Finite Group Databasebeta

← #10692860

Subgroups — #10692860

Order 768.

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

normal subgroups only

This group has 424 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#79314C211yesmaximal, characteristic
2#73719C211yesmaximal, characteristic
3#70830C211yesmaximal, characteristic
4#70814C211yesmaximal, characteristic
5#70814C211yesmaximal, characteristic
6C3×#1285C211yesmaximal, characteristic
7#67621C211yesmaximal, characteristic
8#9450—31nomaximal, Sylow
9C2×#870C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#4981—21no—
15#4981—21no—
16#4936C411yescharacteristic
17#4935C411yescharacteristic
18#4929C411yescharacteristic
19#4929C411yescharacteristic
20#4926C2211yescharacteristic
21#4926C2211yescharacteristic
22#4828—22no—
23#4719—21no—
24#4718—21no—
25#4701—22no—
26#4653—21no—
27#4653—21no—
28#4469—21no—
29#4469—21no—
30#4465—22no—
31#4465—22no—
32#2983—31no—
33#2565—31no—
34#1534—31no—
35#1534—31no—
36#1467—31no—
37#1285—31no—
38#1285D311yescharacteristic
39C22×Dic6C2×C411yescharacteristic
40C2×C3×C4⋊C4—21no—
41C2×C3×C42C2×C411yescharacteristic
42C2×C3×C42C2×C411yescharacteristic
43C2×C3×C42C2311yescharacteristic
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4D411yescharacteristic
52C2×Dic3⋊C4D411yescharacteristic
53C2×C4×Dic3D411yescharacteristic
54C2×C4×Dic3D411yescharacteristic
55#870—21no—
56#870—21no—
57#870—21no—
58#870—21no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#97—21no—
68C3×#97D412yes—
69C3×#95—21no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76#830—21no—
77#830—21no—
78#815—22no—
79Dic3⋊C8—22no—
80Dic3⋊C8—22no—
81C8×Dic3—22no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84#803—21no—
85#803D412yes—
86C2×#125—31no—
87C2×C4×Q8—31no—
88#446—62no—
89#446—62no—
90#432—31no—
91#432—31no—
92#432—61no—
93#432C2×D311yescharacteristic
94#432C2×D311yescharacteristic
95#431—61no—
96#398—61no—
97#398—61no—
98#391—31no—
99#391—31no—
100#387—61no—
101#387—61no—
102#384—31no—
103#382—31no—
104C4×C4⋊C4C2×D311yescharacteristic
105#330—62no—
106#330—62no—
107#324—61no—
108#324—61no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3C22⋊C411yescharacteristic
117C22×Dic3C22⋊C411yescharacteristic
118C22×Dic3C22⋊C411yescharacteristic
119C22×Dic3C22⋊C411yescharacteristic
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—22no—
123C2×Dic6—41no—
124C2×C3×C8—22no—
125C2×C3×C8—41no—
126C3×C4⋊C4—42no—
127C3×C42—41no—
128C3×C42—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C3×C42#4111yescharacteristic
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—42no—
140C4⋊Dic3—42no—
141Dic3⋊C4—21no—
142Dic3⋊C4—21no—
143Dic3⋊C4—21no—
144Dic3⋊C4—21no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—21no—
150C4×Dic3—21no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C22×Q8—31no—
156#125—61no—
157#125—61no—
158#125—61no—
159#125—61no—
160C4×Q8—61no—
161C4×Q8—61no—
162C4×Q8—61no—
163C4×Q8—61no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C42—31no—
175C2×C42—31no—
176C2×C42C4×D311yescharacteristic
177C2×C42C4×D311yescharacteristic
178C2×C42C22×D311yescharacteristic
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—62no—
182C4⋊C8—62no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185C22⋊C8—21no—
186C22⋊C8—21no—
187C22⋊C8—61no—
188C22⋊C8—61no—
189#97—21no—
190#97—32no—
191#97—62no—
192#97—61no—
193#97D1212yes—
194C4×C8—62no—
195#95—21no—
196#95—61no—
197#95—61no—
198#95—61no—
199#95—61no—
200#95—61no—
201#95—61no—
202C23×C3C2×C22⋊C411yescharacteristic
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12311yescharacteristic
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3#10112yes—
234Dic6—42no—
235Dic6—42no—
236C3×C8—42no—
237C3×C8—81no—
238C3⋊C8—42no—
239C3⋊C8—81no—
240C2×Q8—62no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—121no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—62no—
258C2×C8—121no—
259C4⋊C4—42no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—122no—
271C4⋊C4—122no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C42—41no—
277C42—41no—
278C42—61no—
279C42—61no—
280C42—61no—
281C42—61no—
282C42C2×D1211yescharacteristic
283C42#23511yescharacteristic
284C42D3×D411yescharacteristic
285C42#23911yescharacteristic
286C22×C3—21no—
287C22×C3—21no—
288C22×C3#38611yescharacteristic
289C22×C3C2×#10111yescharacteristic
290C22×C3#41311yescharacteristic
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—42no—
298C3×C4—81no—
299Dic3—22no—
300Dic3—41no—
301Dic3—41no—
302Dic3—41no—
303Dic3—81no—
304Dic3—81no—
305C23D3×C22⋊C411yescharacteristic
306Q8—122no—
307Q8—122no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—32no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4C4×D1211yescharacteristic
334C2×C4#88211yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4#89311yescharacteristic
337C2×C4#89411yescharacteristic
338C2×C4#89511yescharacteristic
339C8—42no—
340C8—81no—
341C8—122no—
342C8—241no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3#188411yescharacteristic
346C2×C3#190111yescharacteristic
347C2×C3#191211yescharacteristic
348C22—21no—
349C22—21no—
350C22#466511yescharacteristic
351C22#474511yescharacteristic
352C22D3×#10111yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—62no—
361C4—81no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4—241no—
366C4—241no—
367C3#945011yescharacteristic, complemented, Sylow
368C2—21no—
369C2—21no—
370C2#6870811yescharacteristic
371C2#6877711yescharacteristic
372C2#6899311yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1285D3
hypocenterC3#9450
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
367. C3#9450semidirect

Maximal subgroups

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

subgroupindex
#793142
#737192
#708302
#708142
C3×#12852
#676212
#94503