Finite Group Databasebeta

← #10694360

Subgroups — #10694360

Order 768.

460 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 111 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×#1503C211yesmaximal, characteristic, complemented
3#71841C211yesmaximal, characteristic, complemented
4#70789C211yesmaximal, characteristic
5#70741C211yesmaximal, characteristic, complemented
6#69492C211yesmaximal, characteristic
7C4×#807C211yesmaximal, characteristic, complemented
8#9722—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5088C411yescharacteristic, complemented
14#5085C411yescharacteristic, complemented
15#5080C411yescharacteristic, complemented
16#5079C411yescharacteristic, complemented
17#5048—21no—
18#5048—21no—
19C4×#215—21no—
20C4×#215—21no—
21C2×#807—21no—
22C4×C4⋊Dic3C2211yescharacteristic
23C2×#805—22no—
24C4×#208C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26C3×#328—22no—
27#4532—22no—
28#4529—21no—
29#4470—22no—
30#4455—21no—
31#4454—21no—
32#3167—31no—
33#1986—31no—
34#1980—31no—
35C4×#107—31no—
36#1503D311yescharacteristic, complemented
37#1502—31no—
38#1451—31no—
39C2×#235—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#215—21no—
52C2×#215—21no—
53#930—21no—
54#930—21no—
55C2×C4⋊Dic3—21no—
56C2×C4×Dic3—21no—
57C2×#208D412yes—
58C22×C3⋊C8D411yescharacteristic
59C22×C3⋊C8D411yescharacteristic
60#875—21no—
61#874C2×C411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869C2×C411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C22⋊C8D412yes—
67C3×C4×C8—21no—
68C3×#95—21no—
69#835—21no—
70#835—21no—
71C3⋊#95—21no—
72#809—21no—
73#809—21no—
74#808—21no—
75#808—21no—
76#807—24no—
77#805—44no—
78#803—21no—
79C4×C3⋊C8—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#431C2×D311yescharacteristic
95C2×#107—61no—
96C2×#104—62no—
97C4×M16—31no—
98C2×C4×C8—31no—
99C4×C4⋊C4—31no—
100C4×C4⋊C4C2×D311yescharacteristic
101#340—61no—
102#339—62no—
103#328—22no—
104#327—62no—
105#322—61no—
106#322—61no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C2×#67—41no—
114C22×Dic3—41no—
115#235—42no—
116#235—42no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×C4×D3—41no—
121C2×Dic6—21no—
122C2×C3×C8—22no—
123C3×C4⋊C4—22no—
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×C4⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42—41no—
133C3×C42—42no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C3×C42#4111yescharacteristic
138#215—42no—
139#215—42no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—41no—
146C4×Dic3—42no—
147#208—22no—
148#208—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8C22⋊C411yescharacteristic
153C2×C3⋊C8C22⋊C411yescharacteristic
154C2×C3⋊C8C22⋊C411yescharacteristic
155C2×C3⋊C8C22⋊C411yescharacteristic
156C2×#41—61no—
157C2×SD16—61no—
158C2×SD16—61no—
159C2×M16—32no—
160C22×C8—31no—
161C22×C8—31no—
162C4⋊Q8—31no—
163C4⋊Q8C4×D311yescharacteristic
164C4⋊Q8C4×D311yescharacteristic
165#127—31no—
166#125—21no—
167#124—61no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170#121—61no—
171#121—61no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×D4—61no—
175C4×D4—61no—
176#117—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—61no—
179C2×C42—21no—
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182#107—64no—
183#106—61no—
184#106—61no—
185C4⋊C8—21no—
186#104—124no—
187#103—61no—
188#103—61no—
189#102—61no—
190#102—61no—
191C22⋊C8#6712yes—
192#97—61no—
193C4×C8—21no—
194C4×C8—31no—
195C4×C8—31no—
196C4×C8—31no—
197C4×C8—31no—
198C4×C8—61no—
199#95—21no—
200#95—61no—
201C23×C3#12711yescharacteristic
