Finite Group Databasebeta

← #10705992

Subgroups — #10705992

Order 768.

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

normal subgroups only

This group has 416 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#78131C211yesmaximal, characteristic
2#68997C211yesmaximal, characteristic, complemented
3#68127C211yesmaximal, characteristic, complemented
4C3×#1353C211yesmaximal, characteristic, complemented
5#67652C211yesmaximal, characteristic, complemented
6#67493C211yesmaximal, characteristic
7#67456C211yesmaximal, characteristic
8#11829—31nomaximal, Sylow
9C2×#893C411yescharacteristic
10C2×#887C2211yescharacteristic
11C3×#400C2211yescharacteristic
12C2×#824—21no—
13#5118—21no—
14#4991C2211yescharacteristic
15#4985C411yescharacteristic
16#4719—21no—
17#4675—21no—
18#4673—21no—
19#4672C411yescharacteristic
20#4666—21no—
21#4666—21no—
22#4648C411yescharacteristic
23C3×#343C2211yescharacteristic
24C3×#324C2211yescharacteristic
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4469—21no—
28#2398—31no—
29#1801—31no—
30#1467—31no—
31#1353—31no—
32#1353D311yescharacteristic, complemented
33#1345—31no—
34#1345—31no—
35C22×D12C2×C411yescharacteristic
36C22×C4×D3—21no—
37C22×Dic6D411yescharacteristic
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)⋊C4—21no—
46C2×Dic3⋊C4C2×C411yescharacteristic
47C2×Dic3⋊C4D411yescharacteristic
48C2×C4×Dic3—21no—
49C2×C4×Dic3C2×C411yescharacteristic
50C2×C4×Dic3D411yescharacteristic
51C2×C4×Dic3D411yescharacteristic
52C2×#208—21no—
53#893—41no—
54#893—41no—
55#887—41no—
56#887—41no—
57C3×C22⋊C8—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95C2×C411yescharacteristic
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95C2×C411yescharacteristic
65#830—21no—
66#824—42no—
67#815—41no—
68Dic3⋊C8—41no—
69Dic3⋊C8—41no—
70Dic3⋊C8—41no—
71C2×C4×Q8—31no—
72C2×C4×D4—31no—
73#432—61no—
74#423—61no—
75C2×#101—61no—
76#400—31no—
77#400C2×D311yescharacteristic
78#397—61no—
79#394—31no—
80#394—31no—
81#393—61no—
82C4×C22⋊C4—61no—
83C4×C22⋊C4—61no—
84#376—31no—
85#343—31no—
86#343C2×D311yescharacteristic
87#324—31no—
88#324—61no—
89#324C2×D311yescharacteristic
90C23×D3—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—21no—
97C22×Dic3C22⋊C411yescharacteristic
98C22×Dic3C22⋊C411yescharacteristic
99C22×Dic3C22⋊C411yescharacteristic
100C22×Dic3C22⋊C411yescharacteristic
101C2×D12—41no—
102C2×D12—42no—
103C2×C4×D3—42no—
104C2×C4×D3—42no—
105C2×C4×D3—41no—
106C2×C4×D3—41no—
107C2×Dic6—21no—
108C2×Dic6—21no—
109C2×Dic6—41no—
110C2×Dic6—41no—
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—42no—
118(C2×D3)⋊C4—42no—
119(C2×D3)⋊C4—42no—
120(C2×D3)⋊C4—42no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125C4×Dic3—21no—
126C4×Dic3—21no—
127C4×Dic3—21no—
128C4×Dic3—21no—
129C4×Dic3—21no—
130C4×Dic3—21no—
131C4×Dic3—42no—
132#208—41no—
133#208—41no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C22×Q8—31no—
137C22×D4—31no—
138C23×C4—61no—
139C2×M16—21no—
140C2×M16—61no—
141C4×Q8—121no—
142C4×Q8—121no—
143C4×D4—121no—
144C4×D4—121no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4C4×D311yescharacteristic
148C2×C4⋊C4C22×D311yescharacteristic
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—61no—
154C2×C42—31no—
155C2×C42—31no—
156C2×C42—31no—
157C2×C42—61no—
158C4⋊C8—121no—
159C4⋊C8—121no—
160C4⋊C8—121no—
161#101—122no—
162C22⋊C8—21no—
163C22⋊C8—61no—
164#97—121no—
