Finite Group Databasebeta

← #10823311

Subgroups — #10823311

Order 768.

1054 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86581C211yesmaximal, characteristic
2D3×#428C212yesmaximal
3#74225C211yesmaximal, characteristic
4C3×#2061C211yesmaximal, characteristic, complemented
5#71901C211yesmaximal, characteristic, complemented
6#70666C212yesmaximal, complemented
7#69543C211yesmaximal, characteristic, complemented
8#68989C211yesmaximal, characteristic
9#68979C211yesmaximal, characteristic, complemented
10#68753C212yesmaximal
11#68735C212yesmaximal, complemented
12#23562—31nomaximal, Sylow
13C3×#544C2211yescharacteristic
14C2×D3×M16C2212yes—
15#5729—22no—
16#5728—22no—
17#5689—22no—
18#5688—22no—
19#5634—22no—
20#5622—22no—
21D3×#117C2212yes—
22#5519C2212yes—
23C2×#892C2211yescharacteristic
24C2×D3×C4⋊C4C2211yescharacteristic
25#5500C2211yescharacteristic
26C2×#872C2211yescharacteristic
27C3×#428C2212yes—
28C3×#382C2211yescharacteristic
29#5121C2212yes—
30#4995C2212yes—
31#4977C2211yescharacteristic
32#4975C2211yescharacteristic
33#4964C2212yes—
34#4935C2211yescharacteristic
35#4881—22no—
36D3×#107—22no—
37#4861—22no—
38D3×#106—22no—
39#4668—22no—
40#4647—22no—
41C3×#343C2211yescharacteristic
42C3×#339C2212yes—
43#4555C2212yes—
44#4554C2212yes—
45#4532C2212yes—
46#4530C2211yescharacteristic
47#4523C2212yes—
48#3458—31no—
49C2×#428—32no—
50#2317—31no—
51#2061—31no—
52#2061—31no—
53#2061—31no—
54#2061D311yescharacteristic, complemented
55#2060—32no—
56#2060—32no—
57#1742—31no—
58#1733—32no—
59C22×C4×D3—21no—
60C22×C4×D3C2311yescharacteristic
61C2×C3×M16C2312yes—
62C3×C4⋊Q8—22no—
63C3×#125—22no—
64C3×C22⋊Q8—22no—
65C3×#117C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C42C2311yescharacteristic
69#946—24no—
70Q8×Dic3—22no—
71#944—22no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4D412yes—
74#926C2312yes—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78#924—22no—
79C2×Dic3⋊C4—21no—
80C2×Dic3⋊C4—21no—
81C2×Dic3⋊C4—22no—
82C2×C4×Dic3D412yes—
83C2×C4×Dic3C2311yescharacteristic
84C2×#208C2312yes—
85D3×M16—24no—
86D3×M16Q818yescomplemented
87C2×ΓL(1,𝔽25)—22no—
88C2×D3×C8—22no—
89#897—24no—
90#896—24no—
91#896—24no—
92#892—22no—
93#892—22no—
94#892D412yes—
95#892D412yes—
96D3×C4⋊C4—22no—
97D3×C4⋊C4—22no—
98D3×C4⋊C4D412yes—
99D3×C4⋊C4D412yes—
100#890—22no—
101#890—22no—
102#889—24no—
103#888—22no—
104#888—22no—
105#888—22no—
106#887—22no—
107#887—24no—
108D3×C22⋊C4—22no—
109#878—24no—
110#877—22no—
111#872—22no—
112#872—22no—
113#872Q814yescomplemented
114C42×D3—22no—
115#870—22no—
116#869—22no—
117C3×#107—22no—
118C3×#106—22no—
119C3×#95—22no—
120C3⋊#95—22no—
121#818—22no—
122#817—22no—
123#808—22no—
124#808—22no—
125#807—22no—
126#807—22no—
127C22×M16—32no—
128#544—32no—
129#544—31no—
130#544—62no—
131#544—62no—
132#544C2×D311yescharacteristic
133#543—62no—
134#533—62no—
135#533—62no—
136#533—62no—
137C2×#117—31no—
138C2×#117—31no—
139C2×#117—32no—
140C22×C4⋊C4—31no—
141#428—32no—
142#428—32no—
143#428—32no—
144#428—62no—
145#428—62no—
146#428C2×D312yes—
147C2×#107—62no—
148C2×#106—62no—
149#389—62no—
150#385—62no—
151#384—31no—
152#384—31no—
153#382—31no—
154#382C2×D311yescharacteristic
