Finite Group Databasebeta

← #10900911

Subgroups — #10900911

Order 768.

1358 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1358 conjugacy classes of subgroups — too many to draw. Only its 187 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×#3449C211yesmaximal, characteristic
2#84090C212yesmaximal, complemented
3#82254C211yesmaximal, characteristic, complemented
4#72139C212yesmaximal, complemented
5#71850C214yesmaximal, complemented
6#71849C214yesmaximal, complemented
7#71594C211yesmaximal, characteristic, complemented
8#29637—31nomaximal, Sylow
9C22×C3×#41C2212yes—
10C3×#537—21no—
11C3×#537—21no—
12C3×#536—22no—
13C3×#535—22no—
14C3×#534C2214yes—
15C3×#532—22no—
16C3×#525—24no—
17C2×C3×#117C2211yescharacteristic
18C2×#935C2214yes—
19C22×Dic3⋊C4C2211yescharacteristic
20#5230—22no—
21#5222—22no—
22#5220C2214yes—
23C2×#834C2218yes—
24#5211C2211yescharacteristic
25#5211C2212yes—
26#5210—24no—
27#4985—21no—
28#4983—21no—
29#4946C2212yes—
30#4943C2212yes—
31#4933—22no—
32#4526C2214yes—
33#3449D311yescharacteristic
34#2484—31no—
35#2406—32no—
36#2044—34no—
37#2022—34no—
38#1820—31no—
39#1782—32no—
40C2×C3×#41—22no—
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C2×C3×#41—22no—
44C2×C3×#41—48no—
45C22×C3×Q8D412yes—
46C22×C3×D4D412yes—
47C22×C3×D4C2314yes—
48C23×C3×C4D414yes—
49C23×C3×C4C2311yescharacteristic
50C23×Dic3C2311yescharacteristic
51C23×Dic3C2311yescharacteristic
52C3×C4⋊Q8—22no—
53C3×#127—22no—
54C3×#124—24no—
55C3×#123—28no—
56C3×C22⋊Q8—24no—
57C3×C22⋊Q8—24no—
58C3×#121—24no—
59C3×#121—24no—
60C3×#121—24no—
61C3×#121—24no—
62C3×#120—28no—
63C3×C4×D4—24no—
64C3×C4×D4—24no—
65C3×#117—22no—
66C3×#117—22no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C22⋊C4C2312yes—
70C2×C3×C42D412yes—
71C2×#217—24no—
72C2×#217—24no—
73C2×#217C2312yes—
74C2×#217C2314yes—
75#935—48no—
76C2×Dic3⋊C4—21no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—21no—
82#834—48no—
83C3⋊#95—21no—
84C3⋊#95—22no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95—24no—
89C3⋊#95—22no—
90C3⋊#95—21no—
91C22×#41C2×D312yes—
92#537—21no—
93#537—21no—
94#536—22no—
95#535—22no—
96#534C2×D314yes—
97#532—22no—
98#525—24no—
99C2×#123—34no—
100C2×#117C2×D311yescharacteristic
101C22×C4⋊C4—31no—
102C2×#99—38no—
103#400—61no—
104#398—61no—
105#395—62no—
106#394—62no—
107#387—62no—
108#386—34no—
109#385—32no—
110#385—64no—
111#381—32no—
112#381—31no—
113#381—32no—
114#342—34no—
115C24×C3C2×D414yes—
116C24×C3C2411yescharacteristic
117C3×#41—48no—
118C3×#41—48no—
119C3×#41—48no—
120C3×#41—48no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8—44no—
124C2×C3×D4—24no—
125C2×C3×D4—24no—
126C2×C3×D4—28no—
127C2×C3×D4—28no—
128C2×C3×D4—24no—
129C2×C3×D4—24no—
130C2×C3×D4—22no—
131C2×C3×D4—22no—
132C2×C3×D4—44no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4—44no—
137C22×C3×C4—44no—
138C22×C3×C4C2×D414yes—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D412yes—
145C22×Dic3—21no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—21no—
149C22×Dic3—24no—
150C22×Dic3—22no—
151C22×Dic3—24no—
152C22×Dic3—42no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C22⋊C4—28no—
158C3×C22⋊C4—28no—
159C3×C22⋊C4—24no—
160C3×C42—22no—
161C3×C42—22no—
162#217—24no—
163#217—28no—
164#217—48no—
165#217—48no—
166Dic3⋊C4—44no—
167Dic3⋊C4—48no—
168Dic3⋊C4—41no—
169Dic3⋊C4—41no—
170Dic3⋊C4—42no—
171C2×#41—22no—
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—22no—
175C2×#41—48no—
176C22×Q8#6712yes—
177C22×D4#6712yes—
178C22×D4C22×D314yes—
179C23×C4—31no—
180C23×C4—31no—
181C23×C4#6714yes—
182C23×C4C22×D311yescharacteristic
183C4⋊Q8—22no—
184#127—22no—
185#124—24no—
186#123—28no—
187#123—128no—
188C22⋊Q8—24no—
189C22⋊Q8—24no—
190#121—24no—
191#121—24no—
192#121—24no—
193#121—24no—
194#120—28no—
195C4×D4—24no—
196C4×D4—24no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—64no—
204C2×C4⋊C4—62no—
205C2×C4⋊C4#6711yescharacteristic
206C2×C4⋊C4#6711yescharacteristic
207C2×C22⋊C4—34no—
208C2×C22⋊C4—32no—
209C2×C22⋊C4—64no—
210C2×C22⋊C4—64no—
211C2×C22⋊C4C22×D312yes—
