Finite Group Databasebeta

← #10826869

Subgroups — #10826869

Order 768.

1738 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 1738 conjugacy classes of subgroups — too many to draw. Only its 211 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#87251C211yesmaximal, characteristic
2C22×#944C211yesmaximal, characteristic, complemented
3C2×#4752C212yesmaximal, complemented
4#78645C212yesmaximal, complemented
5C3×#1784C211yesmaximal, characteristic, complemented
6#70119C214yesmaximal, complemented
7#70103C212yesmaximal, complemented
8#69965C212yesmaximal, complemented
9#23911—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C2×#1007C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13#5817—24no—
14#5816—22no—
15#5816C2212yes—
16Q8×#67—24no—
17#5812C2211yescharacteristic
18C2×#944—22no—
19C2×#944—28no—
20C2×#944C2212yes—
21C2×#944C2212yes—
22#5791—22no—
23#5791C2212yes—
24C22×Dic3⋊C4—22no—
25C22×C4×Dic3C2211yescharacteristic
26C2×#909C2214yes—
27C2×C3×#101C2212yes—
28C2×C3×#101C2212yes—
29C3×#407C2212yes—
30C2×#825C2214yes—
31#5120C2214yes—
32#4752—28no—
33#4739—24no—
34#4738—24no—
35#4555C2214yes—
36#3486—31no—
37C22×C4⋊Q8—31no—
38C2×#456—32no—
39#3004—32no—
40#2005—34no—
41#1896—32no—
42#1886—32no—
43#1784D311yescharacteristic, complemented
44C22×C3×Q8—24no—
45C22×C3×Q8—24no—
46C22×C3×Q8—42no—
47C22×C3×Q8D412yes—
48C22×C3×Q8C2311yescharacteristic
49C22×C3×Q8C2311yescharacteristic
50C23×C3×C4D412yes—
51C23×C3×C4C2311yescharacteristic
52C23×Dic3—21no—
53C23×Dic3—21no—
54#1007—24no—
55#1007—42no—
56C2×#239—22no—
57C2×D3×Q8—22no—
58C2×#235—22no—
59C2×#235C2312yes—
60C22×Dic6—21no—
61C22×Dic6—24no—
62C22×Dic6—24no—
63C22×Dic6—42no—
64C22×Dic6C2312yes—
65C2×C3×M16C2314yes—
66#953—22no—
67#947—24no—
68#947—22no—
69#946—28no—
70#946—24no—
71Q8×Dic3—24no—
72#944—216no—
73#944—24no—
74#944—24no—
75#944—24no—
76#944—22no—
77#944—24no—
78#944D418yescomplemented
79#930—24no—
80#930D414yes—
81#929—24no—
82#929—22no—
83C4×#67—28no—
84C4×#67—24no—
85#926—22no—
86#926C2312yes—
87#924—24no—
88#924D414yes—
89C2×Dic3⋊C4—216no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—24no—
94C2×C4×Dic3C2312yes—
95#909—28no—
96#909—44no—
97C2×Dic12—24no—
98C2×#204—24no—
99C3×#101—24no—
100C3×#101—42no—
101C3×C22⋊C8—24no—
102#825—28no—
103#816—28no—
104C2×2−1+4—31no—
105C23×Q8—31no—
106C23×Q8C2×D311yescharacteristic
107C2×#137—34no—
108D4×Q8—64no—
109#548—64no—
110#536—32no—
111#536—62no—
112C2×C4⋊Q8—32no—
113C2×C4⋊Q8—32no—
114C2×C4⋊Q8—68no—
115C2×C4⋊Q8—62no—
116#525—32no—
117#525—62no—
118#519—31no—
119C22×C4⋊C4—62no—
120C22×C42—31no—
121#456—68no—
122#451—64no—
123#448—64no—
124C2×#104—34no—
125#419—34no—
126C2×#101C2×D312yes—
127C2×#101C2×D312yes—
128#407C2×D312yes—
129#339—34no—
130C24×C3C2×D411yescharacteristic
131C2×C3×Q8—24no—
132C2×C3×Q8—24no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—42no—
139C2×C3×Q8—416no—
140C2×C3×Q8C2×D414yes—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—24no—
144C22×C3×C4—24no—