165#95—21no—
166#95—21no—
167#95—31no—
168#95—61no—
169#95—61no—
170#95—61no—
171#95C4×D311yescharacteristic
172C23×C3C2×C22⋊C411yescharacteristic
173C22×D3—22no—
174C22×D3—41no—
175C22×D3—41no—
176C22×D3—41no—
177C22×D3—41no—
178C22×D3—41no—
179C22×D3—41no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
193C2×C3×C4#13711yescharacteristic
194C2×Dic3—22no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3#9911yescharacteristic
207C2×Dic3#9911yescharacteristic
208D12—42no—
209D12—42no—
210C4×D3—82no—
211C4×D3—82no—
212Dic6—42no—
213Dic6—42no—
214C3×C8—81no—
215C3×C8—81no—
216C3⋊C8—81no—
217C3⋊C8—81no—
218C24—61no—
219C2×Q8—61no—
220C2×Q8—61no—
221C2×Q8—121no—
222C2×Q8—121no—
223C2×D4—121no—
224C2×D4—122no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—61no—
232C22×C4—121no—
233C22×C4—121no—
234C22×C4—122no—
235C22×C4—122no—
236C22×C4C2×C4×D311yescharacteristic
237C22×C4D3×D411yescharacteristic
238C22×C4D3×D411yescharacteristic
239M16—41no—
240M16—41no—
241M16—121no—
242M16—121no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—121no—
246C2×C8—121no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C22⋊C4—122no—
252C22⋊C4—122no—
253C22⋊C4—122no—
254C22⋊C4—122no—
255C22⋊C4—121no—
256C22⋊C4—121no—
257C42—61no—
258C42—61no—
259C42—61no—
260C42—61no—
261C42—61no—
262C42—61no—
263C42—122no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3C2×#9911yescharacteristic
267C22×C3#41711yescharacteristic
268C22×C3#42111yescharacteristic
269C2×D3—42no—
270C2×D3—42no—
271C2×D3—42no—
272C2×D3—81no—
273C2×D3—81no—
274C2×D3—81no—
275C2×D3—81no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—81no—
281C3×C4—81no—
282Dic3—21no—
283Dic3—41no—
284Dic3—41no—
285Dic3—41no—
286Dic3—81no—
287Dic3#35512yes—
288C23—62no—
289C23—121no—
290C23—121no—
291C23—121no—
292C23—121no—
293C23—121no—
294C23—121no—
295C23D3×C22⋊C411yescharacteristic
296Q8—122no—
297Q8—122no—
298D4—122no—
299D4—122no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—62no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—242no—
327C2×C4—242no—
328C2×C4#91411yescharacteristic
329C2×C4#91811yescharacteristic
330C8—81no—
331C8—81no—
332C8—241no—
333C8—241no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3#151311yescharacteristic
337C2×C3#212011yescharacteristic
338C2×C3C2×#35511yescharacteristic
339D3—82no—
340C22—21no—
341C22—21no—
342C22—122no—
343C22—122no—
344C22—122no—
345C22—241no—
346C22—241no—
347C22—241no—
348C22—241no—
349C22D3×#9911yescharacteristic
350C22#480311yescharacteristic
351C22#482111yescharacteristic
352C4—21no—
353C4—21no—
354C4—32no—
355C4—41no—
356C4—41no—
357C4—61no—
358C4—81no—
359C4—81no—
360C4—121no—
361C4—121no—
362C4—121no—
363C4—241no—
364C3#1182911yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—242no—
368C2#6833711yescharacteristic
369C2#6914411yescharacteristic
370C2D3×#35511yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1353C2
socleC22×C3C2×#99
prime radicalC22×D12C2×C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1353D3
hypocenterC3#11829
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. #68997C2semidirect
3. #68127C2semidirect
4. C3×#1353C2semidirect
5. #67652C2semidirect
32. #1353D3semidirect
364. C3#11829semidirect

Maximal subgroups

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

subgroupindex
#781312
#689972
#681272
C3×#13532
#676522
#674932
#674562
#118293