Finite Group Databasebeta

← #10698680

Subgroups — #10698680

Order 768.

386 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 91 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#83055C211yesmaximal, characteristic
2C2×C3×#330C211yesmaximal, characteristic
3#71741C211yesmaximal, characteristic
4#69827C211yesmaximal, characteristic
5#69815C211yesmaximal, characteristic
6C2×#4465C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic
8#10428—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13#4986—21no—
14#4985—21no—
15#4983—21no—
16#4982—21no—
17#4982—21no—
18#4981—21no—
19#4936C2211yescharacteristic
20#4935C2211yescharacteristic
21C2×#803C2211yescharacteristic
22C3×#330—22no—
23#4551—21no—
24#4551—21no—
25#4551—21no—
26#4551—21no—
27#4529—21no—
28#4529—21no—
29#4465—22no—
30#2822—31no—
31#1992—31no—
32#1992—31no—
33#1836—31no—
34C2×#330—31no—
35C2×#330D311yescharacteristic
36#1295—31no—
37C22×C3×C8—21no—
38C22×C3×C8—21no—
39C3×#125—21no—
40C3×#125—21no—
41C3×#125—21no—
42C3×#125—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C42C2311yescharacteristic
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3D411yescharacteristic
52C2×C4⋊Dic3D411yescharacteristic
53C2×C4⋊Dic3D411yescharacteristic
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C22×C3⋊C8—21no—
60#870—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64C3×#97—21no—
65C3×#97—21no—
66C3×#97—21no—
67C3×#97D412yes—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76#803—21no—
77#803—21no—
78C2×#125—31no—
79C2×#125C2×D311yescharacteristic
80C2×#97—31no—
81C2×#97C2×D311yescharacteristic
82C2×#97C2×D311yescharacteristic
83#400—61no—
84#400—61no—
85#398—61no—
86#398—61no—
87#398—61no—
88#398—61no—
89#384—31no—
90#382—31no—
91#340—61no—
92#340—61no—
93#340—61no—
94#340—61no—
95#340—61no—
96#340—61no—
97#330—22no—
98#330—62no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—41no—
109C2×C3×C8—22no—
110C2×C3×C8—22no—
111C2×C3×C8—41no—
112C2×C3×C8—41no—
113C2×C3×C8—41no—
114C2×C3×C8—41no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C42C2×D411yescharacteristic
124C3×C42#4111yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—41no—
132C4⋊Dic3—41no—
133C4⋊Dic3—42no—
134C4⋊Dic3—42no—
135C4⋊Dic3—42no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138Dic3⋊C4—41no—
139Dic3⋊C4—41no—
140Dic3⋊C4—81no—
141Dic3⋊C4—81no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—41no—
144C2×C3⋊C8—41no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—61no—
148#125—21no—
149#125—21no—
150#125—21no—
151#125—21no—
152#125—61no—
153#125—61no—
154#125—61no—
155#125—61no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4D1211yescharacteristic
170C2×C4⋊C4D1211yescharacteristic
171C2×C42C22×D311yescharacteristic
172#97—21no—
173#97—21no—
174#97—21no—
175#97—61no—
176#97—61no—
177#97#6712yes—
178#95—61no—
179#95—61no—
180#95—61no—
181#95—61no—
182#95—61no—
183#95—61no—
184#95—61no—
185#95—61no—
186C23×C3#12011yescharacteristic
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12311yescharacteristic
203C2×C3×C4#12311yescharacteristic
204C2×C3×C4#12411yescharacteristic
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219C2×Dic3—81no—
220C3×C8—42no—
221C3×C8—42no—
222C3⋊C8—42no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—121no—
230C22×C4C2×D1211yescharacteristic
231C22×C4D3×D411yescharacteristic
232C22×C4D3×D411yescharacteristic
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—62no—
240C2×C8—121no—
241C2×C8—121no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—122no—
257C4⋊C4—122no—
258C4⋊C4—122no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—241no—
264C4⋊C4—241no—
265C42#23511yescharacteristic
266C42#23711yescharacteristic
267C42#23711yescharacteristic
268C42C2×#6711yescharacteristic
269C22×C3#39411yescharacteristic
270C22×C3#45411yescharacteristic
271C22×C3#45411yescharacteristic
272C22×C3#45411yescharacteristic
273C22×C3#45511yescharacteristic
274C22×C3#45611yescharacteristic
275C22×C3#45611yescharacteristic
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—81no—
283C3×C4—81no—
284Dic3—81no—
285Dic3—81no—
286Dic3—81no—
287Dic3—81no—
288Dic3—161no—
289C23#88211yescharacteristic
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4—241no—
317C2×C4#88311yescharacteristic
318C2×C4#88511yescharacteristic
319C2×C4#88611yescharacteristic
320C2×C4#93011yescharacteristic
321C2×C4#93811yescharacteristic
322C2×C4#93911yescharacteristic
323C8—42no—
324C8—42no—
325C8—122no—
326C2×C3#165811yescharacteristic
327C2×C3#166311yescharacteristic
328C2×C3#166411yescharacteristic
329C2×C3#166411yescharacteristic
330C2×C3#203811yescharacteristic
331C2×C3#203811yescharacteristic
332C2×C3#204011yescharacteristic
333C22#475211yescharacteristic
334C22#475311yescharacteristic
335C22#482511yescharacteristic
336C22#482611yescharacteristic
337C22#482711yescharacteristic
338C22#482711yescharacteristic
339C22#496111yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—81no—
347C4—81no—
348C4—241no—
349C4—241no—
350C4—241no—
351C4—241no—
352C4—481no—
353C3#1042811yescharacteristic, complemented, Sylow
354C2#6837511yescharacteristic
355C2#6837611yescharacteristic
356C2#6837811yescharacteristic
357C2#6837911yescharacteristic
358C2#7010011yescharacteristic
359C2#7036011yescharacteristic
360C2#7036211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#330C2
socle, prime radicalC23×C3#120
hypercenterC2×#330D3
hypocenterC3#10428
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
353. C3#10428semidirect

Maximal subgroups

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

subgroupindex
#830552
C2×C3×#3302
#717412
#698272
#698152
C2×#44652
#676592
#104283