202C22×D3—41no—
203C3×Q8—41no—
204C3×Q8—41no—
205C3×Q8—81no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4C2×C22⋊C411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12411yescharacteristic
220C2×C3×C4AGL(1,ℤ/8)12yes—
221C2×C3×C4#13712yes—
222#67—81no—
223#67—81no—
224C2×Dic3—42no—
225C2×Dic3—41no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228D12—41no—
229D12—41no—
230D12—42no—
231D12—81no—
232C4×D3—82no—
233Dic6—42no—
234C3×C8—42no—
235C3⋊C8—21no—
236C3⋊C8—21no—
237C3⋊C8—21no—
238C3⋊C8—42no—
239C3⋊C8#10412yescomplemented
240#41—122no—
241#41—122no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—121no—
249C22×C4—41no—
250C22×C4—121no—
251C22×C4—121no—
252C22×C4D3×D411yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4C2×#6711yescharacteristic
255SD16—122no—
256SD16—122no—
257M16—62no—
258M16—62no—
259C2×C8—22no—
260C2×C8—31no—
261C2×C8—31no—
262C2×C8—31no—
263C2×C8—31no—
264C2×C8—62no—
265C2×C8—62no—
266C2×C8—121no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—62no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—121no—
279C22⋊C4—41no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C42—41no—
283C42—42no—
284C42—122no—
285C42C2×C4×D311yescharacteristic
286C42#23511yescharacteristic
287C42D3×D411yescharacteristic
288C42#23711yescharacteristic
289C22×C3—21no—
290C22×C3—21no—
291C22×C3#39011yescharacteristic
292C22×C3#49611yescharacteristic
293C22×C3#49711yescharacteristic
294C2×D3—81no—
295C2×D3—81no—
296C2×D3—81no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—82no—
301C3×C4—81no—
302C3×C4—81no—
303C3×C4C2×#10411yescharacteristic
304C3×C4#42111yescharacteristic
305C3×C4#44012yes—
306C3×C4#46812yes—
307Dic3—82no—
308Dic3—81no—
309C23—121no—
310C23#94011yescharacteristic
311Q8—41no—
312Q8—41no—
313Q8—81no—
314Q8—122no—
315D4—122no—
316D4—121no—
317D4—121no—
318D4—241no—
319D4—241no—
320D4—241no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—122no—
330C2×C4—121no—
331C2×C4—241no—
332C2×C4—241no—
333C2×C4—242no—
334C2×C4D3×C22⋊C411yescharacteristic
335C2×C4#67⋊C411yescharacteristic
336C2×C4#88411yescharacteristic
337C2×C4#88511yescharacteristic
338C2×C4#91412yes—
339C2×C4#91812yes—
340C2×C4C4×#6711yescharacteristic
341C2×C4#93911yescharacteristic
342C8—32no—
343C8—42no—
344C8—61no—
345C8—61no—
346C8—61no—
347C8—122no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3#197311yescharacteristic
351C2×C3#197611yescharacteristic
352C2×C3#198011yescharacteristic
353D3—161no—
354C22—21no—
355C22—21no—
356C22—241no—
357C22—241no—
358C22—241no—
359C22#495711yescharacteristic
360C22#517611yescharacteristic
361C22#519011yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—81no—
367C4—81no—
368C4—242no—
369C4—241no—
370C4#481212yes—
371C4#481712yes—
372C4D3×#10411yescharacteristic
373C4#482111yescharacteristic
374C3#972211yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—481no—
378C2#7004211yescharacteristic
379C2#7026011yescharacteristic
380C2#7035811yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#390
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1503D3
hypocenterC3#9722
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. C3×#1503C2semidirect
3. #71841C2semidirect
5. #70741C2semidirect
7. C4×#807C2semidirect
13. #5088C4semidirect
14. #5085C4semidirect
15. #5080C4semidirect
16. #5079C4semidirect
36. #1503D3semidirect
239. C3⋊C8#104semidirect
374. C3#9722semidirect

Maximal subgroups

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

subgroupindex
#833432
C3×#15032
#718412
#707892
#707412
#694922
C4×#8072
#97223