Finite Group Databasebeta

← #10693209

Subgroups — #10693209

Order 768.

644 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84059C211yesmaximal, characteristic
2#78635C211yesmaximal, characteristic
3C3×#1784C211yesmaximal, characteristic, complemented
4#72132C211yesmaximal, characteristic, complemented
5#69979C211yesmaximal, characteristic
6#67687C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9519—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×#942—21no—
11#5791C2211yescharacteristic
12C2×C3×#101—21no—
13C2×C3×#101—21no—
14C3×#407C2211yescharacteristic
15C3×#407C2211yescharacteristic
16#5228—21no—
17#5228C412yescomplemented
18#5227—21no—
19#5227C412yescomplemented
20#5225—21no—
21C2×#835—21no—
22C2×#835—21no—
23#5212C2211yescharacteristic
24#5207C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#5128—21no—
27#5127—21no—
28#4730—21no—
29#4727—21no—
30#67⋊C8—21no—
31#4720—21no—
32#4470—21no—
33#4467—21no—
34#3004—31no—
35#2976—31no—
36#1787—31no—
37#1784D311yescharacteristic, complemented
38#1513—31no—
39#1305—31no—
40#1293—31no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—21no—
45C22×C3×Q8—21no—
46C22×C3×Q8—42no—
47C22×C3×Q8—42no—
48C22×C3×Q8C2×C412yes—
49C23×C3×C4—21no—
50C23×C3×C4C2311yescharacteristic
51C2×#235D411yescharacteristic
52C2×C3×M16—21no—
53C2×C3×M16D411yescharacteristic
54C2×C3×M16D411yescharacteristic
55#953—21no—
56C2×#217—21no—
57C2×#216—22no—
58#942—42no—
59#942—42no—
60C2×#215—22no—
61#930C2×C411yescharacteristic
62#930D411yescharacteristic
63#929—21no—
64C4×#67—21no—
65C4×#67—21no—
66#926D411yescharacteristic
67C2×C4⋊Dic3—21no—
68#924C2×C411yescharacteristic
69#924D411yescharacteristic
70C2×#208—21no—
71C22×C3⋊C8D411yescharacteristic
72C22×C3⋊C8D411yescharacteristic
73C2×ΓL(1,𝔽25)—21no—
74C2×D3×C8—21no—
75C3×#101—42no—
76C3×#101—42no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81#835—22no—
82#835—22no—
83#835—42no—
84#835—42no—
85C3⋊#95—21no—
86#830—21no—
87#830—21no—
88#830—21no—
89#830C2×C412yes—
90(C2×D3)⋊C8—21no—
91(C2×D3)⋊C8—21no—
92#815—21no—
93Dic3⋊C8—21no—
94Dic3⋊C8—21no—
95C8×Dic3—21no—
96C23×Q8C2×D311yescharacteristic
97C2×#137—61no—
98#525—31no—
99#451—32no—
100#451—61no—
101#448—32no—
102#448—61no—
103#436—61no—
104#435—61no—
105#435—61no—
106C8×D4—61no—
107#433—61no—
108#419—61no—
109C2×#103—61no—
110C2×#103—61no—
111C2×#101—21no—
112C2×#101—21no—
113#408—61no—
114#407—31no—
115#407C2×D311yescharacteristic
116#407C2×D311yescharacteristic
117C2×C22⋊C8—31no—
118#380—31no—
119#328—61no—
120#327—61no—
121C24×C3#4111yescharacteristic
122C2×C3×Q8—22no—
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—44no—
131C2×C3×Q8—41no—
132C2×C3×Q8—41no—
133C2×C3×Q8—41no—
134C2×C3×Q8—41no—
135C2×C3×Q8—41no—
136C2×C3×Q8—41no—
137C2×C3×Q8—44no—
138C2×C3×Q8—44no—
139C2×C3×Q8—41no—
140C2×C3×Q8—41no—
141C22×C3×C4—22no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4—44no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4C22×C411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4#4111yescharacteristic
153C2×#67—21no—
154C2×#67—21no—
155C2×#67—21no—
156C22×Dic3—41no—
157#235—41no—
158#235—41no—
159C2×D12C22⋊C411yescharacteristic
160C2×C4×D3—21no—
161C2×Dic6C22⋊C411yescharacteristic
162C3×M16—41no—
163C3×M16—41no—
164C3×M16—41no—
165C3×M16—41no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—41no—
169#217—21no—
170#217—21no—
171#217—21no—
172#217—81no—
173#216—44no—
174#215—44no—
175(C2×D3)⋊C4—21no—
176(C2×D3)⋊C4—21no—
177C4⋊Dic3—41no—
178C4⋊Dic3—41no—
179C4⋊Dic3C22⋊C411yescharacteristic
180C4⋊Dic3C22⋊C411yescharacteristic
181Dic3⋊C4—21no—
182Dic3⋊C4—21no—
183C4×Dic3—21no—
184#208—41no—
185#208—41no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8—21no—
188C2×C3⋊C8—21no—
189C2×C3⋊C8—41no—
190C2×C3⋊C8—41no—
191ΓL(1,𝔽25)—81no—
192D3×C8—81no—
193C2×#41—31no—
194C22×Q8—21no—
195C22×Q8—21no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—21no—
199C22×Q8—42no—
200C22×Q8—42no—
201C22×Q8C4×D312yes—
202C23×C4—21no—
203C23×C4C22×D311yescharacteristic
204#137—122no—
205#137—122no—
206C2×Q16—62no—
207C2×SD16—62no—
208C2×M16—21no—
209C2×M16—61no—
210C2×M16—61no—
211C2×M16#6711yescharacteristic
212C2×M16#6711yescharacteristic
213C22×C8—31no—
214C22×C8—31no—
215C22×C8—61no—
216#123—61no—
217C22⋊Q8—31no—
218C22⋊Q8—31no—
219#121—31no—
220#121—31no—
221#120—61no—
222C4×D4—61no—
223C4×D4—61no—
224#117—31no—
225C2×C4⋊C4—61no—
226C2×C22⋊C4—61no—
227C4⋊C8—61no—
228C4⋊C8—61no—
