Finite Group Databasebeta

← #10694538

Subgroups — #10694538

Order 768.

428 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78463C211yesmaximal, characteristic
2C3×C8×C4⋊C4C211yesmaximal, characteristic, complemented
3#70429C211yesmaximal, characteristic, complemented
4#70426C211yesmaximal, characteristic
5#69729C211yesmaximal, characteristic, complemented
6#69471C211yesmaximal, characteristic, complemented
7#69469C211yesmaximal, characteristic
8#9751—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×C4×C8—21no—
13C2×C3×C4×C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5113—21no—
16#5113—21no—
17#4925C2211yescharacteristic
18#4925C2211yescharacteristic
19#4697—21no—
20#4697—21no—
21#4694C411yescharacteristic, complemented
22#4694C411yescharacteristic, complemented
23#4681C411yescharacteristic, complemented
24#4681C411yescharacteristic, complemented
25C3×#336—21no—
26#4484—22no—
27#4483—22no—
28#4458—21no—
29#4457—21no—
30#4452—22no—
31#4451—22no—
32#3131—31no—
33C8×C4⋊C4D311yescharacteristic, complemented
34#1564—31no—
35#1503—31no—
36#1502—31no—
37#1481—31no—
38#1480—31no—
39C2×#235—21no—
40C22×C3×C8—21no—
41C22×C3×C8D411yescharacteristic
42C22×C3×C8D411yescharacteristic
43C2×C3×C4⋊C4—21no—
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46#930—21no—
47#930—21no—
48#926—21no—
49#924—21no—
50#924—21no—
51#875—21no—
52#874C2×C411yescharacteristic
53C4×D12—21no—
54C4×D12—21no—
55#870—21no—
56#869C2×C411yescharacteristic
57#869C2×C411yescharacteristic
58#869C2×C411yescharacteristic
59C4×Dic6—21no—
60C4×Dic6—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8D412yes—
63C3×C4×C8—42no—
64C3×C4×C8C2×C411yescharacteristic
65C3×C4×C8C2×C411yescharacteristic
66C3×C4×C8D411yescharacteristic
67C3×C4×C8D411yescharacteristic
68C3×#95D412yes—
69#830—21no—
70#830—21no—
71#821—21no—
72#821—21no—
73#821—21no—
74#821—21no—
75#816—21no—
76#816—21no—
77#816—21no—
78#816—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84#532—31no—
85#488—61no—
86#488—61no—
87#487—31no—
88#487—31no—
89#486—31no—
90#486—31no—
91#431—31no—
92#431—31no—
93C2×C4⋊C8C2×D311yescharacteristic
94#416—61no—
95#416—61no—
96C2×C4×C8—21no—
97C2×C4×C8C2×D311yescharacteristic
98C4×C4⋊C4C2×D311yescharacteristic
99#336—21no—
100#335—61no—
101#334—61no—
102#328—62no—
103#327—62no—
104Q8⋊C8—62no—
105D4⋊C8—62no—
106C22×C3×C4—22no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×#67—41no—
111#235—41no—
112#235—41no—
113#235—41no—
114#235—41no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×C3×C8—21no—
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8—21no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C2×C3×C8C22⋊C411yescharacteristic
130C2×C3×C8C22⋊C411yescharacteristic
131C2×C3×C8C22⋊C411yescharacteristic
132C2×C3×C8C22⋊C411yescharacteristic
133C3×C4⋊C4—24no—
134C3×C42—41no—
135C3×C42—41no—
136C3×C42C22×C411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#217—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151C4×Dic3—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×#41—61no—
155C22×C8—21no—
156C22×C8#6711yescharacteristic
157C22×C8#6711yescharacteristic
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161#127—31no—
162#125—61no—
163#124—61no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166#121—61no—
167#121—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170C4×D4—61no—
171C4×D4—61no—
172#117—61no—
173C2×C4⋊C4—21no—
174C2×C42—21no—
175C2×C42C22×D311yescharacteristic
176C4⋊C8—21no—
177C4⋊C8—61no—
178C4⋊C8—61no—
179C4⋊C8D1212yes—
180#103—61no—
181#103—61no—
182#103—61no—
183#103—61no—
184#102—61no—
185#102—61no—
186#102—61no—
187#102—61no—
188C22⋊C8—61no—
189C22⋊C8—61no—
190C4×C8—42no—
191C4×C8—61no—
192C4×C8—61no—
193C4×C8C4×D311yescharacteristic
194C4×C8C4×D311yescharacteristic
195C4×C8D1211yescharacteristic
196C4×C8D1211yescharacteristic
