Finite Group Databasebeta

← #10705580

Subgroups — #10705580

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#77206C211yesmaximal, characteristic
2#68972C211yesmaximal, characteristic
3#68143C211yesmaximal, characteristic
4C3×#1354C211yesmaximal, characteristic, complemented
5#67648C211yesmaximal, characteristic, complemented
6#67495C211yesmaximal, characteristic, complemented
7#67458C211yesmaximal, characteristic, complemented
8#11739—31nomaximal, Sylow
9C2×#946C411yescharacteristic
10C2×#893C2211yescharacteristic
11C3×#395C2211yescharacteristic
12#5231C411yescharacteristic
13C2×#823—21no—
14Dic3×M16—21no—
15#4992—21no—
16#4989—21no—
17#4981C2211yescharacteristic
18D3×C22⋊C8—21no—
19#4674C411yescharacteristic
20#4672—21no—
21#4664C411yescharacteristic
22#4652—21no—
23C3×#343C2211yescharacteristic
24C3×#324C2211yescharacteristic
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4468—21no—
28#2638—31no—
29C4×#100—31no—
30#1459—31no—
31#1354—31no—
32#1354D311yescharacteristic, complemented
33#1346—31no—
34#1346—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×D12D411yescharacteristic
37C22×C4×D3C2×C411yescharacteristic
38C22×C4×D3D411yescharacteristic
39C2×C3×M16—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2311yescharacteristic
42#946—41no—
43#946—41no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4C2×C411yescharacteristic
49C2×(C2×D3)⋊C4D411yescharacteristic
50C2×C4⋊Dic3C2×C411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3D411yescharacteristic
54C2×#208—21no—
55C2×D3×C8—41no—
56#893—41no—
57#893—41no—
58C3×C22⋊C8—21no—
59C3×#95—21no—
60C3×#95C2×C411yescharacteristic
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95C2×C411yescharacteristic
66#830—21no—
67#823—24no—
68(C2×D3)⋊C8—41no—
69#815—41no—
70C8×Dic3—41no—
71C2×C22⋊Q8—31no—
72C2×C4×D4—31no—
73C2×#100—61no—
74C2×C22⋊C8—61no—
75C4×M16—61no—
76#398—31no—
77#397—31no—
78#397—61no—
79#395C2×D311yescharacteristic
80#394—61no—
81#393—31no—
82#387—61no—
83#385—61no—
84#381—31no—
85#343—31no—
86#343C2×D311yescharacteristic
87#324—31no—
88#324C2×D311yescharacteristic
89#323—61no—
90C23×D3C22⋊C411yescharacteristic
91C23×D3C22⋊C411yescharacteristic
92C2×C3×Q8—41no—
93C2×C3×Q8—81no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3C22⋊C411yescharacteristic
102C22×Dic3C22⋊C411yescharacteristic
103C2×D12—22no—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—41no—
107C2×C4×D3—21no—
108C2×C4×D3—21no—
109C2×C4×D3—21no—
110C2×C4×D3—21no—
111C2×C4×D3—81no—
112C2×C4×D3—81no—
113C3×M16—22no—
114C3×M16—41no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—81no—
119(C2×D3)⋊C4—41no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—42no—
122(C2×D3)⋊C4—42no—
123(C2×D3)⋊C4—81no—
124(C2×D3)⋊C4—81no—
125C4⋊Dic3—41no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128Dic3⋊C4—41no—
129Dic3⋊C4—41no—
130C4×Dic3—21no—
131C4×Dic3—21no—
132#208—22no—
133#208—41no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136D3×C8—82no—
137C22×Q8C4×D311yescharacteristic
138C22×D4—31no—
139C23×C4—31no—
140C23×C4—31no—
141C2×M16—21no—
142C2×M16—61no—
143C22×C8—121no—
144C22⋊Q8—121no—
145C22⋊Q8—121no—
146C4×D4—121no—
147C4×D4—121no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4C22×D311yescharacteristic
153C2×C22⋊C4—31no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C42—31no—
160#100—64no—
161C22⋊C8—21no—
162C22⋊C8—61no—
163C22⋊C8—121no—
164#97—121no—
165C4×C8—121no—
166#95—21no—
167#95—31no—
168#95—61no—
169#95—61no—
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—41no—