212C2×C42#6712yes—
213#99—128no—
214#95—62no—
215#95—61no—
216#95—62no—
217#95—62no—
218#95—62no—
219#95—61no—
220#95—62no—
221#95—64no—
222C23×C3—24no—
223C23×C3—24no—
224C23×C3—24no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—44no—
228C23×C3—44no—
229C23×C3C22×D412yes—
230C23×C3C22×D411yescharacteristic
231C23×C3H(5,𝔽2)14yes—
232C3×Q8—44no—
233C3×Q8—44no—
234C3×D4—44no—
235C3×D4—44no—
236C3×D4—48no—
237C3×D4—48no—
238C3×D4—84no—
239C3×D4—84no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—48no—
250C2×C3×C4—48no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12012yes—
256C2×Dic3—22no—
257C2×Dic3—24no—
258C2×Dic3—41no—
259C2×Dic3—42no—
260C2×Dic3—41no—
261C2×Dic3—42no—
262C2×Dic3—44no—
263C2×Dic3—42no—
264C2×Dic3—44no—
265C2×Dic3—41no—
266C2×Dic3—44no—
267C2×Dic3—44no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C24C2×#6714yes—
271C24C23×D311yescharacteristic
272#41—48no—
273#41—48no—
274#41—48no—
275#41—48no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—44no—
279C2×D4—28no—
280C2×D4—28no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—44no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—62no—
294C22×C4—61no—
295C22×C4—64no—
296C22×C4—62no—
297C22×C4—61no—
298C22×C4—64no—
299C22×C4—62no—
300C22×C4—122no—
301C22×C4C2×#6714yes—
302C22×C4C2×#6711yescharacteristic
303C22×C4C2×#6711yescharacteristic
304C22×C4C2×#6712yes—
305C22×C4C2×#6712yes—
306C22×C4C2×#6712yes—
307C22×C4C2×#6712yes—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—128no—
313C4⋊C4—124no—
314C4⋊C4—121no—
315C4⋊C4—121no—
316C4⋊C4—122no—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C22⋊C4—24no—
320C22⋊C4—64no—
321C22⋊C4—68no—
322C22⋊C4—128no—
323C22⋊C4—128no—
324C42—22no—
325C42—22no—
326C22×C3—21no—
327C22×C3—22no—
328C22×C3—21no—
329C22×C3—22no—
330C22×C3—44no—
331C22×C3—48no—
332C22×C3—44no—
333C22×C3—44no—
334C22×C3—44no—
335C22×C3—84no—
336C22×C3C2×#12011yescharacteristic
337C22×C3#53414yes—
338C22×C3#53512yes—
339C3×C4—21no—
340C3×C4—22no—
341C3×C4—21no—
342C3×C4—44no—
343C3×C4—42no—
344C3×C4—42no—
345Dic3—42no—
346Dic3—84no—
347Dic3—82no—
348Dic3—81no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—41no—
353C23—41no—
354C23—44no—
355C23—44no—
356C23#100414yes—
357C23C22×#6712yes—
358C23C22×#6711yescharacteristic
359Q8—44no—
360Q8—44no—
361D4—44no—
362D4—44no—
363D4—48no—
364D4—48no—
365D4—84no—
366D4—84no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—48no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—64no—
381C2×C4—62no—
382C2×C4—121no—
383C2×C4—122no—
384C2×C4—121no—
385C2×C4—124no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—124no—
389C2×C4—124no—
390C2×C4—124no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4#95311yescharacteristic
395C2×C4#95311yescharacteristic
396C2×C4#95312yes—
397C2×C3—24no—
398C2×C3—22no—
399C2×C3—84no—
400C2×C3#261311yescharacteristic
401C2×C3#302511yescharacteristic
402C2×C3#302811yescharacteristic
403C22—21no—
404C22—22no—
405C22—21no—
406C22—22no—
407C22—44no—
408C22—48no—
409C22—44no—
410C22—44no—
411C22—44no—
412C22—84no—
413C22#580414yes—
414C22#580512yes—
415C22C2×#95311yescharacteristic
416C4—21no—
417C4—22no—
418C4—21no—
419C4—44no—
420C4—42no—
421C4—42no—
422C4—122no—
423C4—244no—
424C4—242no—
425C4—241no—
426C3#2963711yescharacteristic, complemented, Sylow
427C2—24no—
428C2—22no—
429C2—84no—
430C2#8405411yescharacteristic
431C2#8412411yescharacteristic
432C2#8412511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
Fitting, prime radicalC3×#3449C2
socleC22×C3C2×#120
hypercenter#3449D3
hypocenterC3#29637
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3449

Complements

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

NH ≅ G/Ntype
2. #84090C2semidirect
3. #82254C2semidirect
4. #72139C2semidirect
5. #71850C2semidirect
6. #71849C2semidirect
7. #71594C2semidirect
426. C3#29637semidirect

Maximal subgroups

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

subgroupindex
C3×#34492
#840902
#822542
#721392
#718502
#718492
#715942
#296373