197#95D1212yes—
198C23×C3#12111yescharacteristic
199C22×D3—41no—
200C2×C3×C4—22no—
201C2×C3×C4—24no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4C2×C22⋊C411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12411yescharacteristic
210C2×C3×C4#12711yescharacteristic
211C2×C3×C4#13511yescharacteristic
212C2×C3×C4#13511yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214C2×C3×C4#13711yescharacteristic
215#67—81no—
216#67—81no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220D12—41no—
221D12—41no—
222D12—42no—
223D12—81no—
224C4×D3—81no—
225C4×D3—81no—
226Dic6—41no—
227Dic6—41no—
228Dic6—42no—
229Dic6—81no—
230C3×C8—21no—
231C3×C8—21no—
232C3×C8—21no—
233C3×C8—21no—
234C3×C8—42no—
235C3⋊C8—81no—
236C3⋊C8—81no—
237#41—121no—
238#41—121no—
239#41—121no—
240#41—121no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—121no—
248C22×C4—22no—
249C22×C4—121no—
250C22×C4C2×D1211yescharacteristic
251C22×C4#23511yescharacteristic
252C22×C4C2×#6711yescharacteristic
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—121no—
261C2×C8—121no—
262C2×C8(C2×D3)⋊C411yescharacteristic
263C2×C8(C2×D3)⋊C411yescharacteristic
264C2×C8(C2×D3)⋊C411yescharacteristic
265C2×C8(C2×D3)⋊C411yescharacteristic
266C4⋊C4—24no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C42—41no—
279C42—41no—
280C42—121no—
281C42C2×C4×D311yescharacteristic
282C42C2×D1211yescharacteristic
283C42C2×D1211yescharacteristic
284C42#23511yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3#39011yescharacteristic
288C22×C3#46311yescharacteristic
289C22×C3#46411yescharacteristic
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—44no—
297C3×C4#41611yescharacteristic
298C3×C4#41711yescharacteristic
299C3×C4#44312yes—
300C3×C4#46311yescharacteristic
301C3×C4#46411yescharacteristic
302Dic3—81no—
303Dic3—81no—
304Dic3—81no—
305C23—121no—
306C23#93011yescharacteristic
307Q8—122no—
308Q8—121no—
309Q8—121no—
310Q8—241no—
311D4—122no—
312D4—121no—
313D4—121no—
314D4—241no—
315D4—241no—
316D4—241no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—241no—
326C2×C4—241no—
327C2×C4C4×D1211yescharacteristic
328C2×C4C4×D1211yescharacteristic
329C2×C4#87411yescharacteristic
330C2×C4#87511yescharacteristic
331C2×C4C2×(C2×D3)⋊C411yescharacteristic
332C2×C4#93011yescharacteristic
333C2×C4#93211yescharacteristic
334C2×C4#94211yescharacteristic
335C2×C4#95011yescharacteristic
336C2×C4#95011yescharacteristic
337C8—21no—
338C8—21no—
339C8—21no—
340C8—21no—
341C8—42no—
342C8—241no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3#197411yescharacteristic
347C2×C3#197611yescharacteristic
348C2×C3#198311yescharacteristic
349D3—161no—
350C22—21no—
351C22—21no—
352C22—241no—
353C22—241no—
354C22—241no—
355C22#493911yescharacteristic
356C22#501611yescharacteristic
357C22#502811yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—44no—
362C4—241no—
363C4—241no—
364C4—241no—
365C4#470012yes—
366C4#496611yescharacteristic
367C4#500211yescharacteristic
368C4#501611yescharacteristic
369C4#502811yescharacteristic
370C3#975111yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—481no—
374C2#6948811yescharacteristic
375C2#6979511yescharacteristic
376C2#6989711yescharacteristic
subgroupG/H
centerC2×C4#930
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×C8×C4⋊C4C2
socleC22×C3#390
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenterC8×C4⋊C4D3
hypocenterC3#9751
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C8×C4⋊C4C2semidirect
3. #70429C2semidirect
5. #69729C2semidirect
6. #69471C2semidirect
21. #4694C4semidirect
22. #4694C4semidirect
23. #4681C4semidirect
24. #4681C4semidirect
33. C8×C4⋊C4D3semidirect
370. C3#9751semidirect

Maximal subgroups

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

subgroupindex
#784632
C3×C8×C4⋊C42
#704292
#704262
#697292
#694712
#694692
#97513