Finite Group Databasebeta

← #10693206

Subgroups — #10693206

Order 768.

644 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C3×#2976C211yesmaximal, characteristic
2#82933C211yesmaximal, characteristic
3#71709C211yesmaximal, characteristic
4#69990C211yesmaximal, characteristic, complemented
5#69963C211yesmaximal, characteristic, complemented
6#67684C211yesmaximal, characteristic, complemented
7#67679C211yesmaximal, characteristic, complemented
8#9519—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C3×#525C2211yescharacteristic
11C2×#951—21no—
12C3×#436—21no—
13C3×#435—21no—
14C3×#435—21no—
15C3×C8×D4—21no—
16C3×#433—21no—
17C3×#408—21no—
18C3×#407C2211yescharacteristic
19C2×C3×C22⋊C8C2211yescharacteristic
20#5212C2211yescharacteristic
21#5207C2211yescharacteristic
22#5207C2211yescharacteristic
23C2×#824—21no—
24C2×#824—21no—
25C2×#816—21no—
26C2×#816—21no—
27#5050—21no—
28#5050C412yescomplemented
29#5040—21no—
30#5040C412yescomplemented
31#5004—21no—
32#4538—21no—
33#4537—21no—
34#3004—31no—
35#2976D311yescharacteristic
36#1787—31no—
37#1784—31no—
38#1513—31no—
39#1305—31no—
40#1293—31no—
41C2×C3×#41D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C22×Dic6—21no—
45C22×Dic6—21no—
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—42no—
50C22×Dic6—42no—
51C22×Dic6C2×C412yes—
52C2×C3×M16—21no—
53C2×C3×M16D411yescharacteristic
54C2×C3×M16D411yescharacteristic
55C22×C3×C8—21no—
56C22×C3×C8D411yescharacteristic
57C22×C3×C8D411yescharacteristic
58C3×#123—21no—
59C3×C22⋊Q8C2×C411yescharacteristic
60C3×C22⋊Q8D411yescharacteristic
61C3×#121C2×C411yescharacteristic
62C3×#121D411yescharacteristic
63C3×#120—21no—
64C3×C4×D4—21no—
65C3×C4×D4—21no—
66C3×#117D411yescharacteristic
67C2×#217—21no—
68#951—42no—
69#951—42no—
70C2×#216—22no—
71C2×#214—22no—
72C2×C4⋊Dic3—21no—
73C2×#208—21no—
74C2×#208—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8—21no—
83C3×#97—21no—
84C3×C4×C8—21no—
85C3⋊#95—21no—
86#830—21no—
87#830—21no—
88#830—21no—
89#830C2×C412yes—
90#824—42no—
91#824—42no—
92#816—42no—
93#816—42no—
94#810—22no—
95#810—22no—
96C23×Q8—31no—
97C2×#137—61no—
98#525C2×D311yescharacteristic
99#451—32no—
100#451—61no—
101#448—32no—
102#448—61no—
103#436—21no—
104#435—21no—
105#435—21no—
106C8×D4—21no—
107#433—21no—
108#419—61no—
109C2×#103—61no—
110C2×#103—61no—
111C2×#101—61no—
112C2×#101—61no—
113#408—21no—
114#407—31no—
115#407—31no—
116#407C2×D311yescharacteristic
117C2×C22⋊C8C2×D311yescharacteristic
118#380—31no—
119#328—61no—
120#327—61no—
121C24×C3#4111yescharacteristic
122C3×#41—41no—
123C3×#41—41no—
124C2×C3×Q8C22⋊C411yescharacteristic
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4C22⋊C411yescharacteristic
129C22×C3×C4—21no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4C22×C411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×Dic3—22no—
139C22×Dic3—44no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C2×Dic6—22no—
144C2×Dic6—22no—
145C2×Dic6—22no—
146C2×Dic6—22no—
147C2×Dic6—22no—
148C2×Dic6—22no—
149C2×Dic6—22no—
150C2×Dic6—22no—
151C2×Dic6—44no—
152C2×Dic6—44no—
153C2×Dic6—44no—
154C2×Dic6—41no—
155C2×Dic6—41no—
156C2×Dic6—41no—
157C2×Dic6—41no—
158C2×Dic6—41no—
159C2×Dic6—41no—
160C2×Dic6—41no—
161C2×Dic6—41no—
162C3×M16—41no—
163C3×M16—41no—
164C3×M16—81no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8—41no—
170C2×C3×C8—41no—
171C2×C3×C8—81no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4C22⋊C411yescharacteristic
175C3×C4⋊C4C22⋊C411yescharacteristic
176C3×C22⋊C4—21no—
177C3×C22⋊C4—21no—
178C3×C22⋊C4—21no—
179C3×C22⋊C4—21no—
180C3×C22⋊C4—21no—
181C3×C42—21no—
182#217—81no—
183#216—44no—
184#214—44no—
185C4⋊Dic3—41no—
186C4⋊Dic3—41no—
187#208—41no—
188#208—41no—
189#208—41no—
190#208—41no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—41no—
193C2×#41#6711yescharacteristic
194C22×Q8—32no—
195C22×Q8—61no—
196C22×Q8—61no—
197C22×Q8—61no—
198C22×Q8—61no—
199C22×Q8—61no—
200C22×Q8—122no—
201C22×Q8—122no—
202C23×C4—61no—
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—21no—
214C22×C8#6711yescharacteristic
215C22×C8#6711yescharacteristic
216#123—21no—
217C22⋊Q8C4×D311yescharacteristic
218C22⋊Q8D1211yescharacteristic
219#121C4×D311yescharacteristic
220#121D1211yescharacteristic
221#120—21no—
222C4×D4—21no—
223C4×D4—21no—
224#117#6711yescharacteristic
225C2×C4⋊C4—61no—
226C2×C22⋊C4—61no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229#103—62no—
230#103—62no—
231#103—122no—
232#103—122no—
233#101—122no—
