Finite Group Databasebeta

← #10705997

Subgroups — #10705997

Order 768.

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

normal subgroups only

This group has 420 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#78129C211yesmaximal, characteristic
2#68994C211yesmaximal, characteristic
3#68145C211yesmaximal, characteristic
4C3×#1353C211yesmaximal, characteristic, complemented
5#67652C211yesmaximal, characteristic, complemented
6#67495C211yesmaximal, characteristic, complemented
7#67458C211yesmaximal, characteristic, complemented
8#11830—31nomaximal, Sylow
9C2×#893C2211yescharacteristic
10C2×#892C411yescharacteristic
11C3×#400C2211yescharacteristic
12C2×#823—21no—
13#5118—21no—
14#4993C411yescharacteristic
15#4981C2211yescharacteristic
16#4726—21no—
17#4675—21no—
18#4674C411yescharacteristic
19#4672—21no—
20#4666—21no—
21#4666C411yescharacteristic
22#4649—21no—
23C3×#343C2211yescharacteristic
24C3×#324C2211yescharacteristic
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4468—21no—
28#2395—31no—
29#1800—31no—
30#1465—31no—
31#1353—31no—
32#1353D311yescharacteristic, complemented
33#1346—31no—
34#1346—31no—
35C22×D12D411yescharacteristic
36C22×C4×D3C2×C411yescharacteristic
37C22×C4×D3D411yescharacteristic
38C2×C3×M16—21no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4C2311yescharacteristic
41C2×(C2×D3)⋊C4—21no—
42C2×(C2×D3)⋊C4—21no—
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4C2×C411yescharacteristic
46C2×(C2×D3)⋊C4D411yescharacteristic
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×C4×Dic3—21no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3D411yescharacteristic
53C2×#208—21no—
54C2×ΓL(1,𝔽25)—41no—
55#893—41no—
56#893—41no—
57#892—41no—
58#892—41no—
59C3×C22⋊C8—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95C2×C411yescharacteristic
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95C2×C411yescharacteristic
66#830—21no—
67#823—42no—
68(C2×D3)⋊C8—41no—
69Dic3⋊C8—41no—
70Dic3⋊C8—41no—
71C2×C4×D4—31no—
72C2×#117—31no—
73#423—61no—
74C2×#100—61no—
75#407—61no—
76#400C2×D311yescharacteristic
77#399—31no—
78#398—31no—
79#397—31no—
80#397—61no—
81#394—61no—
82C4×C4⋊C4—61no—
83C4×C22⋊C4—31no—
84C4×C22⋊C4—61no—
85#343—31no—
86#343C2×D311yescharacteristic
87#324—31no—
88#324C2×D311yescharacteristic
89#323—61no—
90C23×D3C22⋊C411yescharacteristic
91C23×D3C22⋊C411yescharacteristic
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3C22⋊C411yescharacteristic
100C22×Dic3C22⋊C411yescharacteristic
101C2×D12—21no—
102C2×D12—21no—
103C2×D12—41no—
104C2×D12—41no—
105C2×C4×D3—21no—
106C2×C4×D3—21no—
107C2×C4×D3—21no—
108C2×C4×D3—21no—
109C2×C4×D3—42no—
110C2×C4×D3—42no—
111C3×M16—41no—
112C3×M16—41no—
113C2×C3×C8—41no—
114C2×C3×C8—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117(C2×D3)⋊C4—41no—
118(C2×D3)⋊C4—41no—
119(C2×D3)⋊C4—42no—
120(C2×D3)⋊C4—42no—
121(C2×D3)⋊C4—42no—
122(C2×D3)⋊C4—42no—
123C4⋊Dic3—41no—
124Dic3⋊C4—42no—
125Dic3⋊C4—41no—
126Dic3⋊C4—41no—
127C4×Dic3—21no—
128C4×Dic3—21no—
129C4×Dic3—42no—
130C4×Dic3—41no—
131C4×Dic3—41no—
132#208—41no—
133#208—41no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136ΓL(1,𝔽25)—82no—
137C22×D4—31no—
138C23×C4—31no—
139C23×C4—31no—
140C2×M16—21no—
141C2×M16—61no—
142C2×M16—121no—
143C4×D4—121no—
144C4×D4—121no—
145#117—121no—
146#117—121no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4C4×D311yescharacteristic
151C2×C4⋊C4C22×D311yescharacteristic
152C2×C22⋊C4—31no—
153C2×C22⋊C4—31no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C42—31no—
159C2×C42—61no—
160C2×C42—61no—
161C4⋊C8—121no—