155#343—32no—
156#343—31no—
157#343C2×D311yescharacteristic
158#339—32no—
159#339—32no—
160#339—32no—
161#339C2×D312yes—
162C23×D3C2×D411yescharacteristic
163C2×C3×Q8—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3C2×D412yes—
171C22×Dic3C2×D411yescharacteristic
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×C4×D3—42no—
175C2×C4×D3—82no—
176C2×C4×D3C2×D412yes—
177C2×C4×D3C2×D412yes—
178C2×C4×D3C2×Q814yes—
179C2×C4×D3C2×Q812yes—
180C2×C4×D3#4112yes—
181C2×Dic6—24no—
182C2×Dic6—22no—
183C3×M16—22no—
184C3×M16C2×Q814yes—
185C2×C3×C8—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4C2×D412yes—
191C3×C4⋊C4C2×D412yes—
192C3×C22⋊C4—22no—
193C3×C42—22no—
194C3×C42—21no—
195C3×C42C2×Q812yes—
196#217—22no—
197(C2×D3)⋊C4—22no—
198(C2×D3)⋊C4—22no—
199(C2×D3)⋊C4—24no—
200(C2×D3)⋊C4—24no—
201C4⋊Dic3—24no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—82no—
206C4⋊Dic3C2×D412yes—
207C4⋊Dic3C2×D412yes—
208Dic3⋊C4—24no—
209Dic3⋊C4—24no—
210Dic3⋊C4—24no—
211Dic3⋊C4—22no—
212Dic3⋊C4—22no—
213Dic3⋊C4—24no—
214Dic3⋊C4—41no—
215Dic3⋊C4—41no—
216Dic3⋊C4—82no—
217C4×Dic3—22no—
218C4×Dic3—21no—
219C4×Dic3—22no—
220C4×Dic3—22no—
221C4×Dic3—22no—
222C4×Dic3—22no—
223C4×Dic3C2×Q812yes—
224#208—22no—
225#208C2×Q814yes—
226C2×C3⋊C8—22no—
227ΓL(1,𝔽25)—28no—
228D3×C8—28no—
229C23×C4—31no—
230C23×C4—61no—
231C2×M16—32no—
232C2×M16—38no—
233C2×M16—62no—
234C2×M16—64no—
235C2×M16C22×D312yes—
236C22×C8—62no—
237C4⋊Q8—22no—
238C4⋊Q8—62no—
239C4⋊Q8—62no—
240C4⋊Q8—62no—
241C4⋊Q8—62no—
242C4⋊Q8—62no—
243#125—22no—
244#125—62no—
245#125—62no—
246#125—64no—
247#125—62no—
248C22⋊Q8—22no—
249C22⋊Q8—64no—
250C22⋊Q8—64no—
251C22⋊Q8—64no—
252C22⋊Q8—64no—
253C22⋊Q8—64no—
254C22⋊Q8—62no—
255C4×Q8—62no—
256C4×Q8—64no—
257C4×Q8—62no—
258#117—34no—
259#117—32no—
260#117—32no—
261#117—32no—
262#117—62no—
263#117—62no—
264#117—62no—
265#117—62no—
266#117—62no—
267#117C22×D312yes—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4—62no—
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C22⋊C4—32no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—61no—
283C2×C42—31no—
284C2×C42—32no—
285C2×C42—62no—
286C2×C42C22×D311yescharacteristic
287#107—22no—
288#107—62no—
289#107—62no—
290#107—62no—
291#106—22no—
292#106—62no—
293#106—62no—
294#106—62no—
295#95—22no—
296#95—62no—
297C23×C3C22×D411yescharacteristic
298C22×D3—41no—
299C22×D3—41no—
300C22×D3#12012yes—
301C22×D3C22⋊Q812yes—
302C22×D3C4⋊Q812yes—
303C3×Q8—82no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4C22×D412yes—
314C2×C3×C4C22×Q812yes—
315C2×C3×C4C22×Q811yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—41no—
326C2×Dic3#12011yescharacteristic
327C2×Dic3#12011yescharacteristic
328C2×Dic3C22⋊Q812yes—
329C2×Dic3C4⋊Q812yes—
330C4×D3—82no—
331C4×D3—82no—
332C4×D3—84no—
333C4×D3C22⋊Q814yes—
334C4×D3C22⋊Q814yes—
335C4×D3C4⋊Q814yes—
336C4×D3C4⋊Q814yes—
337Dic6—84no—
338Dic6—82no—
339C3×C8—24no—
340C3⋊C8—24no—
341C24—31no—
342C2×Q8—22no—
343C2×Q8—64no—
344C2×Q8—62no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—31no—
348C22×C4—32no—
349C22×C4—32no—
350C22×C4—32no—
351C22×C4—34no—
352C22×C4—32no—
353C22×C4—32no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—122no—