234#101—122no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—32no—
242C22⋊C8—61no—
243C22⋊C8—61no—
244C22⋊C8—61no—
245#97—21no—
246C4×C8—21no—
247#95—61no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3C4×D411yescharacteristic
254C23×C3#12111yescharacteristic
255C23×C3#12311yescharacteristic
256C3×Q8—41no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—81no—
261C3×D4—81no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
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—21no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4C2×C22⋊C411yescharacteristic
277C2×C3×C4C4×D411yescharacteristic
278C2×C3×C4#12011yescharacteristic
279C2×C3×C4#12111yescharacteristic
280C2×Dic3—24no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—81no—
288C2×Dic3—81no—
289C2×Dic3—81no—
290Dic6—24no—
291Dic6—24no—
292Dic6—42no—
293Dic6—42no—
294Dic6—42no—
295Dic6—42no—
296Dic6—42no—
297Dic6—42no—
298C3×C8—41no—
299C3×C8—41no—
300C3×C8—41no—
301C3×C8—41no—
302C3⋊C8—42no—
303C3⋊C8—81no—
304C24#23511yescharacteristic
305#41—41no—
306#41—41no—
307C2×Q8—62no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×Q8—62no—
313C2×Q8—62no—
314C2×Q8—62no—
315C2×Q8—121no—
316C2×Q8—121no—
317C2×Q8—121no—
318C2×Q8—121no—
319C2×Q8—121no—
320C2×Q8—121no—
321C2×Q8—124no—
322C2×Q8—124no—
323C2×Q8—124no—
324C2×Q8—121no—
325C2×Q8—121no—
326C2×Q8(C2×D3)⋊C411yescharacteristic
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4(C2×D3)⋊C411yescharacteristic
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—62no—
335C22×C4—124no—
336C22×C4—121no—
337C22×C4—121no—
338C22×C4—121no—
339C22×C4C2×C4×D311yescharacteristic
340C22×C4C2×D1211yescharacteristic
341C22×C4#23511yescharacteristic
342C22×C4C2×#6711yescharacteristic
343C22×C4C2×#6711yescharacteristic
344C22×C4C2×#6711yescharacteristic
345Q16—124no—
346SD16—124no—
347M16—41no—
348M16—41no—
349M16—81no—
350M16—121no—
351M16—121no—
352M16—121no—
353M16—121no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—62no—
361C2×C8—81no—
362C2×C8—121no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C4⋊C4(C2×D3)⋊C411yescharacteristic
368C4⋊C4(C2×D3)⋊C411yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—241no—
375C42—21no—
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—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3#38611yescharacteristic
390C22×C3#47111yescharacteristic
391C22×C3#47311yescharacteristic
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—41no—
396C3×C4—41no—
397C3×C4—41no—
398C3×C4B(4,𝔽2)11yescharacteristic
399C3×C4B(4,𝔽2)11yescharacteristic
400Dic3—44no—
401Dic3—161no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—41no—
406C23—41no—
407C23C4×#6711yescharacteristic
408C23#92911yescharacteristic
409C23#93011yescharacteristic
410Q8—41no—
411Q8—64no—
412Q8—64no—
413Q8—122no—
414Q8—122no—
415Q8—122no—
416Q8—122no—
417Q8—122no—
418Q8—122no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—81no—
423D4—81no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—64no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4—122no—
445C2×C4—241no—
446C2×C4—241no—
447C2×C4—241no—
448C2×C4C2×(C2×D3)⋊C411yescharacteristic
449C2×C4C4×#6711yescharacteristic
450C2×C4#93011yescharacteristic
451C2×C4#95311yescharacteristic
452C8—41no—
453C8—41no—
454C8—41no—
455C8—41no—
456C8—122no—
457C8—241no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—81no—
463C2×C3#188911yescharacteristic
464C2×C3#190211yescharacteristic
465C2×C3#190811yescharacteristic
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22#514111yescharacteristic
480C22#514311yescharacteristic
481C22#521311yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—124no—
489C4—481no—
490C4#524411yescharacteristic
491C4#524411yescharacteristic
492C3#951911yescharacteristic, complemented, Sylow
493C2—21no—
494C2—21no—
495C2—41no—
496C2—41no—
497C2—81no—
498C2#7165411yescharacteristic
499C2#7201911yescharacteristic
500C2#7202211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2976C2
socleC22×C3#386
prime radicalC3×#525C22
cosocleC23×C3×C4C23
hypercenter#2976D3
hypocenterC3#9519
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2976

Complements

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

NH ≅ G/Ntype
4. #69990C2semidirect
5. #69963C2semidirect
6. #67684C2semidirect
7. #67679C2semidirect
28. #5050C4semidirect
30. #5040C4semidirect
492. C3#9519semidirect

Maximal subgroups

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

subgroupindex
C3×#29762
#829332
#717092
#699902
#699632
#676842
#676792
#95193