Finite Group Databasebeta

← #10704366

Subgroups — #10704366

Order 768.

434 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 93 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#74382C211yesmaximal, characteristic
2#68100C211yesmaximal, characteristic
3#68100C211yesmaximal, characteristic
4C3×#1326C211yesmaximal, characteristic
5#67636C211yesmaximal, characteristic
6#67447C211yesmaximal, characteristic
7#67446C211yesmaximal, characteristic
8#11454—31nomaximal, Sylow
9C22×Dic3⋊C4C411yescharacteristic
10C2×#877C411yescharacteristic
11C2×#877C2211yescharacteristic
12C3×#385C2211yescharacteristic
13#5210C411yescharacteristic
14#4976—21no—
15#4975—21no—
16#4950C411yescharacteristic
17#4946C2211yescharacteristic
18#4718—21no—
19#4718—21no—
20#4650—21no—
21#4647—21no—
22C3×#323C2211yescharacteristic
23C3×#323C2211yescharacteristic
24#4525C2211yescharacteristic
25#4525C2211yescharacteristic
26#4469—21no—
27#4469—21no—
28#2317—31no—
29#1458—31no—
30#1458—31no—
31#1326—31no—
32#1326—31no—
33#1326—31no—
34#1326D311yescharacteristic
35C23×C3×C4C2×C411yescharacteristic
36C23×Dic3C2×C411yescharacteristic
37C23×Dic3C2×C411yescharacteristic
38C2×C3×C4⋊C4—21no—
39C2×C3×C22⋊C4C2×C411yescharacteristic
40C2×C3×C22⋊C4C2311yescharacteristic
41C2×#217C2×C411yescharacteristic
42C2×#217C2×C411yescharacteristic
43C2×C4⋊Dic3—21no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—41no—
50C2×Dic3⋊C4—41no—
51C2×Dic3⋊C4D411yescharacteristic
52C2×Dic3⋊C4D411yescharacteristic
53C2×C4×Dic3—21no—
54C2×C4×Dic3D411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56#877—22no—
57#877—41no—
58#877—41no—
59#877C814yescomplemented
60C3×C22⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66#830—21no—
67#830—21no—
68Dic3⋊C8—41no—
69Dic3⋊C8—41no—
70C8×Dic3—41no—
71C8×Dic3—41no—
72C2×#117—31no—
73C2×#117—31no—
74C22×C4⋊C4—31no—
75#431—61no—
76#431—61no—
77#389—61no—
78#389—61no—
79#385—31no—
80#385—31no—
81#385—31no—
82#385C2×D311yescharacteristic
83#384—61no—
84#382—61no—
85#324—61no—
86#324—61no—
87#323—31no—
88#323—31no—
89#323C2×D311yescharacteristic
90#323C2×D311yescharacteristic
91C24×C3C22×C411yescharacteristic
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—41no—
95C22×C3×C4—81no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—22no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—41no—
104C22×Dic3—81no—
105C22×Dic3C22⋊C411yescharacteristic
106C22×Dic3C22⋊C411yescharacteristic
107C22×Dic3C22⋊C411yescharacteristic
108C22×Dic3C22⋊C411yescharacteristic
109C22×Dic3C2×C812yes—
110C22×Dic3M1612yes—
111C2×C3×C8—41no—
112C2×C3×C8—41no—
113C3×C4⋊C4—81no—
114C3×C22⋊C4—21no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4C2×C812yes—
117#217—21no—
118#217—41no—
119#217C2×C812yes—
120C4⋊Dic3—81no—
121Dic3⋊C4—24no—
122Dic3⋊C4—41no—
123Dic3⋊C4—41no—
124Dic3⋊C4—42no—
125Dic3⋊C4—41no—
126Dic3⋊C4—41no—
127Dic3⋊C4—81no—
128Dic3⋊C4—81no—
129C4×Dic3—24no—
130C4×Dic3—21no—
131C4×Dic3—21no—
132C4×Dic3—21no—
133C4×Dic3—21no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C23×C4—31no—
137C23×C4—31no—
138C23×C4C4×D311yescharacteristic
139#117—34no—
140#117—62no—
141#117—121no—
142#117—121no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—121no—
153C2×C4⋊C4—121no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4C4×D311yescharacteristic
157C2×C22⋊C4C22×D311yescharacteristic
158C2×C42—31no—
159C2×C42—31no—
160C2×C42—61no—
161C4⋊C8—121no—
162C4⋊C8—121no—
163C22⋊C8—21no—
164C22⋊C8—21no—
165C22⋊C8—61no—
166C22⋊C8—61no—
167C4×C8—121no—
168C4×C8—121no—
169#95—21no—
170#95—61no—
