Finite Group Databasebeta

← #10691621

Subgroups — #10691621

Order 768.

524 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#4722C211yesmaximal, characteristic
2#73879C211yesmaximal, characteristic
3#69529C211yesmaximal, characteristic
4#69510C211yesmaximal, characteristic
5#69502C211yesmaximal, characteristic
6C3×#1289C211yesmaximal, characteristic
7#67677C211yesmaximal, characteristic
8#9257—31nomaximal, Sylow
9C22×C4⋊Dic3C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C4×#217C2211yescharacteristic
15C2×#830C2211yescharacteristic
16C2×#818C412yescomplemented
17C2×#816C412yescomplemented
18Dic3×C4⋊C4—21no—
19#4972—21no—
20#4942—21no—
21Dic3×C22⋊C4—21no—
22#4722—21no—
23#4722—21no—
24#4722—22no—
25#4653—22no—
26#4529—22no—
27#4527—21no—
28#4527—21no—
29#4452—22no—
30C2×#484—31no—
31#2345—31no—
32#1996—31no—
33#1824—31no—
34#1520—31no—
35#1289D311yescharacteristic
36#1282—31no—
37C23×C3×C4C2311yescharacteristic
38C23×Dic3—21no—
39C22×Dic6C2×C411yescharacteristic
40C22×C3×C8C2×C412yes—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×#217—21no—
46C2×#217D411yescharacteristic
47C2×#217D411yescharacteristic
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3C2×C411yescharacteristic
52C2×C4⋊Dic3C2×C412yes—
53C2×C4⋊Dic3D411yescharacteristic
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4⋊Dic3D412yes—
56#924—21no—
57#924—21no—
58#924—21no—
59#924—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—22no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—22no—
64C22×C3⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#95—21no—
68C3×#95D412yes—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—21no—
72#830—21no—
73#830—21no—
74#818—22no—
75#818—22no—
76#816—22no—
77#816—22no—
78C2×C22⋊Q8—31no—
79C22×C4⋊C4—31no—
80#484—62no—
81#484—61no—
82#484—61no—
83C2×#107—32no—
84C2×#103—32no—
85C2×C22⋊C8—31no—
86C2×C22⋊C8C2×D311yescharacteristic
87#389—61no—
88#387—62no—
89#380—61no—
90#380C2×D311yescharacteristic
91#380C2×D311yescharacteristic
92C4×C4⋊C4—61no—
93C4×C22⋊C4—31no—
94C4×C22⋊C4—61no—
95#342—61no—
96#342—61no—
97#340—62no—
98#328—62no—
99C24×C3C2×D411yescharacteristic
100C22×C3×C4—22no—
101C22×C3×C4—42no—
102C22×C3×C4—41no—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—22no—
113C22×Dic3—22no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—82no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6—42no—
121C2×C3×C8—22no—
122C2×C3×C8—42no—
123C3×C4⋊C4—42no—
124C3×C4⋊C4—42no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129#217—21no—
130#217—21no—
131#217—21no—
132#217—41no—
133#217—41no—
134#217Q1612yes—
135C4⋊Dic3—22no—
136C4⋊Dic3—22no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3C22⋊C412yes—
142C4⋊Dic3C22⋊C412yes—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—82no—
146C4×Dic3—42no—
147C4×Dic3—82no—
148C2×C3⋊C8—42no—
149C2×C3⋊C8—41no—
150C22×Q8—31no—
151C23×C4—61no—
152C23×C4C22×D311yescharacteristic
153C22×C8—61no—
154C22×C8C4×D312yes—
155C22⋊Q8—61no—
156C22⋊Q8—61no—
157C22⋊Q8—61no—
158C22⋊Q8—61no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—32no—
162C2×C4⋊C4—32no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—62no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—31no—
175C2×C22⋊C4—61no—
176C2×C42—61no—
177C2×C42—62no—
178#107—62no—
179#107—62no—
180#103—62no—
181#103—62no—
182C22⋊C8—21no—
183C22⋊C8—21no—
184C22⋊C8—61no—
185C22⋊C8—61no—
186#95—21no—
187#95—62no—
188#95—62no—
189#95—61no—
190#95D1212yes—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3C4×D411yescharacteristic
194C23×C3#12011yescharacteristic
195C23×C3#12111yescharacteristic
196C23×C3C2×Q1611yescharacteristic
197C23×C3#13511yescharacteristic
198C23×C3#13711yescharacteristic
199C23×C3#13711yescharacteristic
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4C2×C22⋊C411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12311yescharacteristic
216C2×Dic3—22no—
217C2×Dic3—22no—
218C2×Dic3—22no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—82no—
229C2×Dic3—82no—
230C2×Dic3—81no—
231C2×Dic3—81no—
232Dic6—42no—
233Dic6—81no—
234C3×C8—42no—
235C3⋊C8—81no—
236C24D3×D411yescharacteristic
237C2×Q8—61no—
238C2×Q8—61no—
239C2×Q8—122no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—62no—
247C22×C4—62no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—242no—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4C2×D1211yescharacteristic
254C22×C4#23511yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4#23911yescharacteristic
258C2×C8—22no—
259C2×C8—42no—
260C2×C8—122no—
261C2×C8—121no—
262C4⋊C4—32no—
263C4⋊C4—32no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—62no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—62no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—242no—
275C22⋊C4—32no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C22⋊C4—121no—
284C22⋊C4—121no—
285C42—122no—
286C42—242no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3#38611yescharacteristic
298C22×C3C4×Q1611yescharacteristic
299C22×C3#44111yescharacteristic
300C22×C3#45111yescharacteristic
301C22×C3#45211yescharacteristic
302C22×C3#46211yescharacteristic
303C22×C3#46411yescharacteristic
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—82no—
308C3×C4—161no—
309C3×C4#41012yes—
310Dic3—42no—
311Dic3—42no—
312Dic3—42no—
313Dic3—81no—
314Dic3—81no—
315Dic3—81no—
316C23—41no—
317C23—41no—
318C23#88211yescharacteristic
319C23#88411yescharacteristic
320C23#89311yescharacteristic
321C23#91211yescharacteristic
322C23#91511yescharacteristic
323C23#91511yescharacteristic
324C23D3×Q1611yescharacteristic
325Q8—122no—
326Q8—241no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—242no—
352C2×C4—242no—
353C2×C4—241no—
354C2×C4—241no—
355C2×C4C4×D1211yescharacteristic
356C2×C4D3×C22⋊C411yescharacteristic
357C2×C4#89411yescharacteristic
358C2×C4#89511yescharacteristic
359C8—42no—
360C8—241no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3#187811yescharacteristic
368C2×C3#189911yescharacteristic
369C2×C3#190411yescharacteristic
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22#466511yescharacteristic
381C22#476711yescharacteristic
382C22#478311yescharacteristic
383C22#479511yescharacteristic
384C22#480911yescharacteristic
385C22#485311yescharacteristic
386C22#487311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—82no—
391C4—122no—
392C4—122no—
393C4—122no—
394C4—161no—
395C4—241no—
396C4—241no—
397C4—241no—
398C4#474012yes—
399C3#925711yescharacteristic, complemented, Sylow
400C2—21no—
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2#6870411yescharacteristic
407C2#6890811yescharacteristic
408C2#6895411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1289C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1289D3
hypocenterC3#9257
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
16. C2×#818C4semidirect
17. C2×#816C4semidirect
399. C3#9257semidirect

Maximal subgroups

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

subgroupindex
C2×#47222
#738792
#695292
#695102
#695022
C3×#12892
#676772
#92573