180C22×D3—41no—
181C22×D3—41no—
182C22×D3—41no—
183C22×D3#10412yes—
184C3×Q8—82no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
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×C4—41no—
197C2×C3×C4#13511yescharacteristic
198C2×C3×C4#13511yescharacteristic
199C2×Dic3—21no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3#9911yescharacteristic
209C2×Dic3#9911yescharacteristic
210D12—42no—
211D12—42no—
212C4×D3—42no—
213C4×D3—81no—
214C4×D3—81no—
215C4×D3—81no—
216C3×C8—42no—
217C3×C8—81no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220C24—31no—
221C24—31no—
222C2×Q8—41no—
223C2×Q8—81no—
224C2×D4—62no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—121no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—241no—
239C22×C4—241no—
240C22×C4C2×C4×D311yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4D3×D411yescharacteristic
243M16—22no—
244M16—41no—
245M16—62no—
246M16—121no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—121no—
250C2×C8—121no—
251C2×C8—242no—
252C4⋊C4—41no—
253C4⋊C4—81no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C22⋊C4—121no—
260C22⋊C4—121no—
261C22⋊C4—122no—
262C22⋊C4—122no—
263C22⋊C4—241no—
264C22⋊C4—241no—
265C42—61no—
266C42—61no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3C2×#9911yescharacteristic
270C22×C3#41611yescharacteristic
271C22×C3C2×#10411yescharacteristic
272C2×D3—22no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—42no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—41no—
279C2×D3—81no—
280C2×D3—81no—
281C2×D3—81no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—81no—
287C3×C4—81no—
288Dic3—21no—
289Dic3—21no—
290Dic3—41no—
291Dic3—81no—
292Dic3—81no—
293C23—32no—
294C23—61no—
295C23—61no—
296C23—61no—
297C23—61no—
298C23—61no—
299C23—121no—
300C23—121no—
301C23—121no—
302C23—121no—
303C23D3×C22⋊C411yescharacteristic
304Q8—82no—
305D4—122no—
306D4—122no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—61no—
322C2×C4—122no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—241no—
332C2×C4—241no—
333C2×C4—241no—
334C2×C4#91611yescharacteristic
335C2×C4#91911yescharacteristic
336C8—42no—
337C8—81no—
338C8—122no—
339C8—241no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3#150111yescharacteristic
343C2×C3#212111yescharacteristic
344C2×C3#213511yescharacteristic
345D3—42no—
346D3—81no—
347C22—21no—
348C22—21no—
349C22—62no—
350C22—122no—
351C22—121no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22—121no—
356C22—241no—
357C22—241no—
358C22—241no—
359C22D3×#9911yescharacteristic
360C22#480411yescharacteristic
361C22D3×#10411yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—61no—
367C4—61no—
368C4—81no—
369C4—81no—
370C4—121no—
371C4—241no—
372C4—241no—
373C3#1173911yescharacteristic, complemented, Sylow
374C2—21no—
375C2—21no—
376C2—122no—
377C2—241no—
378C2#6833311yescharacteristic
379C2#6913511yescharacteristic
380C2#6917311yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1354C2
socleC22×C3C2×#99
prime radicalC22×D12D4
cosocleC2×C3×C4⋊C4C23
hypercenter#1354D3
hypocenterC3#11739
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1354

Complements

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

NH ≅ G/Ntype
4. C3×#1354C2semidirect
5. #67648C2semidirect
6. #67495C2semidirect
7. #67458C2semidirect
32. #1354D3semidirect
373. C3#11739semidirect

Maximal subgroups

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

subgroupindex
#772062
#689722
#681432
C3×#13542
#676482
#674952
#674582
#117393