162C4⋊C8—121no—
163#100—122no—
164C22⋊C8—21no—
165C22⋊C8—61no—
166C22⋊C8—121no—
167#95—21no—
168#95—21no—
169#95—31no—
170#95—61no—
171#95—61no—
172#95C4×D311yescharacteristic
173C23×C3C2×C22⋊C411yescharacteristic
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—21no—
179C22×D3—21no—
180C22×D3—41no—
181C22×D3—41no—
182C22×D3—41no—
183C22×D3—41no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
197C2×C3×C4#13711yescharacteristic
198C2×Dic3—22no—
199C2×Dic3—22no—
200C2×Dic3—21no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3#9911yescharacteristic
208C2×Dic3#9911yescharacteristic
209D12—42no—
210D12—42no—
211C4×D3—42no—
212C4×D3—42no—
213C4×D3—42no—
214C4×D3—81no—
215C3×C8—81no—
216C3×C8—81no—
217C3⋊C8—81no—
218C3⋊C8—81no—
219C24—31no—
220C24—31no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—121no—
224C2×D4—121no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—122no—
236C22×C4—122no—
237C22×C4C2×C4×D311yescharacteristic
238C22×C4D3×D411yescharacteristic
239C22×C4D3×D411yescharacteristic
240M16—41no—
241M16—41no—
242M16—121no—
243M16—121no—
244M16—242no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—121no—
248C2×C8—121no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—121no—
252C4⋊C4—122no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C22⋊C4—121no—
256C22⋊C4—121no—
257C22⋊C4—122no—
258C22⋊C4—122no—
259C22⋊C4—122no—
260C22⋊C4—122no—
261C42—61no—
262C42—61no—
263C42—121no—
264C42—121no—
265C42—122no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3C2×#9911yescharacteristic
269C22×C3#41711yescharacteristic
270C22×C3#42111yescharacteristic
271C2×D3—42no—
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—81no—
279C2×D3—81no—
280C2×D3—81no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—81no—
286C3×C4—81no—
287Dic3—21no—
288Dic3—42no—
289Dic3—41no—
290Dic3—81no—
291Dic3#35212yes—
292C23—61no—
293C23—61no—
294C23—61no—
295C23—61no—
296C23—61no—
297C23—61no—
298C23—121no—
299C23—121no—
300C23—121no—
301C23—121no—
302C23D3×C22⋊C411yescharacteristic
303D4—122no—
304D4—122no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—62no—
320C2×C4—62no—
321C2×C4—61no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—241no—
332C2×C4#91411yescharacteristic
333C2×C4#91811yescharacteristic
334C8—81no—
335C8—81no—
336C8—241no—
337C8—241no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3#151311yescharacteristic
341C2×C3C2×#35211yescharacteristic
342C2×C3#213211yescharacteristic
343D3—81no—
344D3—81no—
345C22—21no—
346C22—21no—
347C22—122no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—121no—
352C22—121no—
353C22—121no—
354C22—241no—
355C22—241no—
356C22—241no—
357C22D3×#9911yescharacteristic
358C22#480311yescharacteristic
359C22#482111yescharacteristic
360C4—21no—
361C4—21no—
362C4—32no—
363C4—41no—
364C4—41no—
365C4—61no—
366C4—81no—
367C4—81no—
368C4—122no—
369C4—121no—
370C4—241no—
371C3#1183011yescharacteristic, complemented, Sylow
372C2—21no—
373C2—21no—
374C2—241no—
375C2—241no—
376C2#6833711yescharacteristic
377C2D3×#35211yescharacteristic
378C2#6916511yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1353C2
socleC22×C3C2×#99
prime radicalC22×D12D4
cosocleC2×C3×C4⋊C4C23
hypercenter#1353D3
hypocenterC3#11830
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
4. C3×#1353C2semidirect
5. #67652C2semidirect
6. #67495C2semidirect
7. #67458C2semidirect
32. #1353D3semidirect
371. C3#11830semidirect

Maximal subgroups

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

subgroupindex
#781292
#689942
#681452
C3×#13532
#676522
#674952
#674582
#118303