Finite Group Databasebeta

← #10705682

Subgroups — #10705682

Order 768.

484 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 71 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#78203C211yesmaximal, characteristic
2#68989C211yesmaximal, characteristic, complemented
3#68126C211yesmaximal, characteristic, complemented
4C3×#1353C211yesmaximal, characteristic, complemented
5#67652C211yesmaximal, characteristic, complemented
6#67493C211yesmaximal, characteristic, complemented
7#67456C211yesmaximal, characteristic, complemented
8#11761—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×#893C411yescharacteristic
11C2×#892C2211yescharacteristic
12C3×#400C2211yescharacteristic
13#4993C411yescharacteristic
14#4991C2211yescharacteristic
15#4719—21no—
16#4671C411yescharacteristic
17#4669—21no—
18#4668—21no—
19#4665—21no—
20#4665—21no—
21#4651C411yescharacteristic
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4554—21no—
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4468—21no—
28#2598—31no—
29#1742—31no—
30#1464—31no—
31#1353—31no—
32#1353D311yescharacteristic, complemented
33#1345—31no—
34#1345—31no—
35C22×D12—21no—
36C22×D12C2×C411yescharacteristic
37C22×C4×D3—21no—
38C22×C4×D3D411yescharacteristic
39C2×C3×M16—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2311yescharacteristic
42C2×(C2×D3)⋊C4—21no—
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4C2×C411yescharacteristic
48C2×(C2×D3)⋊C4D411yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3C2×C411yescharacteristic
52C2×C4×Dic3D411yescharacteristic
53C2×#208—21no—
54D3×M16—24no—
55D3×M16—42no—
56C2×ΓL(1,𝔽25)—41no—
57C2×D3×C8—41no—
58#893—41no—
59#893—41no—
60#892—41no—
61#892—41no—
62C3×C22⋊C8—21no—
63C3×#95—21no—
64C3×#95—21no—
65C3×#95C2×C411yescharacteristic
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95C2×C411yescharacteristic
69#830—21no—
70#815—41no—
71Dic3⋊C8—41no—
72C22×M16—61no—
73C2×C4×D4—31no—
74C2×#117—31no—
75#432—61no—
76#400C2×D311yescharacteristic
77#399—31no—
78#394—31no—
79#392—31no—
80#388—61no—
81#386—61no—
82#386—61no—
83#385—61no—
84#383—31no—
85#343—31no—
86#343—61no—
87#343C2×D311yescharacteristic
88#324—31no—
89#324C2×D311yescharacteristic
90#323—61no—
91C23×D3—21no—
92C23×D3C22⋊C411yescharacteristic
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3C22⋊C411yescharacteristic
100C22×Dic3C22⋊C411yescharacteristic
101C22×Dic3C22⋊C411yescharacteristic
102C2×D12—41no—
103C2×D12—42no—
104C2×D12—42no—
105C2×D12—42no—
106C2×D12—81no—
107C2×C4×D3—22no—
108C2×C4×D3—21no—
109C2×C4×D3—21no—
110C2×C4×D3—21no—
111C2×C4×D3—21no—
112C2×C4×D3—42no—
113C2×C4×D3—81no—
114C2×C4×D3—81no—
115C3×M16—22no—
116C3×M16—41no—
117C2×C3×C8—41no—
118C2×C3×C8—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—42no—
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126(C2×D3)⋊C4—41no—
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4—81no—
129(C2×D3)⋊C4—81no—
130C4⋊Dic3—41no—
131Dic3⋊C4—81no—
132C4×Dic3—21no—
133C4×Dic3—21no—
134C4×Dic3—21no—
135C4×Dic3—21no—
136#208—22no—
137#208—41no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140ΓL(1,𝔽25)—44no—
141D3×C8—44no—
142C22×D4—31no—
143C22×D4—61no—
144C23×C4—31no—
145C23×C4—61no—
146C2×M16—21no—
147C2×M16—64no—
148C2×M16—61no—
149C2×M16—121no—
150C2×M16—122no—
151C22×C8—121no—
152C4×D4—121no—
153C4×D4—121no—
154#117—121no—
155#117—121no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4C4×D311yescharacteristic
159C2×C4⋊C4C22×D311yescharacteristic
160C2×C22⋊C4—31no—
161C2×C22⋊C4—31no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C42—31no—
168C2×C42—31no—
169C4⋊C8—121no—
170C22⋊C8—21no—
171C22⋊C8—61no—
172#97—121no—
173#95—21no—
174#95—21no—
175#95—31no—
176#95—61no—
177#95—61no—
178#95C4×D311yescharacteristic
