Finite Group Databasebeta

← #10691143

Subgroups — #10691143

Order 768.

500 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2356C211yesmaximal, characteristic, complemented
2#82441C211yesmaximal, characteristic
3#71621C211yesmaximal, characteristic, complemented
4#69545C211yesmaximal, characteristic
5#69469C211yesmaximal, characteristic
6#67631C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9182—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#942—21no—
11#5477C2211yescharacteristic
12C3×#391—21no—
13C3×#387—22no—
14C3×#383—21no—
15C3×#382C2211yescharacteristic
16C3×C4×C4⋊C4—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18C3×#376—21no—
19#5031C412yescomplemented
20#5028—22no—
21#5027C412yescomplemented
22#4966—21no—
23#4925C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4535—22no—
27#4451—21no—
28#4451—21no—
29#4451—22no—
30#3037—31no—
31#2356D311yescharacteristic, complemented
32#1517—31no—
33#1502—31no—
34#1479—31no—
35#1303—31no—
36#1286—31no—
37C22×C3×Q8D411yescharacteristic
38C2×#235—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8C2×C412yes—
42C3×C4×Q8D412yes—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C42—21no—
48C2×C3×C42—22no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#216—22no—
52#942—42no—
53#942—42no—
54C2×#215—22no—
55#930—21no—
56#930—21no—
57C2×#208—21no—
58#875—21no—
59#874C2×C411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869C2×C411yescharacteristic
63C3×#95—22no—
64C3×#95—22no—
65C3×#95—22no—
66C3×#95—21no—
67C3×#95D411yescharacteristic
68C3×#95D411yescharacteristic
69C3×#95D412yes—
70#830—21no—
71#830—21no—
72#810—22no—
73#809—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)C2×C412yes—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C2×#137—61no—
81#532—31no—
82C2×C4×Q8C2×D311yescharacteristic
83#464—62no—
84#462—32no—
85#460—32no—
86#431—31no—
87#431—31no—
88#423—31no—
89#417—61no—
90#391—21no—
91#387—22no—
92#383—21no—
93#382C2×D311yescharacteristic
94C4×C4⋊C4—21no—
95C4×C4⋊C4C2×D311yescharacteristic
96#376—21no—
97#327—61no—
98#327—61no—
99#327—62no—
100Q8⋊C8—62no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8C22⋊C412yes—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C2×#67—41no—
116#235—41no—
117#235—41no—
118#235—41no—
119#235—41no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×C4×D3—41no—
124C2×Dic6—21no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—42no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4C22⋊C412yes—
132C3×C42—22no—
133C3×C42—22no—
134C3×C42—44no—
135C3×C42—41no—
136C3×C42—41no—
137C3×C42C22×C411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140C3×C42#4111yescharacteristic
141#216—44no—
142#215—44no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147#208—41no—
148#208—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—41no—
152C2×#41—61no—
153C22×Q8#6711yescharacteristic
154#137—122no—
155#137—122no—
156C2×Q16—62no—
157C2×SD16—62no—
158C2×M16—61no—
159C4⋊Q8—31no—
160#127—31no—
161#124—61no—
162#121—61no—
163#121—61no—
164C4×Q8—21no—
165C4×Q8—21no—
166C4×Q8C4×D312yes—
167C4×Q8D1212yes—
168C4×D4—61no—
169C4×D4—61no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4#6711yescharacteristic
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—32no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182#103—62no—
183#102—62no—
184C22⋊C8—61no—
185C22⋊C8—61no—
186C4×C8—61no—
187C4×C8—61no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—21no—
192#95#6711yescharacteristic
193#95#6711yescharacteristic
194#95#6712yes—
195C23×C3#12011yescharacteristic
196C22×D3—41no—
197C3×Q8—22no—
198C3×Q8—22no—
199C3×Q8—41no—
200C3×Q8—41no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4C2×C22⋊C411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12311yescharacteristic
223C2×C3×C4#13511yescharacteristic
224C2×C3×C4#13511yescharacteristic
225C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
226C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
227#67—81no—
228#67—81no—
229C2×Dic3—41no—
230D12—41no—
231D12—41no—
232D12—81no—
233D12—81no—
234C4×D3—81no—
235C4×D3—81no—
236Dic6—41no—
237Dic6—41no—
238C3⋊C8—42no—
239C3⋊C8—81no—
240C3⋊C8—81no—
241#41—121no—
242#41—121no—
243#41—121no—
244#41—121no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—61no—
251C2×Q8(C2×D3)⋊C412yes—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—121no—
262C22×C4C2×#6711yescharacteristic
263C22×C4C2×#6711yescharacteristic
264C22×C4C2×#6711yescharacteristic
265Q16—124no—
266SD16—124no—
267M16—121no—
268M16—121no—
269C2×C8—62no—
270C2×C8—121no—
271C2×C8—121no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4(C2×D3)⋊C412yes—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C42—22no—
284C42—22no—
285C42—44no—
286C42—41no—
287C42—41no—
288C42C2×C4×D311yescharacteristic
289C42C2×D1211yescharacteristic
290C42#23511yescharacteristic
291C42#23511yescharacteristic
292C22×C3—21no—
293C22×C3—21no—
294C22×C3#38611yescharacteristic
295C22×C3#44911yescharacteristic
296C22×C3#44911yescharacteristic
297C2×D3—81no—
298C2×D3—81no—
299C2×D3—81no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—42no—
305C3×C4—42no—
306C3×C4—42no—
307C3×C4—81no—
308C3×C4—82no—
309C3×C4#41611yescharacteristic
310C3×C4#41811yescharacteristic
311C3×C4#48211yescharacteristic
312C3×C4#48211yescharacteristic
313Dic3—81no—
314C23—121no—
315C23#95311yescharacteristic
316Q8—22no—
317Q8—22no—
318Q8—41no—
319Q8—41no—
320Q8—121no—
321Q8—121no—
322D4—121no—
323D4—121no—
324D4—241no—
325D4—241no—
326D4—241no—
327D4—241no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—121no—
345C2×C4—241no—
346C2×C4—241no—
347C2×C4C2×(C2×D3)⋊C411yescharacteristic
348C2×C4C4×#6711yescharacteristic
349C2×C4C4×#6711yescharacteristic
350C2×C4#92911yescharacteristic
351C2×C4#93011yescharacteristic
352C2×C4#93011yescharacteristic
353C2×C4#94911yescharacteristic
354C2×C4#94911yescharacteristic
355C2×C4#95011yescharacteristic
356C2×C4#95011yescharacteristic
357C8—122no—
358C8—241no—
359C8—241no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3#187511yescharacteristic
363C2×C3#190511yescharacteristic
364C2×C3#190611yescharacteristic
365D3—161no—
366C22—21no—
367C22—21no—
368C22—241no—
369C22—241no—
370C22—241no—
371C22#521311yescharacteristic
372C22#523811yescharacteristic
373C22#523811yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—81no—
382C4—82no—
383C4—241no—
384C4#500111yescharacteristic
385C4#500211yescharacteristic
386C4#500311yescharacteristic
387C4#500311yescharacteristic
388C3#918211yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—481no—
392C2#7158611yescharacteristic
393C2#7164511yescharacteristic
394C2#7166211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2356C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2356D3
hypocenterC3#9182
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
1. C3×#2356C2semidirect
3. #71621C2semidirect
6. #67631C2semidirect
7. #67620C2semidirect
19. #5031C4semidirect
21. #5027C4semidirect
31. #2356D3semidirect
388. C3#9182semidirect

Maximal subgroups

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

subgroupindex
C3×#23562
#824412
#716212
#695452
#694692
#676312
#676202
#91823