Finite Group Databasebeta

← #10943914

Subgroups — #10943914

Order 768.

3580 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 3580 conjugacy classes of subgroups — too many to draw. Only its 255 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
1C22×#1007C211yesmaximal, characteristic
2C3×#2434C211yesmaximal, characteristic, complemented
3C2×#5095C214yesmaximal, complemented
4#82735C212yesmaximal, complemented
5#72133C211yesmaximal, characteristic, complemented
6#69692C214yesmaximal, complemented
7#69454C212yesmaximal, complemented
8#32892—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×#1007—22no—
11C2×#1007—216no—
12C2×#1007—24no—
13C2×#1007C2214yes—
14C22×#239—22no—
15C22×D3×Q8—22no—
16C22×#235—22no—
17C22×#235C2212yes—
18C2×C3×C4⋊Q8C2212yescomplemented
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×#942C2214yescomplemented
21C3×#395—22no—
22C3×#391C2212yescomplemented
23C3×#385—22no—
24C3×C4×C22⋊C4C2211yescharacteristic
25C2×#836C2214yescomplemented
26#5207C2212yes—
27#5095—216no—
28#5050—22no—
29#5047—24no—
30#5046—22no—
31#4966C2214yescomplemented
32C2×#805C2218yescomplemented
33C22×2−1+4—31no—
34C2×#456—34no—
35#3007—32no—
36#2434D311yescharacteristic, complemented
37#1793—34no—
38#1789—32no—
39#1784—31no—
40C22×C3×Q8—28no—
41C22×C3×Q8—22no—
42C22×C3×Q8—22no—
43C22×C3×Q8D412yes—
44C22×C3×Q8C2311yescharacteristic
45C22×C3×Q8C2312yes—
46C23×C3×C4D412yes—
47C23×C3×C4C2311yescharacteristic
48C22×#67—22no—
49#1007—232no—
50#1007—232no—
51#1007—232no—
52#1007—28no—
53#1007—216no—
54#1007D4116yes—
55C2×#239—232no—
56C2×#239—28no—
57C2×#239—28no—
58C2×D3×Q8—28no—
59C2×D3×Q8—232no—
60C2×D3×Q8—28no—
61C2×#235—28no—
62C2×#235—232no—
63C2×#235—28no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—216no—
67C2×#235D414yes—
68C2×#235C2314yes—
69C22×D12—22no—
70C22×D12D412yes—
71C22×C4×D3—24no—
72C22×C4×D3—22no—
73C22×Dic6—22no—
74C22×Dic6D412yes—
75C3×C4⋊Q8—28no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C42C2312yes—
84C2×#216—24no—
85#942—216no—
86C2×#215—24no—
87C2×#208C2314yes—
88C3×#95—21no—
89C3×#95—24no—
90#836—28no—
91#830—22no—
92#830—22no—
93#810—24no—
94#809—24no—
95#805—216no—
96C2×2−1+4—34no—
97C2×2−1+4—64no—
98C2×2−1+4—616no—
99C2×2−1+4—62no—
100C22×#41—32no—
101C22×#41—62no—
102C22×#41—62no—
103C23×Q8—62no—
104C23×Q8C2×D311yescharacteristic
105C2×#137—34no—
106C2×C4⋊Q8C2×D312yescomplemented
107C2×C22⋊Q8C2×D311yescharacteristic
108#456—616no—
109#452—64no—
110#451—62no—
111#450—62no—
112C2×#104—38no—
113#417—34no—
114C2×#101—34no—
115#407—32no—
116#395—22no—
117#391C2×D312yescomplemented
118#385—22no—
119C4×C22⋊C4C2×D311yescharacteristic
120C24×C3C2×D411yescharacteristic
121C23×D3—22no—
122C2×C3×Q8—216no—
123C2×C3×Q8—216no—
124C2×C3×Q8—24no—
125C2×C3×Q8—28no—
126C2×C3×Q8—24no—
127C2×C3×Q8—24no—
128C2×C3×Q8—22no—
129C2×C3×Q8—24no—
130C2×C3×Q8C2×D418yes—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—28no—
134C22×C3×C4—24no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D414yes—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C2×#67—216no—
