Finite Group Databasebeta

← #10693150

Subgroups — #10693150

Order 768.

542 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 95 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#78634C211yesmaximal, characteristic
2#74027C211yesmaximal, characteristic
3#69973C211yesmaximal, characteristic, complemented
4#69973C211yesmaximal, characteristic, complemented
5#69968C211yesmaximal, characteristic
6C3×#1305C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9508—31nomaximal, Sylow
9C2×C4×#67C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#886C412yes—
12C3×#407C2211yescharacteristic
13C3×#407C2211yescharacteristic
14C3×#380C2211yescharacteristic
15#5207C2211yescharacteristic
16#5207C2211yescharacteristic
17#5128—22no—
18#4987—21no—
19#4984—21no—
20#4943C412yes—
21#4729—22no—
22#4726—21no—
23#4718—21no—
24#4674—21no—
25#4661—21no—
26#4554—22no—
27#4554—22no—
28#4469—21no—
29#4468—21no—
30#2975—31no—
31#2486—31no—
32#1821—31no—
33#1821—31no—
34#1465—31no—
35#1305—31no—
36#1305D311yescharacteristic, complemented
37C23×C3×C4C2311yescharacteristic
38C22×#67C2×C411yescharacteristic
39C23×Dic3C2×C412yes—
40C22×C4×D3D411yescharacteristic
41C2×C3×M16—21no—
42C2×C3×M16D412yes—
43C2×C3×C4⋊C4—21no—
44C2×C3×C22⋊C4C2×C412yes—
45C2×#217C2×C411yescharacteristic
46C4×#67—21no—
47C4×#67—21no—
48C4×#67—21no—
49C4×#67—21no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4D411yescharacteristic
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—22no—
59C2×Dic3⋊C4D411yescharacteristic
60C2×C4×Dic3D411yescharacteristic
61C2×#208—21no—
62C2×#208D412yes—
63C2×ΓL(1,𝔽25)—22no—
64#886—44no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#95—21no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72#830—21no—
73#830—21no—
74#830—21no—
75#830—21no—
76(C2×D3)⋊C8—22no—
77Dic3⋊C8—22no—
78C8×Dic3—22no—
79C2×#123—32no—
80C2×C4×D4—31no—
81C22×C4⋊C4—31no—
82#436—62no—
83#436—62no—
84#431—61no—
85#407—31no—
86#407—31no—
87#407—61no—
88#407C2×D311yescharacteristic
89#407C2×D311yescharacteristic
90#398—61no—
91#397—61no—
92#389—61no—
93#381—32no—
94#380—61no—
95#380C2×D311yescharacteristic
96#343—62no—
97#343—62no—
98#324—61no—
99#323—61no—
100C24×C3C22×C411yescharacteristic
101C23×D3C22⋊C411yescharacteristic
102C22×C3×C4—22no—
103C22×C3×C4—42no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×#67—21no—
111C2×#67—21no—
112C2×#67—22no—
113C2×#67—41no—
114C22×Dic3—22no—
115C22×Dic3—22no—
116C22×Dic3—22no—
117C22×Dic3—42no—
118C22×Dic3—42no—
119C22×Dic3C22⋊C411yescharacteristic
120C22×Dic3C22⋊C411yescharacteristic
121C22×Dic3C22⋊C411yescharacteristic
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3—42no—
125C3×M16—42no—
126C3×M16—42no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C3×C4⋊C4—42no—
130C3×C22⋊C4—42no—
131#217—21no—
132#217—21no—
133(C2×D3)⋊C4—21no—
134(C2×D3)⋊C4—21no—
135(C2×D3)⋊C4—44no—
136C4⋊Dic3—41no—
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140C4⋊Dic3—44no—
141Dic3⋊C4—21no—
142Dic3⋊C4—21no—
143C4×Dic3—21no—
144C4×Dic3—21no—
145#208—42no—
146#208—42no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—41no—
149ΓL(1,𝔽25)—82no—
150C22×D4—31no—
151C23×C4—31no—
152C23×C4—32no—
153C23×C4C22×D311yescharacteristic
154C2×M16—21no—
155C2×M16—32no—
156C2×M16—62no—
157C2×M16—61no—
158C2×M16D1212yes—
159#123—124no—
160C4×D4—61no—
161C4×D4—61no—
162C4×D4—61no—
163C4×D4—61no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—62no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—31no—
175C2×C22⋊C4—62no—
176C2×C22⋊C4C4×D312yes—
177C2×C42—31no—
178C4⋊C8—62no—
179C22⋊C8—21no—
180C22⋊C8—21no—
181C22⋊C8—21no—
182C22⋊C8—21no—
183C22⋊C8—62no—
184C22⋊C8—61no—
185C22⋊C8—61no—