229#103—62no—
230#103—62no—
231#103—122no—
232#103—122no—
233#101—42no—
234#101—42no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8—32no—
240C22⋊C8—61no—
241C22⋊C8—61no—
242C22⋊C8—61no—
243C22⋊C8—61no—
244C22⋊C8—61no—
245#97—61no—
246C4×C8—61no—
247#95—61no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3C4×D411yescharacteristic
253C23×C3#12111yescharacteristic
254C23×C3#12311yescharacteristic
255C22×D3—21no—
256C3×Q8—24no—
257C3×Q8—24no—
258C3×Q8—42no—
259C3×Q8—42no—
260C3×Q8—42no—
261C3×Q8—42no—
262C3×Q8—42no—
263C3×Q8—42no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—24no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4C2×C22⋊C411yescharacteristic
281C2×C3×C4C4×D411yescharacteristic
282C2×C3×C4#12011yescharacteristic
283C2×C3×C4#12111yescharacteristic
284#67—41no—
285#67—41no—
286#67—81no—
287#67—81no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—81no—
292C2×Dic3—81no—
293C2×Dic3—81no—
294D12—41no—
295C4×D3—41no—
296C4×D3—41no—
297Dic6—41no—
298C3×C8—41no—
299C3×C8—41no—
300C3×C8—81no—
301C3⋊C8—42no—
302C3⋊C8—41no—
303C3⋊C8—41no—
304C24#23511yescharacteristic
305#41—121no—
306#41—121no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—31no—
316C2×Q8—44no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×D4—31no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—61no—
331C22×C4—22no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—44no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—61no—
338C22×C4—121no—
339C22×C4C2×C4×D311yescharacteristic
340C22×C4D3×D411yescharacteristic
341C22×C4D3×D411yescharacteristic
342C22×C4D3×D411yescharacteristic
343C22×C4#23711yescharacteristic
344C22×C4C2×#6711yescharacteristic
345Q16—124no—
346SD16—124no—
347M16—41no—
348M16—41no—
349M16—41no—
350M16—41no—
351M16—121no—
352M16—121no—
353M16—241no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—41no—
357C2×C8—62no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8—121no—
361C2×C8—121no—
362C2×C8—241no—
363C4⋊C4—31no—
364C4⋊C4—31no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—121no—
368C4⋊C4—121no—
369C22⋊C4—61no—
370C22⋊C4—61no—
371C22⋊C4—61no—
372C22⋊C4—61no—
373C22⋊C4—61no—
374C22⋊C4—241no—
375C42—61no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3#38611yescharacteristic
387C22×C3#47111yescharacteristic
388C22×C3#47311yescharacteristic
389C2×D3—41no—
390C2×D3—41no—
391C2×D3—41no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—44no—
396C3×C4B(4,𝔽2)11yescharacteristic
397C3×C4B(4,𝔽2)11yescharacteristic
398Dic3—41no—
399Dic3—41no—
400Dic3—41no—
401Dic3—161no—
402C23—21no—
403C23—21no—
404C23—41no—
405C23—41no—
406C23—61no—
407C23#88311yescharacteristic
408C23#88411yescharacteristic
409C23C4×#6711yescharacteristic
410Q8—24no—
411Q8—24no—
412Q8—42no—
413Q8—42no—
414Q8—42no—
415Q8—42no—
416Q8—42no—
417Q8—42no—
418Q8—121no—
419D4—121no—
420D4—121no—
421D4—121no—
422D4—241no—
423D4—241no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—241no—
446C2×C4—241no—
447C2×C4—241no—
448C2×C4D3×C22⋊C411yescharacteristic
449C2×C4#67⋊C411yescharacteristic
450C2×C4#93711yescharacteristic
451C2×C4#93811yescharacteristic
452C8—41no—
453C8—41no—
454C8—81no—
455C8—122no—
456C8—121no—
457C8—121no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3#188911yescharacteristic
463C2×C3#190211yescharacteristic
464C2×C3#190811yescharacteristic
465D3—81no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—42no—
476C22—121no—
477C22—121no—
478C22—121no—
479C22#475411yescharacteristic
480C22#475511yescharacteristic
481C22#495611yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—44no—
486C4—121no—
487C4—121no—
488C4—121no—
489C4—481no—
490C4#503211yescharacteristic
491C4#503211yescharacteristic
492C3#951911yescharacteristic, complemented, Sylow
493C2—21no—
494C2—21no—
495C2—41no—
496C2—41no—
497C2—241no—
498C2#6949911yescharacteristic
499C2#7012211yescharacteristic
500C2#7012311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1784C2
socleC22×C3#386
prime radical#5791C22
cosocleC23×C3×C4C23
hypercenter#1784D3
hypocenterC3#9519
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
3. C3×#1784C2semidirect
4. #72132C2semidirect
6. #67687C2semidirect
7. #67684C2semidirect
17. #5228C4semidirect
19. #5227C4semidirect
37. #1784D3semidirect
492. C3#9519semidirect

Maximal subgroups

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

subgroupindex
#840592
#786352
C3×#17842
#721322
#699792
#676872
#676842
#95193