144C2×#67—24no—
145C2×#67—24no—
146C22×Dic3—22no—
147#239—264no—
148#239—232no—
149#239—232no—
150D3×Q8—264no—
151D3×Q8—232no—
152D3×Q8—232no—
153#235—264no—
154#235—232no—
155#235—216no—
156#235—216no—
157#235—232no—
158#235—28no—
159#235—28no—
160#235—28no—
161#235C2×D4116yes—
162C2×D12—216no—
163C2×D12—28no—
164C2×D12—22no—
165C2×D12—28no—
166C2×D12C2×D414yes—
167C2×C4×D3—232no—
168C2×C4×D3—216no—
169C2×C4×D3—216no—
170C2×C4×D3—24no—
171C2×C4×D3—24no—
172C2×Dic6—216no—
173C2×Dic6—28no—
174C2×Dic6—22no—
175C2×Dic6—28no—
176C2×Dic6C2×D414yes—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—48no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—41no—
182C3×C22⋊C4—41no—
183C3×C42—24no—
184#216—48no—
185#215—48no—
186#208—28no—
187C2×C3⋊C8—24no—
1882−1+4—316no—
1892−1+4—616no—
1902−1+4—632no—
1912−1+4—632no—
1922−1+4—632no—
1932−1+4—68no—
194C2×#41—34no—
195C2×#41—34no—
196C2×#41—616no—
197C2×#41—68no—
198C2×#41—632no—
199C2×#41—68no—
200C2×#41—632no—
201C2×#41—68no—
202C2×#41—62no—
203C2×#41—62no—
204C2×#41—68no—
205C22×Q8—28no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—32no—
209C22×Q8—62no—
210C22×Q8—68no—
211C22×Q8—632no—
212C22×Q8—68no—
213C22×Q8#6712yes—
214C22×Q8C22×D311yescharacteristic
215C22×Q8C22×D312yes—
216C22×D4—32no—
217C22×D4—62no—
218C22×D4—62no—
219C23×C4—64no—
220C23×C4—62no—
221C23×C4#6712yes—
222C23×C4C22×D311yescharacteristic
223#137—616no—
224C2×Q16—64no—
225C2×SD16—64no—
226C2×M16—34no—
227C4⋊Q8—28no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4C22×D311yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C42C22×D312yes—
236#104—616no—
237#103—64no—
238#102—64no—
239#101—68no—
240C22⋊C8—62no—
241C22⋊C8—62no—
242#95—21no—
243#95—24no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—41no—
247C23×C3#12011yescharacteristic
248C23×C3#12011yescharacteristic
249C23×C3C22×D411yescharacteristic
250C22×D3—28no—
251C22×D3—24no—
252C22×D3—44no—
253C22×D3—44no—
254C3×Q8—216no—
255C3×Q8—216no—
256C3×Q8—44no—
257C2×C3×C4—28no—
258C2×C3×C4—28no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—22no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4#12012yes—
272C2×C3×C4#12018yes—
273C2×C3×C4C22×D412yes—
274C2×C3×C4C22×D414yes—
275#67—216no—
276#67—216no—
277C2×Dic3—28no—
278C2×Dic3—24no—
279D12—232no—
280D12—28no—
281D12—24no—
282D12#12018yes—
283C4×D3—264no—
284C4×D3—216no—
285C4×D3—216no—
286Dic6—232no—
287Dic6—28no—
288Dic6—24no—
289Dic6#12018yes—
290C3⋊C8—44no—
291C24—62no—
292C24C2×#6711yescharacteristic
293#41—316no—
294#41—68no—
295#41—68no—
296#41—664no—
297#41—616no—
298#41—632no—
299#41—616no—
300#41—632no—
301#41—664no—
302#41—632no—
303#41—632no—
304#41—68no—
305C2×Q8—216no—
306C2×Q8—216no—
307C2×Q8—24no—
308C2×Q8—28no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—34no—
314C2×Q8—68no—
315C2×Q8—616no—
316C2×Q8—68no—
317C2×Q8—632no—
318C2×Q8—664no—
319C2×Q8—632no—
320C2×Q8—62no—
321C2×Q8D3×D418yes—
322C2×D4—34no—
323C2×D4—68no—