145C22×C3×C4—42no—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D414yes—
150C22×C3×C4C2411yescharacteristic
151C2×#67—24no—
152C2×#67—22no—
153C22×Dic3—24no—
154C22×Dic3—28no—
155C22×Dic3—22no—
156C22×Dic3—24no—
157C22×Dic3—22no—
158C22×Dic3—42no—
159C22×Dic3—42no—
160#239—48no—
161D3×Q8—48no—
162#235—24no—
163#235—24no—
164#235—44no—
165#235—44no—
166C2×D12—22no—
167C2×D12C2×D412yes—
168C2×C4×D3—24no—
169C2×C4×D3—22no—
170C2×Dic6—24no—
171C2×Dic6—24no—
172C2×Dic6—22no—
173C2×Dic6—24no—
174C2×Dic6—22no—
175C2×Dic6—416no—
176C2×Dic6—42no—
177C2×Dic6—42no—
178C2×Dic6C2×D412yes—
179C2×Dic6C2×D414yes—
180C3×M16—24no—
181C3×M16—24no—
182C2×C3×C8—24no—
183#217—24no—
184#217—22no—
185(C2×D3)⋊C4—28no—
186(C2×D3)⋊C4—24no—
187C4⋊Dic3—24no—
188C4⋊Dic3C2×D414yes—
189Dic3⋊C4—216no—
190Dic3⋊C4—28no—
191Dic3⋊C4—216no—
192Dic3⋊C4—24no—
193C4×Dic3—24no—
194C4×Dic3—24no—
195C4×Dic3—22no—
196C4×Dic3—22no—
197C4×Dic3C2×D414yes—
198Dic12—48no—
199#204—48no—
2002−1+4—64no—
2012−1+4—122no—
202C2×#41—32no—
203C2×#41—62no—
204C2×#41—62no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—32no—
208C22×Q8—42no—
209C22×Q8—64no—
210C22×Q8—62no—
211C22×Q8—61no—
212C22×Q8—64no—
213C22×Q8—122no—
214C22×Q8D1212yes—
215C22×Q8C22×D311yescharacteristic
216C22×Q8C22×D311yescharacteristic
217C23×C4—61no—
218C23×C4—61no—
219C23×C4D1212yes—
220C23×C4C22×D311yescharacteristic
221#137—68no—
222#137—124no—
223C2×Q16—64no—
224C2×SD16—64no—
225C2×M16C22×D314yes—
226C4⋊Q8—38no—
227C4⋊Q8—64no—
228C4⋊Q8—64no—
229C4⋊Q8—62no—
230C4⋊Q8—616no—
231C4⋊Q8—64no—
232C4⋊Q8—64no—
233#124—62no—
234#124—64no—
235#123—62no—
236#123—64no—
237C22⋊Q8—34no—
238C22⋊Q8—64no—
239C22⋊Q8—64no—
240C22⋊Q8—68no—
241#121—34no—
242#121—64no—
243#120—62no—
244C4×Q8—64no—
245C4×D4—68no—
246C4×D4—64no—
247#117—32no—
248#117—62no—
249C2×C4⋊C4—616no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C42—32no—
253C2×C42—64no—
254C2×C42—61no—
255#104—68no—
256#103—68no—
257#101—24no—
258#101—42no—
259C22⋊C8—24no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—24no—
263C23×C3#12012yes—
264C23×C3C22×D411yescharacteristic
265C22×D3—22no—
266C3×Q8—28no—
267C3×Q8—48no—
268C3×Q8—44no—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—21no—
272C2×C3×C4—22no—
273C2×C3×C4—28no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—48no—
278C2×C3×C4#12014yes—
279C2×C3×C4#12012yes—
280C2×C3×C4C22×D412yes—
281C2×C3×C4C22×D412yes—
282C2×C3×C4H(5,𝔽2)12yes—
283#67—42no—
284#67—42no—
285#67—48no—
286C2×Dic3—216no—
287C2×Dic3—28no—
288C2×Dic3—24no—
289C2×Dic3—24no—
290C2×Dic3—24no—
291C2×Dic3—22no—
292C2×Dic3—48no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295D12—24no—
296D12—44no—
297C4×D3—48no—
298C4×D3—44no—
299Dic6—28no—
300Dic6—24no—
301Dic6—48no—
302Dic6—44no—
303Dic6—44no—
304C3×C8—44no—
305C24C2×D1211yescharacteristic
306#41—64no—
307#41—64no—
308#41—124no—
309#41—128no—
310#41—124no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—32no—
319C2×Q8—34no—
320C2×Q8—42no—
321C2×Q8—416no—
322C2×Q8—62no—
323C2×Q8—64no—
324C2×Q8—62no—