171#95—61no—
172#95—61no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3C2×C22⋊C411yescharacteristic
178C23×C3C22×C811yescharacteristic
179C23×C3C2×M1611yescharacteristic
180C2×C3×C4—22no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—41no—
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—82no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—22no—
194C2×Dic3—22no—
195C2×Dic3—22no—
196C2×Dic3—22no—
197C2×Dic3—22no—
198C2×Dic3—21no—
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—82no—
208C2×Dic3C22⋊C812yes—
209C2×Dic3C22⋊C812yes—
210C2×Dic3#9911yescharacteristic
211C2×Dic3#9911yescharacteristic
212C2×Dic3#10112yes—
213C3×C8—81no—
214C3×C8—81no—
215C3⋊C8—81no—
216C3⋊C8—81no—
217C24C2×C4×D311yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—32no—
223C22×C4—32no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—41no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—62no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—81no—
233C22×C4—121no—
234C22×C4—241no—
235C22×C4D3×D411yescharacteristic
236C22×C4D3×D411yescharacteristic
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—121no—
240C2×C8—121no—
241C4⋊C4—64no—
242C4⋊C4—81no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C4⋊C4—122no—
248C4⋊C4—241no—
249C4⋊C4—241no—
250C4⋊C4—241no—
251C22⋊C4—21no—
252C22⋊C4—32no—
253C22⋊C4—41no—
254C22⋊C4—61no—
255C22⋊C4—121no—
256C22⋊C4D3×C812yes—
257C42—61no—
258C42—61no—
259C42—61no—
260C42—61no—
261C42—64no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—42no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3C2×C22⋊C811yescharacteristic
272C22×C3C2×#9911yescharacteristic
273C22×C3C2×#10111yescharacteristic
274C3×C4—42no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—81no—
278Dic3—22no—
279Dic3—42no—
280Dic3—41no—
281Dic3—41no—
282Dic3—81no—
283Dic3#32412yes—
284Dic3#32412yes—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23D3×C22⋊C411yescharacteristic
290C23C2×D3×C811yescharacteristic
291C23D3×M1611yescharacteristic
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—32no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—32no—
299C2×C4—32no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—62no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—82no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—242no—
325C8—81no—
326C8—81no—
327C8—241no—
328C8—241no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3C2×#32411yescharacteristic
334C2×C3#212211yescharacteristic
335C2×C3#212311yescharacteristic
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22D3×C22⋊C811yescharacteristic
346C22D3×#9911yescharacteristic
347C22D3×#10111yescharacteristic
348C4—32no—
349C4—32no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—62no—
354C4—81no—
355C4—122no—
356C4—121no—
357C4—121no—
358C4—241no—
359C3#1145411yescharacteristic, complemented, Sylow
360C2—21no—
361C2—21no—
362C2—41no—
363C2—41no—
364C2D3×#32411yescharacteristic
365C2#6913711yescharacteristic
366C2#6914511yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC23×C3C22×C8
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1326C2
socleC22×C3C2×#99
prime radicalC24×C3C22×C4
cosocleC2×C3×C22⋊C4C23
hypercenter#1326D3
hypocenterC3#11454
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1326

Complements

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

NH ≅ G/Ntype
59. #877C8semidirect
359. C3#11454semidirect

Maximal subgroups

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

subgroupindex
#743822
#681002
C3×#13262
#676362
#674472
#674462
#114543