324C2×D4—616no—
325C2×D4—68no—
326C2×D4—616no—
327C2×D4—64no—
328C2×D4—64no—
329C2×D4—62no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—28no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—632no—
338C22×C4—616no—
339C22×C4—616no—
340C22×C4—62no—
341C22×C4—64no—
342C22×C4—64no—
343C22×C4D3×D412yes—
344C22×C4D3×D412yes—
345C22×C4C2×#6714yes—
346C22×C4C2×#6711yescharacteristic
347C22×C4C23×D311yescharacteristic
348Q16—128no—
349SD16—128no—
350M16—68no—
351C2×C8—64no—
352C4⋊C4—22no—
353C4⋊C4—42no—
354C4⋊C4—48no—
355C22⋊C4—24no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C42—24no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—44no—
367C22×C3#45612yescomplemented
368C22×C3C2×#12012yes—
369C22×C3C2×#12011yescharacteristic
370C2×D3—28no—
371C2×D3—416no—
372C2×D3—44no—
373C2×D3—44no—
374C3×C4—28no—
375C3×C4—22no—
376C3×C4—44no—
377C3×C4—81no—
378C3×C4—81no—
379C3×C4C2×#12014yes—
380Dic3—28no—
381C23—22no—
382C23—22no—
383C23—41no—
384C23—68no—
385C23—64no—
386C23—124no—
387C23—124no—
388C23#93711yescharacteristic
389C23#93711yescharacteristic
390C23C22×#6711yescharacteristic
391Q8—216no—
392Q8—216no—
393Q8—38no—
394Q8—44no—
395Q8—68no—
396Q8—632no—
397Q8—64no—
398D4—38no—
399D4—68no—
400D4—632no—
401D4—616no—
402D4—616no—
403D4—64no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—664no—
419C2×C4—616no—
420C2×C4—68no—
421C2×C4—64no—
422C2×C4—616no—
423C2×C4#93718yes—
424C2×C4#93712yes—
425C2×C4C2×D3×D414yes—
426C2×C4C2×D3×D412yes—
427C8—124no—
428C2×C3—22no—
429C2×C3—22no—
430C2×C3—41no—
431C2×C3#284611yescharacteristic
432C2×C3C2×#45611yescharacteristic
433C2×C3#302011yescharacteristic
434D3—48no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—44no—
443C22—68no—
444C22—1216no—
445C22—124no—
446C22—124no—
447C22#509512yescomplemented
448C22C2×#93712yes—
449C22C2×#93711yescharacteristic
450C4—28no—
451C4—22no—
452C4—44no—
453C4—68no—
454C4—81no—
455C4—81no—
456C4D3×#12014yes—
457C3#3289211yescharacteristic, complemented, Sylow
458C2—22no—
459C2—22no—
460C2—41no—
461C2—128no—
462C2#8261211yescharacteristic
463C2C2×#509511yescharacteristic
464C2#8341511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2434C2
socleC22×C3C2×#120
prime radicalC22×#1007C2
hypercenter#2434D3
hypocenterC3#32892
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2434

Complements

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

NH ≅ G/Ntype
2. C3×#2434C2semidirect
3. C2×#5095C2semidirect
4. #82735C2semidirect
5. #72133C2semidirect
6. #69692C2semidirect
7. #69454C2semidirect
18. C2×C3×C4⋊Q8C22semidirect
20. C2×#942C22semidirect
22. C3×#391C22semidirect
25. C2×#836C22semidirect
31. #4966C22semidirect
32. C2×#805C22semidirect
36. #2434D3semidirect
106. C2×C4⋊Q8C2×D3semidirect
117. #391C2×D3semidirect
367. C22×C3#456semidirect
447. C22#5095semidirect
457. C3#32892semidirect

Maximal subgroups

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

subgroupindex
C22×#10072
C3×#24342
C2×#50952
#827352
#721332
#696922
#694542
#328923