Finite Group Databasebeta

← #10698685

Subgroups — #10698685

Order 768.

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

normal subgroups only

This group has 402 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#69816C211yesmaximal, characteristic
6C2×#4465C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic
8#10429—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×#818—21no—
14C2×#817—21no—
15#4986—21no—
16#4985—21no—
17#4983—21no—
18#4982—21no—
19#4982—21no—
20#4981—21no—
21#4936C2211yescharacteristic
22#4935C2211yescharacteristic
23C2×#803C2211yescharacteristic
24C3×#330—22no—
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#1847—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#818—42no—
77#817—42no—
78#803—21no—
79#803—21no—
80C2×#125—31no—
81C2×#125C2×D311yescharacteristic
82C2×#107—61no—
83C2×#106—61no—
84C2×#97—31no—
85C2×#97C2×D311yescharacteristic
86C2×#97C2×D311yescharacteristic
87#400—61no—
88#400—61no—
89#398—61no—
90#398—61no—
91#398—61no—
92#398—61no—
93#384—31no—
94#382—31no—
95#340—61no—
96#340—61no—
97#340—61no—
98#340—61no—
99#330—22no—
100#330—62no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—41no—
111C2×C3×C8—22no—
112C2×C3×C8—22no—
113C2×C3×C8—22no—
114C2×C3×C8—22no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127C3×C42#4111yescharacteristic
128C3×C42#4111yescharacteristic
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C4⋊Dic3—42no—
138C4⋊Dic3—42no—
139Dic3⋊C4—41no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143Dic3⋊C4—81no—
144Dic3⋊C4—81no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—41no—
147C2×C3⋊C8—41no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—61no—
151#125—21no—
152#125—21no—
153#125—21no—
154#125—21no—
155#125—61no—
156#125—61no—
157#125—61no—
158#125—61no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4D1211yescharacteristic
173C2×C4⋊C4D1211yescharacteristic
174C2×C42C22×D311yescharacteristic
175#107—122no—
176#106—122no—
177#97—21no—
178#97—21no—
179#97—21no—
180#97—61no—
181#97—61no—
182#97#6712yes—
183#95—61no—
184#95—61no—
185#95—61no—
186#95—61no—
187#95—61no—
188#95—61no—
189#95—61no—
190#95—61no—
191C23×C3#12011yescharacteristic
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12311yescharacteristic
208C2×C3×C4#12311yescharacteristic
209C2×C3×C4#12411yescharacteristic
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224C2×Dic3—81no—
225C3×C8—24no—
226C3×C8—42no—
227C3⋊C8—42no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—121no—
235C22×C4C2×D1211yescharacteristic
236C22×C4D3×D411yescharacteristic
237C22×C4D3×D411yescharacteristic
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—62no—
245C2×C8—121no—
246C2×C8—121no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—122no—
264C4⋊C4—122no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—241no—
270C4⋊C4—241no—
271C42#23511yescharacteristic
272C42#23711yescharacteristic
273C42#23711yescharacteristic
274C42C2×#6711yescharacteristic
275C22×C3#39411yescharacteristic
276C22×C3#45311yescharacteristic
277C22×C3#45411yescharacteristic
278C22×C3#45411yescharacteristic
279C22×C3#45611yescharacteristic
280C22×C3#45611yescharacteristic
281C22×C3#45611yescharacteristic
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—81no—
289C3×C4—81no—
290Dic3—81no—
291Dic3—81no—
292Dic3—81no—
293Dic3—81no—
294Dic3—161no—
295C23#88211yescharacteristic
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—241no—
321C2×C4—241no—
322C2×C4—241no—
323C2×C4#88311yescharacteristic
324C2×C4#88511yescharacteristic
325C2×C4#88611yescharacteristic
326C2×C4#93011yescharacteristic
327C2×C4#93811yescharacteristic
328C2×C4#93911yescharacteristic
329C8—24no—
330C8—42no—
331C8—122no—
332C2×C3#165711yescharacteristic
333C2×C3#166011yescharacteristic
334C2×C3#166311yescharacteristic
335C2×C3#166311yescharacteristic
336C2×C3#203811yescharacteristic
337C2×C3#203811yescharacteristic
338C2×C3#203911yescharacteristic
339C22#475211yescharacteristic
340C22#475311yescharacteristic
341C22#482211yescharacteristic
342C22#482611yescharacteristic
343C22#482611yescharacteristic
344C22#482711yescharacteristic
345C22#496111yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—81no—
353C4—81no—
354C4—241no—
355C4—241no—
356C4—241no—
357C4—241no—
358C4—481no—
359C3#1042911yescharacteristic, complemented, Sylow
360C2#6837111yescharacteristic
361C2#6837211yescharacteristic
362C2#6837611yescharacteristic
363C2#6837711yescharacteristic
364C2#7010011yescharacteristic
365C2#7036011yescharacteristic
366C2#7036111yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#330C2
socle, prime radicalC23×C3#120
hypercenterC2×#330D3
hypocenterC3#10429
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
359. C3#10429semidirect

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
#698162
C2×#44652
#676592
#104293