Finite Group Databasebeta

← #10692364

Subgroups — #10692364

Order 768.

500 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 107 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×#3095C211yesmaximal, characteristic
2#82239C211yesmaximal, characteristic
3#71701C211yesmaximal, characteristic
4#71688C211yesmaximal, characteristic
5#71590C211yesmaximal, characteristic
6#67684C211yesmaximal, characteristic
7#67683C211yesmaximal, characteristic
8#9363—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C3×#485—22no—
12C3×#484—21no—
13C3×#483—21no—
14C3×#428C412yescomplemented
15C3×#419C412yescomplemented
16C3×#407C2211yescharacteristic
17#5212C2211yescharacteristic
18C4×#217C2211yescharacteristic
19C4×#217C2211yescharacteristic
20#5207C2211yescharacteristic
21#4984—21no—
22Dic3×C4⋊C4—21no—
23#4951—22no—
24#4936—21no—
25#4932—21no—
26#4538—21no—
27#4538—21no—
28#4532—22no—
29#4526—22no—
30#3095D311yescharacteristic
31#2346—31no—
32#1996—31no—
33#1826—31no—
34#1508—31no—
35#1305—31no—
36#1297—31no—
37C22×C3×Q8C2×C411yescharacteristic
38C23×C3×C4C2311yescharacteristic
39C2×C3×M16C2×C412yes—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×#117—21no—
45C3×#117—21no—
46C3×#117C2×C412yes—
47C3×#117D412yes—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C42—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217D412yes—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—22no—
60C2×C4×Dic3—22no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3—21no—
63C2×#208—21no—
64C3×#107—22no—
65C3×#106—22no—
66C3×#103—22no—
67C3×#103—22no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95D411yescharacteristic
75C3⋊#95D411yescharacteristic
76#830—21no—
77#830—21no—
78C2×C22⋊Q8C2×D311yescharacteristic
79C2×#117C2×D311yescharacteristic
80#485—22no—
81#484—21no—
82#483—21no—
83#428Dic312yescomplemented
84#419Dic312yescomplemented
85#407—31no—
86#407C2×D311yescharacteristic
87#389—61no—
88#388—62no—
89#382—61no—
90#380—31no—
91C4×C4⋊C4—61no—
92C4×C22⋊C4—31no—
93C4×C22⋊C4—31no—
94#376—61no—
95#342—62no—
96#339—62no—
97#328—61no—
98#328—61no—
99C24×C3C2×D411yescharacteristic
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—42no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—42no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—22no—
116C22×Dic3—21no—
117C22×Dic3—41no—
118C3×M16—42no—
119C2×C3×C8—22no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4C22⋊C412yes—
127C3×C4⋊C4C22⋊C412yes—
128C3×C22⋊C4—22no—
129C3×C22⋊C4—22no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42—22no—
133C3×C42—22no—
134#217—22no—
135#217—42no—
136#217—41no—
137#217—42no—
138C4⋊Dic3—42no—
139C4⋊Dic3—82no—
140C4×Dic3—44no—
141C4×Dic3—42no—
142C4×Dic3—42no—
143#208—42no—
144C2×C3⋊C8—41no—
145C22×Q8C2×Dic311yescharacteristic
146C23×C4C22×D311yescharacteristic
147C2×M16—61no—
148C2×M16C2×Dic312yes—
149C22⋊Q8—21no—
150C22⋊Q8—21no—
151C22⋊Q8—21no—
152C22⋊Q8—21no—
153#117—21no—
154#117—21no—
155#117C2×Dic312yes—
156#117#6712yes—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—62no—
160C2×C4⋊C4C2×Dic311yescharacteristic
161C2×C4⋊C4#6711yescharacteristic
162C2×C4⋊C4#6711yescharacteristic
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—32no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C42—21no—
169C2×C42—62no—
170C2×C42—61no—
171C2×C42—61no—
172#107—22no—
173#106—22no—
174#103—22no—
175#103—22no—
176C22⋊C8—21no—
177C22⋊C8—21no—
178C22⋊C8—61no—
179C22⋊C8—61no—
180#95—31no—
181#95—31no—
182#95—62no—
183#95—62no—
184#95—61no—
185#95—61no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3C4×D411yescharacteristic
191C23×C3#12011yescharacteristic
192C23×C3#12111yescharacteristic
193C3×Q8—42no—
194C3×Q8—81no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4C2×C22⋊C411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×Dic3—22no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226C3×C8—42no—
227C3⋊C8—81no—
228C24D3×D411yescharacteristic
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—42no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—62no—
240C22×C4—121no—
241C22×C4D3×D411yescharacteristic
242C22×C4#23711yescharacteristic
243C22×C4#23711yescharacteristic
244C22×C4C22×Dic311yescharacteristic
245C22×C4C2×#6711yescharacteristic
246C22×C4C2×#6711yescharacteristic
247M16—42no—
248M16—122no—
249C2×C8—22no—
250C2×C8—121no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—122no—
258C4⋊C4—242no—
259C4⋊C4#21712yes—
260C4⋊C4#21712yes—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—62no—
266C22⋊C4—122no—
267C22⋊C4—121no—
268C22⋊C4—122no—
269C42—22no—
270C42—22no—
271C42—122no—
272C42—122no—
273C42—124no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3#38611yescharacteristic
285C22×C3#45511yescharacteristic
286C22×C3#45611yescharacteristic
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—42no—
291C3×C4—42no—
292C3×C4—81no—
293C3×C4—81no—
294C3×C4#41012yes—
295Dic3—42no—
296Dic3—82no—
297Dic3—81no—
298Dic3—82no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23D4×Dic311yescharacteristic
304C23#93711yescharacteristic
305C23#93911yescharacteristic
306Q8—42no—
307Q8—81no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—62no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—121no—
333C2×C4—241no—
334C2×C4—241no—
335C2×C4D4×Dic311yescharacteristic
336C2×C4#93511yescharacteristic
337C2×C4#93911yescharacteristic
338C2×C4C2×#21711yescharacteristic
339C8—42no—
340C8—241no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3#188211yescharacteristic
346C2×C3#189911yescharacteristic
347C2×C3#190611yescharacteristic
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—81no—
357C22—81no—
358C22#520211yescharacteristic
359C22#520511yescharacteristic
360C22#522011yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—122no—
369C4—241no—
370C4—242no—
371C4—242no—
372C4#524012yes—
373C3#936311yescharacteristic, complemented, Sylow
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#7170511yescharacteristic
379C2#7210011yescharacteristic
380C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#3095C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#3095D3
hypocenterC3#9363
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3095

Complements

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

NH ≅ G/Ntype
14. C3×#428C4semidirect
15. C3×#419C4semidirect
83. #428Dic3semidirect
84. #419Dic3semidirect
373. C3#9363semidirect

Maximal subgroups

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

subgroupindex
C3×#30952
#822392
#717012
#716882
#715902
#676842
#676832
#93633