244#123—62no—
245C22⋊Q8—32no—
246C22⋊Q8—32no—
247C22⋊Q8—62no—
248C22⋊Q8—64no—
249C22⋊Q8—62no—
250C22⋊Q8—62no—
251#121—32no—
252#121—32no—
253#121—62no—
254#121—62no—
255#121—62no—
256#120—62no—
257C4×Q8—62no—
258C4×D4—62no—
259C4×D4—64no—
260C4×D4—62no—
261C4×D4—62no—
262C4×D4—64no—
263#117—22no—
264#117—22no—
265#117—22no—
266#117—22no—
267#117—32no—
268#117—62no—
269#117—62no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4#6712yes—
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C42—21no—
284C2×C42C22×D311yescharacteristic
285C2×C42C22×D311yescharacteristic
286#107—62no—
287#107—62no—
288#106—62no—
289#106—62no—
290#104—64no—
291#104—64no—
292#103—62no—
293#103—62no—
294#102—62no—
295#102—62no—
296C22⋊C8—62no—
297C22⋊C8—62no—
298#95—22no—
299#95—22no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3#12011yescharacteristic
305C23×C3#12011yescharacteristic
306C23×C3C22×D411yescharacteristic
307C22×D3—22no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—44no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—82no—
330C2×C3×C4#12012yes—
331C2×C3×C4C22×D412yes—
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C42−1+412yes—
335#67—44no—
336#67—44no—
337#67—42no—
338#67—42no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342D12—22no—
343D12—22no—
344D12—82no—
345C4×D3—42no—
346C4×D3—42no—
347C4×D3—84no—
348Dic6—22no—
349Dic6—22no—
350Dic6—82no—
351C3⋊C8—42no—
352C3⋊C8—42no—
353C24C2×#6711yescharacteristic
354#41—62no—
355#41—62no—
356#41—62no—
357#41—62no—
358#41—244no—
359C2×Q8—32no—
360C2×Q8—62no—
361C2×D4—32no—
362C2×D4—62no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—62no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—64no—
376C22×C4—62no—
377C22×C4—82no—
378C22×C4D3×D412yes—
379C22×C4D3×D412yes—
380C22×C4C2×#6711yescharacteristic
381C22×C4C2×#6712yes—
382C22×C4C2×#6712yes—
383C22×C4C23×D311yescharacteristic
384M16—62no—
385M16—62no—
386M16—62no—
387M16—62no—
388C2×C8—62no—
389C2×C8—62no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—32no—
397C4⋊C4—32no—
398C4⋊C4—64no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—64no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C22⋊C4—24no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—64no—
410C22⋊C4—64no—
411C22⋊C4—62no—
412C22⋊C4—62no—
413C22⋊C4—62no—
414C22⋊C4—62no—
415C22⋊C4—62no—
416C42—21no—
417C42—21no—
418C42—42no—
419C42—64no—
420C42—62no—
421C42#23512yescomplemented
422C42#23512yescomplemented
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3C2×#12011yescharacteristic
434C22×C3#52511yescharacteristic
435C22×C3#53611yescharacteristic
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—42no—
439C3×C4—22no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4—81no—
445C3×C4—81no—
446C3×C4#54712yes—
447C3×C4#54712yes—
448Dic3—42no—
449Dic3—42no—
450Dic3—42no—
451C23—21no—
452C23—21no—
453C23—41no—
454C23—41no—
455C23—62no—
456C23#93711yescharacteristic
457C23#93711yescharacteristic
458C23C22×#6711yescharacteristic
459Q8—62no—
460Q8—62no—
461Q8—242no—
462D4—62no—
463D4—62no—
464D4—124no—
465D4—124no—
466D4—122no—
467D4—122no—
468D4—242no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—82no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—244no—
497C2×C4#93712yes—
498C2×C4C2×#23511yescharacteristic
499C2×C4C2×#23511yescharacteristic
500C2×C4C2×D3×D412yes—
501C2×C4#100712yes—
502C8—122no—
503C8—122no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3#240611yescharacteristic
509C2×C3#301911yescharacteristic
510C2×C3#302011yescharacteristic
511D3—82no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—42no—
521C22—42no—
522C22—122no—
523C22—122no—
524C22—122no—
525C22#579111yescharacteristic
526C22C2×#93711yescharacteristic
527C22#581611yescharacteristic
528C4—22no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—81no—
534C4—81no—
535C4—122no—
536C4—122no—
537C4—122no—
538C4#550712yes—
539C4#550712yes—
540C3#2407911yescharacteristic, complemented, Sylow
541C2—21no—
542C2—21no—
543C2—41no—
544C2—41no—
545C2—242no—
546C2#8219211yescharacteristic
547C2#8339411yescharacteristic
548C2#8341511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2317C2
socleC22×C3C2×#120
prime radical#85718C2
hypercenter#2317D3
hypocenterC3#24079
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C3×#2317C2semidirect
3. #82216C2semidirect
4. #71591C2semidirect
5. #71562C2semidirect
6. #69593C2semidirect
7. #69592C2semidirect
8. #69573C2semidirect
9. #69563C2semidirect
10. #69562C2semidirect
11. #69543C2semidirect
50. #2317D3semidirect
207. C3×C42#41semidirect
208. C3×C42#41semidirect
421. C42#235semidirect
422. C42#235semidirect
540. C3#24079semidirect

Maximal subgroups

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

subgroupindex
#857182
C3×#23172
#822162
#715912
#715622
#695932
#695922
#695732
#695632
#695622
#695432
#240793