179C23×C3C2×C22⋊C411yescharacteristic
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—22no—
183C22×D3—41no—
184C22×D3—41no—
185C22×D3—41no—
186C22×D3—41no—
187C22×D3—41no—
188C22×D3—41no—
189C22×D3—41no—
190C22×D3—41no—
191C22×D3#10412yes—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
205C2×C3×C4#13711yescharacteristic
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3#9911yescharacteristic
216C2×Dic3#9911yescharacteristic
217D12—42no—
218D12—42no—
219D12—84no—
220C4×D3—22no—
221C4×D3—22no—
222C4×D3—22no—
223C4×D3—22no—
224C4×D3—84no—
225C3×C8—42no—
226C3×C8—81no—
227C3⋊C8—42no—
228C3⋊C8—81no—
229C24—31no—
230C24—61no—
231C2×D4—121no—
232C2×D4—122no—
233C2×D4—122no—
234C2×D4—122no—
235C2×D4—241no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—62no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—122no—
248C22×C4—241no—
249C22×C4—241no—
250C22×C4C2×C4×D311yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253M16—22no—
254M16—41no—
255M16—62no—
256M16—124no—
257M16—121no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—124no—
261C2×C8—121no—
262C2×C8—121no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—121no—
266C4⋊C4—241no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C22⋊C4—121no—
270C22⋊C4—121no—
271C22⋊C4—122no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4—241no—
275C22⋊C4—241no—
276C42—61no—
277C42—61no—
278C42—61no—
279C42—61no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3C2×#9911yescharacteristic
283C22×C3#41711yescharacteristic
284C22×C3C2×#10411yescharacteristic
285C2×D3—22no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—42no—
289C2×D3—42no—
290C2×D3—42no—
291C2×D3—42no—
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C2×D3—81no—
296C2×D3—81no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—81no—
302C3×C4—81no—
303Dic3—21no—
304Dic3—21no—
305Dic3—41no—
306Dic3—41no—
307Dic3—81no—
308C23—32no—
309C23—62no—
310C23—61no—
311C23—61no—
312C23—121no—
313C23—121no—
314C23—121no—
315C23—121no—
316C23—121no—
317C23—121no—
318C23—121no—
319C23—121no—
320C23D3×C22⋊C411yescharacteristic
321D4—122no—
322D4—122no—
323D4—244no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—244no—
352C2×C4#91411yescharacteristic
353C2×C4#91811yescharacteristic
354C8—42no—
355C8—81no—
356C8—122no—
357C8—241no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3#150411yescharacteristic
361C2×C3#212011yescharacteristic
362C2×C3#213411yescharacteristic
363D3—42no—
364D3—82no—
365C22—21no—
366C22—21no—
367C22—62no—
368C22—122no—
369C22—122no—
370C22—122no—
371C22—121no—
372C22—121no—
373C22—122no—
374C22—241no—
375C22—241no—
376C22—241no—
377C22—241no—
378C22—241no—
379C22D3×#9911yescharacteristic
380C22#480311yescharacteristic
381C22D3×#10411yescharacteristic
382C4—21no—
383C4—21no—
384C4—41no—
385C4—41no—
386C4—61no—
387C4—61no—
388C4—81no—
389C4—81no—
390C4—121no—
391C4—121no—
392C4—241no—
393C3#1176111yescharacteristic, complemented, Sylow
394C2—21no—
395C2—21no—
396C2—122no—
397C2—242no—
398C2#6833411yescharacteristic
399C2#6914411yescharacteristic
400C2#6916611yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1353C2
socleC22×C3C2×#99
prime radical#4671C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1353D3
hypocenterC3#11761
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1353

Complements

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

NH ≅ G/Ntype
2. #68989C2semidirect
3. #68126C2semidirect
4. C3×#1353C2semidirect
5. #67652C2semidirect
6. #67493C2semidirect
7. #67456C2semidirect
32. #1353D3semidirect
393. C3#11761semidirect

Maximal subgroups

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

subgroupindex
#782032
#689892
#681262
C3×#13532
#676522
#674932
#674562
#117613