Finite Group Databasebeta

← #10694589

Subgroups — #10694589

Order 768.

440 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 97 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#83343C211yesmaximal, characteristic
2C3×#1531C211yesmaximal, characteristic, complemented
3#71836C211yesmaximal, characteristic, complemented
4#70832C211yesmaximal, characteristic
5#70794C211yesmaximal, characteristic, complemented
6#69804C211yesmaximal, characteristic, complemented
7#69798C211yesmaximal, characteristic
8#9761—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×#815—21no—
14C2×C8×Dic3—21no—
15#5088C411yescharacteristic, complemented
16#5085C411yescharacteristic, complemented
17#5080C411yescharacteristic, complemented
18#5079C411yescharacteristic, complemented
19#5048—21no—
20#5048—21no—
21C4×#215—21no—
22C4×#215—21no—
23#4932C2211yescharacteristic
24#4926C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26C3×#330—22no—
27#4538—22no—
28#4480—21no—
29#4479—21no—
30#4470—22no—
31#4464—22no—
32#3167—31no—
33#1726—31no—
34#1568—31no—
35#1531D311yescharacteristic, complemented
36#1530—31no—
37#1506—31no—
38#1505—31no—
39C2×#235—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×C4⋊Q8C2×C411yescharacteristic
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#215—21no—
51C2×#215—21no—
52#930—21no—
53#930—21no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8D411yescharacteristic
56C22×C3⋊C8D411yescharacteristic
57#875—21no—
58#874C2×C411yescharacteristic
59C4×D12—21no—
60C4×D12—21no—
61#869C2×C411yescharacteristic
62C3×C4⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#97—21no—
65C3×#97—21no—
66C3×#97D412yes—
67#835—21no—
68#835—21no—
69C3⋊#95—21no—
70C3⋊#95D412yes—
71#830—21no—
72#815—24no—
73C8×Dic3—24no—
74#809—21no—
75#809—21no—
76#808—21no—
77#808—21no—
78C4⋊(C3⋊C8)—21no—
79#803—21no—
80C4×C3⋊C8C2×C411yescharacteristic
81C4×C3⋊C8C2×C411yescharacteristic
82C4×C3⋊C8D411yescharacteristic
83C4×C3⋊C8D411yescharacteristic
84#533C2×D311yescharacteristic
85#532—31no—
86#498—31no—
87#492—31no—
88#487—31no—
89#486—31no—
90#465—61no—
91#465—61no—
92C4×SD16—61no—
93C4×SD16—61no—
94#432—31no—
95#432C2×D311yescharacteristic
96C2×#97—61no—
97C2×#97C2×D311yescharacteristic
98C2×C4×C8—31no—
99C2×C4×C8—61no—
100#376—31no—
101#335—61no—
102#334—61no—
103#330—22no—
104#329—62no—
105#328—62no—
106#327—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—41no—
113C22×Dic3—22no—
114#235—82no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—21no—
120C2×C3×C8—22no—
121C2×C3×C8—22no—
122C2×C3×C8—22no—
123C2×C3×C8—41no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C42—41no—
132C3×C42C22×C411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#215—42no—
137#215—42no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4×Dic3—22no—
143C4×Dic3—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—41no—
147C2×C3⋊C8C22⋊C411yescharacteristic
148C2×C3⋊C8C22⋊C411yescharacteristic
149C2×C3⋊C8C22⋊C411yescharacteristic
150C2×C3⋊C8C22⋊C411yescharacteristic
151C2×#41—61no—
152C2×SD16—61no—
153C2×SD16—61no—
154C22×C8—21no—
155C22×C8—31no—
156C22×C8—31no—
157C4⋊Q8—31no—
158C4⋊Q8C4×D311yescharacteristic
159C4⋊Q8C4×D311yescharacteristic
160#127—31no—
161#125—21no—
162#124—61no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165#121—61no—
166#121—61no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×D4—61no—
170C4×D4—61no—
171#117—21no—
172C2×C42—61no—
173C2×C42C22×D311yescharacteristic
174#106—61no—
175#106—61no—
176C4⋊C8—21no—
177C4⋊C8—61no—
178#103—61no—