186C22⋊C8—61no—
187C22⋊C8—61no—
188C4×C8—62no—
189#95—21no—
190#95—62no—
191#95—62no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3C2×C22⋊C411yescharacteristic
197C23×C3C4×D411yescharacteristic
198C23×C3C4×D411yescharacteristic
199C22×D3—21no—
200C22×D3—21no—
201C22×D3—21no—
202C22×D3—41no—
203C22×D3—41no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12311yescharacteristic
217#67—42no—
218#67—42no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—82no—
229C2×Dic3—82no—
230C2×Dic3#10012yes—
231C4×D3—42no—
232C4×D3—81no—
233C3×C8—42no—
234C3×C8—81no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237C24—31no—
238C24C2×C4×D311yescharacteristic
239C2×D4—62no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—121no—
243C22×C4—22no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—42no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—62no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—122no—
254C22×C4—122no—
255C22×C4—122no—
256C22×C4C2×D1211yescharacteristic
257C22×C4#23511yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4#23911yescharacteristic
262M16—42no—
263M16—42no—
264M16—122no—
265M16—122no—
266M16—242no—
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—62no—
270C2×C8—121no—
271C4⋊C4—42no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—124no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—42no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—124no—
285C42—61no—
286C42—61no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3#38611yescharacteristic
298C22×C3C2×#10011yescharacteristic
299C22×C3#41311yescharacteristic
300C2×D3—42no—
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—81no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—82no—
309C3×C4#45812yes—
310Dic3—22no—
311Dic3—41no—
312Dic3—41no—
313Dic3—82no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—61no—
319C23—61no—
320C23—61no—
321C23—121no—
322C23—121no—
323C23C4×D1211yescharacteristic
324C23D3×C22⋊C411yescharacteristic
325C23#89311yescharacteristic
326D4—122no—
327D4—122no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—32no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—122no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—241no—
349C2×C4—242no—
350C2×C4—242no—
351C2×C4#88211yescharacteristic
352C2×C4#88411yescharacteristic
353C2×C4#89411yescharacteristic
354C2×C4#89511yescharacteristic
355C8—42no—
356C8—81no—
357C8—122no—
358C8—241no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3#189011yescharacteristic
364C2×C3#190011yescharacteristic
365C2×C3#191511yescharacteristic
366D3—81no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—42no—
376C22—42no—
377C22—122no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22—241no—
382C22#466511yescharacteristic
383C22D3×#10011yescharacteristic
384C22#475111yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—62no—
389C4—82no—
390C4—121no—
391C4—121no—
392C4—242no—
393C4#474212yes—
394C3#950811yescharacteristic, complemented, Sylow
395C2—21no—
396C2—21no—
397C2—41no—
398C2—41no—
399C2—241no—
400C2#6896311yescharacteristic
401C2#6899211yescharacteristic
402C2#6904111yescharacteristic
subgroupG/H
centerC22#4665
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1305C2
socleC22×C3#386
prime radicalC22×#67C2×C4
cosocleC23×C3×C4C23
hypercenter#1305D3
hypocenterC3#9508
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
3. #69973C2semidirect
4. #69973C2semidirect
6. C3×#1305C2semidirect
7. #67626C2semidirect
36. #1305D3semidirect
394. C3#9508semidirect

Maximal subgroups

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

subgroupindex
#786342
#740272
#699732
#699682
C3×#13052
#676262
#95083