325C2×Q8—64no—
326C2×Q8—64no—
327C2×Q8—1216no—
328C2×Q8—122no—
329C2×Q8—128no—
330C2×Q8—122no—
331C2×Q8D3×D414yes—
332C2×D4—32no—
333C2×D4—62no—
334C2×D4—64no—
335C2×D4—62no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—42no—
341C22×C4—64no—
342C22×C4—64no—
343C22×C4—68no—
344C22×C4—62no—
345C22×C4—64no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—122no—
349C22×C4—122no—
350C22×C4C2×D1212yes—
351C22×C4C2×D1211yescharacteristic
352C22×C4C2×D1212yes—
353C22×C4D3×D414yes—
354C22×C4C23×D311yescharacteristic
355Q16—128no—
356SD16—128no—
357M16—24no—
358M16—24no—
359C2×C8—24no—
360C4⋊C4—34no—
361C4⋊C4—616no—
362C4⋊C4—64no—
363C4⋊C4—68no—
364C4⋊C4—616no—
365C4⋊C4—64no—
366C22⋊C4—68no—
367C22⋊C4—64no—
368C22⋊C4—62no—
369C22⋊C4—64no—
370C42—34no—
371C42—64no—
372C42—64no—
373C42—62no—
374C42—62no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3#45614yescomplemented
381C22×C3C2×#12011yescharacteristic
382C22×C3C2×#12011yescharacteristic
383C22×C3#53411yescharacteristic
384C2×D3—42no—
385C2×D3—44no—
386C3×C4—22no—
387C3×C4—24no—
388C3×C4—42no—
389C3×C4D4214yes—
390Dic3—28no—
391Dic3—44no—
392Dic3—42no—
393Dic3—44no—
394C23—21no—
395C23—21no—
396C23—24no—
397C23—62no—
398C23#88212yes—
399C23C22×D1211yescharacteristic
400Q8—28no—
401Q8—48no—
402Q8—44no—
403Q8—68no—
404Q8—64no—
405Q8—128no—
406Q8—124no—
407Q8—124no—
408D4—64no—
409D4—124no—
410D4—122no—
411D4—122no—
412D4—128no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—48no—
422C2×C4—616no—
423C2×C4—68no—
424C2×C4—64no—
425C2×C4—64no—
426C2×C4—64no—
427C2×C4—62no—
428C2×C4—128no—
429C2×C4—128no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—124no—
433C2×C4#88214yes—
434C2×C4#88212yes—
435C2×C4C2×D3×D412yes—
436C2×C4C2×D3×D412yes—
437C2×C4#100412yes—
438C8—44no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—24no—
442C2×C3#240311yescharacteristic
443C2×C3C2×#45612yes—
444D3—82no—
445C22—24no—
446C22—24no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—124no—
451C22—122no—
452C22#475214yescomplemented
453C22C2×#88211yescharacteristic
454C22C2×#88211yescharacteristic
455C22#549311yescharacteristic
456C4—22no—
457C4—24no—
458C4—42no—
459C4—68no—
460C4—124no—
461C4—122no—
462C4—124no—
463C4#559014yes—
464C3#2391111yescharacteristic, complemented, Sylow
465C2—21no—
466C2—21no—
467C2—24no—
468C2—242no—
469C2#7385211yescharacteristic
470C2C2×#475212yes—
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1784C2
socleC22×C3C2×#120
prime radical#87251C2
hypercenter#1784D3
hypocenterC3#23911
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. C22×#944C2semidirect
3. C2×#4752C2semidirect
4. #78645C2semidirect
5. C3×#1784C2semidirect
6. #70119C2semidirect
7. #70103C2semidirect
8. #69965C2semidirect
43. #1784D3semidirect
78. #944D4semidirect
380. C22×C3#456semidirect
452. C22#4752semidirect
464. C3#23911semidirect

Maximal subgroups

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

subgroupindex
#872512
C22×#9442
C2×#47522
#786452
C3×#17842
#701192
#701032
#699652
#239113