359C22×C4—242no—
360C22×C4D3×D412yes—
361C22×C4C23×D311yescharacteristic
362M16—22no—
363M16—34no—
364M16—68no—
365M16—62no—
366M16D3×Q814yes—
367C2×C8—22no—
368C2×C8—68no—
369C2×C8—62no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—32no—
375C4⋊C4—32no—
376C4⋊C4—62no—
377C4⋊C4—62no—
378C4⋊C4—64no—
379C4⋊C4—64no—
380C4⋊C4—64no—
381C4⋊C4—64no—
382C4⋊C4—64no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C4⋊C4—62no—
386C4⋊C4—121no—
387C4⋊C4—121no—
388C4⋊C4—242no—
389C4⋊C4—242no—
390C4⋊C4D3×D412yes—
391C4⋊C4D3×D412yes—
392C22⋊C4—22no—
393C22⋊C4—64no—
394C22⋊C4—62no—
395C22⋊C4—62no—
396C22⋊C4—64no—
397C22⋊C4—62no—
398C42—22no—
399C42—21no—
400C42—32no—
401C42—62no—
402C42—62no—
403C42—62no—
404C42—62no—
405C42—62no—
406C42—61no—
407C42D3×Q812yes—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3C2×#12011yescharacteristic
411C22×C3C2×C22⋊Q811yescharacteristic
412C22×C3C2×C4⋊Q811yescharacteristic
413C2×D3—22no—
414C2×D3—22no—
415C2×D3—42no—
416C2×D3—42no—
417C2×D3#39514yes—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—42no—
421C3×C4—42no—
422C3×C4—81no—
423C3×C4C2×C22⋊Q812yes—
424C3×C4C2×C4⋊Q812yes—
425Dic3—42no—
426Dic3—42no—
427Dic3—42no—
428Dic3—42no—
429Dic3—81no—
430Dic3#39512yes—
431Dic3#39512yes—
432C23—32no—
433C23—32no—
434C23—32no—
435C23—121no—
436C23—121no—
437C23C2×D3×D411yescharacteristic
438Q8—82no—
439Q8—244no—
440Q8—242no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—32no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—32no—
449C2×C4—34no—
450C2×C4—34no—
451C2×C4—34no—
452C2×C4—34no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—121no—
467C2×C4—242no—
468C2×C4—242no—
469C2×C4—244no—
470C2×C4C2×D3×D412yes—
471C2×C4C2×D3×Q811yescharacteristic
472C2×C4C2×D3×Q812yes—
473C2×C4D3×#4111yescharacteristic
474C8—24no—
475C8—64no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3C2×#39511yescharacteristic
479C2×C3#302111yescharacteristic
480C2×C3#302211yescharacteristic
481D3—24no—
482C22—21no—
483C22—21no—
484C22—34no—
485C22—62no—
486C22—62no—
487C22—122no—
488C22—122no—
489C22D3×#12011yescharacteristic
490C22D3×C22⋊Q811yescharacteristic
491C22D3×C4⋊Q811yescharacteristic
492C4—32no—
493C4—32no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—81no—
499C4—122no—
500C4—122no—
501C4—122no—
502C4—122no—
503C4—241no—
504C4D3×C22⋊Q812yes—
505C4D3×C4⋊Q812yes—
506C3#2356211yescharacteristic, complemented, Sylow
507C2—21no—
508C2—21no—
509C2—64no—
510C2D3×#39511yescharacteristic
511C2#8028611yescharacteristic
512C2#8029311yescharacteristic
subgroupG/H
centerC22D3×#120
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2061C2
socleC22×C3C2×#120
prime radicalC23×D3C2×D4
hypercenter#2061D3
hypocenterC3#23562
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
4. C3×#2061C2semidirect
5. #71901C2semidirect
6. #70666C2semidirect
7. #69543C2semidirect
9. #68979C2semidirect
11. #68735C2semidirect
54. #2061D3semidirect
86. D3×M16Q8semidirect
113. #872Q8semidirect
506. C3#23562semidirect

Maximal subgroups

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

subgroupindex
#865812
D3×#4282
#742252
C3×#20612
#719012
#706662
#695432
#689892
#689792
#687532
#687352
#235623