179#103—61no—
180#102—61no—
181#102—61no—
182C22⋊C8—21no—
183C22⋊C8—61no—
184#97—21no—
185#97—21no—
186#97—64no—
187#97—61no—
188#97#6712yes—
189C4×C8—31no—
190C4×C8—31no—
191C4×C8—31no—
192C4×C8—31no—
193C4×C8—64no—
194#95—32no—
195#95—61no—
196C23×C3#12111yescharacteristic
197C22×D3—41no—
198C3×Q8—41no—
199C3×Q8—41no—
200C3×Q8—81no—
201C2×C3×C4—22no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4C2×C22⋊C411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12411yescharacteristic
212C2×C3×C4#12711yescharacteristic
213#67—82no—
214C2×Dic3—24no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217C2×Dic3—41no—
218D12—41no—
219D12—41no—
220D12—81no—
221D12—81no—
222C4×D3—82no—
223Dic6—81no—
224C3×C8—24no—
225C3×C8—81no—
226C3⋊C8—21no—
227C3⋊C8—21no—
228C3⋊C8—81no—
229C3⋊C8#10412yes—
230C3⋊C8#10412yes—
231#41—242no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—121no—
239C22×C4—62no—
240C22×C4—121no—
241C22×C4#23511yescharacteristic
242C22×C4D3×D411yescharacteristic
243C22×C4D3×D411yescharacteristic
244SD16—122no—
245SD16—122no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8—41no—
254C2×C8—62no—
255C2×C8—62no—
256C2×C8—121no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C22⋊C4—41no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C42—41no—
269C42—62no—
270C42—62no—
271C42C2×C4×D311yescharacteristic
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C42C2×#6711yescharacteristic
275C22×C3—21no—
276C22×C3—21no—
277C22×C3#39011yescharacteristic
278C22×C3#47112yes—
279C2×D3—81no—
280C2×D3—81no—
281C2×D3—81no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4C2×#10411yescharacteristic
290C3×C4C2×#10411yescharacteristic
291Dic3—44no—
292Dic3—81no—
293C23—121no—
294C23#88411yescharacteristic
295Q8—41no—
296Q8—41no—
297Q8—81no—
298Q8—241no—
299D4—121no—
300D4—121no—
301D4—241no—
302D4—241no—
303D4—242no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—64no—
311C2×C4—122no—
312C2×C4—122no—
313C2×C4—121no—
314C2×C4—242no—
315C2×C4D3×C22⋊C411yescharacteristic
316C2×C4#67⋊C411yescharacteristic
317C2×C4#88511yescharacteristic
318C2×C4C4×#6711yescharacteristic
319C2×C4#93911yescharacteristic
320C2×C4#94011yescharacteristic
321C8—24no—
322C8—32no—
323C8—32no—
324C8—61no—
325C8—61no—
326C8—81no—
327C8—241no—
328C2×C3—22no—
329C2×C3#196211yescharacteristic
330C2×C3#197611yescharacteristic
331C2×C3#197711yescharacteristic
332D3—161no—
333C22—21no—
334C22—21no—
335C22—241no—
336C22—241no—
337C22—241no—
338C22#475412yes—
339C22#495711yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—81no—
347C4—124no—
348C4—241no—
349C4D3×#10411yescharacteristic
350C4D3×#10411yescharacteristic
351C3#976111yescharacteristic, complemented, Sylow
352C2—22no—
353C2—481no—
354C2#7010711yescharacteristic
355C2#7035611yescharacteristic
356C2#7039711yescharacteristic
subgroupG/H
centerC22#4957
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1531C2
socleC22×C3#390
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1531D3
hypocenterC3#9761
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1531

Complements

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

NH ≅ G/Ntype
2. C3×#1531C2semidirect
3. #71836C2semidirect
5. #70794C2semidirect
6. #69804C2semidirect
15. #5088C4semidirect
16. #5085C4semidirect
17. #5080C4semidirect
18. #5079C4semidirect
35. #1531D3semidirect
351. C3#9761semidirect

Maximal subgroups

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

subgroupindex
#833432
C3×#15312
#718362
#708322
#707942
#698042
#697982
#97613