Finite Group Databasebeta

← #10692641

Subgroups — #10692641

Order 768.

606 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 93 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#73605C211yesmaximal, characteristic
2#73567C211yesmaximal, characteristic
3#69980C211yesmaximal, characteristic, complemented
4#69967C211yesmaximal, characteristic
5#69445C211yesmaximal, characteristic, complemented
6C3×#1297C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9411—31nomaximal, Sylow
9C22×(C2×D3)⋊C4C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14C4×#217C2211yescharacteristic
15#5207C2211yescharacteristic
16#4955—21no—
17#4955—21no—
18#4943—21no—
19#4942—21no—
20#4940—22no—
21#4940—22no—
22#4674C412yescomplemented
23#4673C412yescomplemented
24#4669—22no—
25#4468—22no—
26#4467—22no—
27#4452—22no—
28#2561—31no—
29#2312—31no—
30#1508—31no—
31#1508—31no—
32#1462—31no—
33#1297—31no—
34#1297D311yescharacteristic, complemented
35C24×D3D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3D411yescharacteristic
38C22×Dic6C2×C411yescharacteristic
39C2×C3×M16—21no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C42—21no—
43C2×C3×C42—21no—
44C2×#217—21no—
45C2×#217—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—22no—
54C2×(C2×D3)⋊C4C2×C412yes—
55C2×(C2×D3)⋊C4D412yes—
56C2×C4⋊Dic3C2×C411yescharacteristic
57#924—21no—
58#924—21no—
59#924—21no—
60#924—21no—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—22no—
63C2×C4×Dic3—21no—
64C2×#208—21no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#95C2×C412yes—
68C3×#95D412yes—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95D412yes—
73#830—21no—
74#830—21no—
75C2×C22⋊Q8—31no—
76C22×C22⋊C4—31no—
77#407—31no—
78#407C2×D311yescharacteristic
79#397—32no—
80#393—32no—
81#388—62no—
82#388—62no—
83#388—62no—
84#381—61no—
85#380—61no—
86#379—61no—
87#379—61no—
88C4×C22⋊C4—31no—
89C4×C22⋊C4C2×D311yescharacteristic
90C4×C22⋊C4C2×D311yescharacteristic
91#328—62no—
92#328—62no—
93#323—62no—
94C24×C3C2×D411yescharacteristic
95C23×D3—21no—
96C23×D3—21no—
97C23×D3—42no—
98C23×D3—42no—
99C23×D3—81no—
100C23×D3C22⋊C412yes—
101C22×C3×C4—22no—
102C22×C3×C4—22no—
103C22×C3×C4—81no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—22no—
115C22×Dic3—42no—
116C22×Dic3C22⋊C412yes—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—81no—
120C3×M16—81no—
121C2×C3×C8—41no—
122C3×C22⋊C4—42no—
123C3×C22⋊C4—42no—
124C3×C42—81no—
125C3×C42—81no—
126#217—42no—
127#217—41no—
128#217—41no—
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—44no—
134(C2×D3)⋊C4—82no—
135(C2×D3)⋊C4—82no—
136(C2×D3)⋊C4—82no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139Dic3⋊C4—41no—
140Dic3⋊C4—41no—
141Dic3⋊C4—82no—
142C4×Dic3—81no—
143#208—81no—
144C2×C3⋊C8—41no—
145C25—31no—
146C22×Q8—31no—
147C23×C4—31no—
148C23×C4C22×D311yescharacteristic
149C2×M16—21no—
150C2×M16—61no—
151C22⋊Q8—61no—
152C22⋊Q8—61no—
153C22⋊Q8—61no—
154C22⋊Q8—61no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—62no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—32no—
161C2×C22⋊C4—32no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—62no—
170C2×C22⋊C4—62no—
171C2×C22⋊C4—62no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—61no—
175C22⋊C8—21no—
176C22⋊C8—21no—
177C22⋊C8—61no—
178C22⋊C8—61no—
179#95—32no—
180#95—62no—
181#95—62no—
182#95—62no—
183#95C4×D312yes—
184#95D1212yes—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—41no—
188C23×C3C2×C22⋊C411yescharacteristic
189C23×C3#12011yescharacteristic
190C23×C3#12311yescharacteristic
191C22×D3—22no—
192C22×D3—22no—
193C22×D3—22no—
194C22×D3—42no—
195C22×D3—42no—
196C22×D3—42no—
197C22×D3—42no—
198C22×D3—42no—
199C22×D3—42no—
200C22×D3—81no—
201C22×D3—81no—
202C22×D3—88no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×Dic3—22no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—81no—
230Dic6—81no—
231Dic6—81no—
232C3×C8—81no—
233C3⋊C8—81no—
234C24—32no—
235C24—61no—
236C24—61no—
237C24—122no—
238C24—122no—
239C24—241no—
240C24D3×D411yescharacteristic
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—241no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—32no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—62no—
252C22×C4—81no—
253C22×C4—122no—
254C22×C4C2×C4×D311yescharacteristic
255C22×C4C2×D1211yescharacteristic
256C22×C4#23511yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4#23911yescharacteristic
260M16—81no—
261M16—241no—
262C2×C8—41no—
263C2×C8—121no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—242no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—62no—
272C22⋊C4—62no—
273C22⋊C4—62no—
274C22⋊C4—62no—
275C22⋊C4—124no—
276C22⋊C4—122no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—242no—
280C22⋊C4—242no—
281C22⋊C4—242no—
282C42—81no—
283C42—81no—
284C42—241no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—44no—
293C22×C3#38611yescharacteristic
294C22×C3#42112yes—
295C22×C3#45512yes—
296C2×D3—44no—
297C2×D3—42no—
298C2×D3—42no—
299C2×D3—42no—
300C2×D3—84no—
301C2×D3—84no—
302C2×D3—82no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—82no—
308C3×C4—82no—
309Dic3—42no—
310Dic3—82no—
311Dic3—81no—
312Dic3—81no—
313C23—22no—
314C23—22no—
315C23—41no—
316C23—62no—
317C23—62no—
318C23—62no—
319C23—122no—
320C23—122no—
321C23—122no—
322C23—122no—
323C23—122no—
324C23—122no—
325C23—241no—
326C23—241no—
327C23—248no—
328C23D3×C22⋊C411yescharacteristic
329C23#88211yescharacteristic
330C23#89411yescharacteristic
331Q8—241no—
332Q8—241no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—62no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—241no—
356C2×C4C4×D1211yescharacteristic
357C2×C4#88411yescharacteristic
358C2×C4#89311yescharacteristic
359C2×C4#89511yescharacteristic
360C8—81no—
361C8—241no—
362C2×C3—22no—
363C2×C3—22no—
364C2×C3—41no—
365C2×C3#188511yescharacteristic
366C2×C3#189711yescharacteristic
367C2×C3#191311yescharacteristic
368D3—82no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—44no—
377C22—124no—
378C22—122no—
379C22—122no—
380C22—122no—
381C22—242no—
382C22—244no—
383C22—244no—
384C22#466511yescharacteristic
385C22#482112yes—
386C22#482512yes—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—82no—
392C4—82no—
393C4—122no—
394C4—242no—
395C4—241no—
396C4—241no—
397C3#941111yescharacteristic, complemented, Sylow
398C2—22no—
399C2—22no—
400C2—41no—
401C2—242no—
402C2#6875711yescharacteristic
403C2#6875911yescharacteristic
404C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1297C2
socleC22×C3#386
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1297D3
hypocenterC3#9411
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
3. #69980C2semidirect
5. #69445C2semidirect
6. C3×#1297C2semidirect
7. #67618C2semidirect
22. #4674C4semidirect
23. #4673C4semidirect
34. #1297D3semidirect
397. C3#9411semidirect

Maximal subgroups

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

subgroupindex
#736052
#735672
#699802
#699672
#694452
C3×#12972
#676182
#94113