Finite Group Databasebeta

← #10691703

Subgroups — #10691703

Order 768.

480 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79269C211yesmaximal, characteristic
2#78522C211yesmaximal, characteristic
3C2×#4482C211yesmaximal, characteristic, complemented
4#70432C211yesmaximal, characteristic
5#69879C211yesmaximal, characteristic, complemented
6C3×#1283C211yesmaximal, characteristic, complemented
7#67745C211yesmaximal, characteristic, complemented
8#9269—31nomaximal, Sylow
9C2×#907—21no—
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5129—21no—
16#4924C2211yescharacteristic
17C2×C4⋊(C3⋊C8)C2211yescharacteristic
18#4842—22no—
19#4735C414yes—
20#4706—22no—
21#4694—21no—
22#4688—21no—
23C8×D12—21no—
24#4680—21no—
25#4554—21no—
26#4554—21no—
27#4486—21no—
28#4485—21no—
29#4482—21no—
30#4482—21no—
31#3227—31no—
32#2935—31no—
33#1997—31no—
34#1547—31no—
35C2×#332—31no—
36#1301—31no—
37#1283D311yescharacteristic, complemented
38C2×#235—21no—
39C22×Dic6—21no—
40C2×C3×M16D411yescharacteristic
41C2×C3×M16D411yescharacteristic
42C22×C3×C8—21no—
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930C2×C412yes—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3C2×C412yes—
48C2×#208—21no—
49C22×C3⋊C8—21no—
50#907—42no—
51#907—42no—
52C2×ΓL(1,𝔽25)—22no—
53C2×D3×C8—22no—
54#875—21no—
55#874D411yescharacteristic
56C4×D12—21no—
57C4×D12D412yes—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C22⋊C8C2×C412yes—
67C3×#97—21no—
68C3×C4×C8—21no—
69C3×C4×C8—21no—
70#821—24no—
71(C2×D3)⋊C8—22no—
72(C2×D3)⋊C8—22no—
73#817—24no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D411yescharacteristic
78C4⋊(C3⋊C8)D411yescharacteristic
79C2×#131—61no—
80#532—31no—
81C2×C4⋊Q8—31no—
82#498—61no—
83#486—61no—
84#481—34no—
85#476—62no—
86#436—61no—
87#435—62no—
88C8×D4—61no—
89#431C2×D311yescharacteristic
90#423—31no—
91C2×C4⋊C8—31no—
92C2×C4⋊C8C2×D311yescharacteristic
93#408—61no—
94C4×M16C2×D311yescharacteristic
95#343—61no—
96#343—61no—
97#332—61no—
98#332—61no—
99#332—61no—
100#331—61no—
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C2×#67—22no—
105C22×Dic3—22no—
106#235—41no—
107#235—41no—
108#235—41no—
109#235—41no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12C22⋊C412yes—
113C2×C4×D3—22no—
114C2×Dic6—21no—
115C2×Dic6—41no—
116C2×Dic6—41no—
117C2×Dic6—81no—
118C2×Dic6—81no—
119C3×M16—21no—
120C3×M16—21no—
121C3×M16—21no—
122C3×M16SD1612yes—
123C2×C3×C8—21no—
124C2×C3×C8—22no—
125C2×C3×C8—21no—
126C2×C3×C8—41no—
127C2×C3×C8—81no—
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134C4⋊Dic3—21no—
135C4⋊Dic3—22no—
136C4⋊Dic3—22no—
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139C4⋊Dic3C22⋊C412yes—
140#208—41no—
141#208—41no—
142C2×C3⋊C8—21no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—41no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8—41no—
147ΓL(1,𝔽25)—44no—
148D3×C8—44no—
149C2×#41—61no—
150C22×Q8—61no—
151#131—122no—
152#131—122no—
153C2×M16—62no—
154C2×M16—61no—
155C2×M16D1211yescharacteristic
156C2×M16D1211yescharacteristic
157C22×C8—21no—
158C22×C8—62no—
159C22×C8—61no—
160C4⋊Q8—31no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165#127—31no—
166#124—61no—
167#121—32no—
168#121—61no—
169C4×D4—32no—
170C4×D4—61no—
171C2×C4⋊C4—32no—
172C2×C4⋊C4—61no—
173C2×C42C22×D311yescharacteristic
174#106—64no—
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—31no—
179C4⋊C8—31no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183#102—64no—
184C22⋊C8—62no—
185C22⋊C8—62no—
186C22⋊C8C4×D312yes—
187#97—21no—
188C4×C8—21no—
189C4×C8—21no—
190C23×C3C4×D411yescharacteristic
191C22×D3—22no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—41no—
195C2×C3×C4C2×C22⋊C411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12311yescharacteristic
201C2×C3×C4C2×SD1611yescharacteristic
202C2×C3×C4#13511yescharacteristic
203C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
204C2×C3×C4#13711yescharacteristic
205#67—42no—
206#67—42no—
207C2×Dic3—22no—
208C2×Dic3—42no—
209C2×Dic3—82no—
210D12—42no—
211D12—41no—
212D12—41no—
213D12—41no—
214D12—41no—
215C4×D3—42no—
216C4×D3—42no—
217Dic6—41no—
218Dic6—41no—
219Dic6—82no—
220Dic6—81no—
221Dic6—81no—
222C3×C8—22no—
223C3×C8—42no—
224C3×C8—41no—
225C3×C8—81no—
226C3⋊C8—41no—
227C3⋊C8—41no—
228C3⋊C8—81no—
229#41—121no—
230#41—121no—
231#41—121no—
232#41—121no—
233C2×Q8—61no—
234C2×Q8—121no—
235C2×Q8—121no—
236C2×Q8—241no—
237C2×Q8—241no—
238C2×D4—32no—
239C2×D4—62no—
240C2×D4—61no—
241C2×D4—61no—
242C22×C4—62no—
243C22×C4—62no—
244C22×C4C2×C4×D311yescharacteristic
245C22×C4C2×D1211yescharacteristic
246C22×C4#23511yescharacteristic
247M16—21no—
248M16—21no—
249M16—21no—
250M16—124no—
251M16—121no—
252M16—121no—
253M16#20412yes—
254C2×C8—21no—
255C2×C8—22no—
256C2×C8—21no—
257C2×C8—41no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—81no—
261C2×C8—124no—
262C2×C8—121no—
263C2×C8—121no—
264C2×C8—121no—
265C4⋊C4—32no—
266C4⋊C4—61no—
267C4⋊C4—62no—
268C4⋊C4—62no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C22⋊C4—62no—
272C22⋊C4—62no—
273C42D3×D411yescharacteristic
274C42D3×D411yescharacteristic
275C42D3×D411yescharacteristic
276C42#23911yescharacteristic
277C22×C3—21no—
278C22×C3—21no—
279C22×C3#38611yescharacteristic
280C22×C3C4×SD1611yescharacteristic
281C22×C3#44011yescharacteristic
282C2×D3—42no—
283C2×D3—42no—
284C2×D3—42no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4#45011yescharacteristic
289C3×C4#45211yescharacteristic
290C3×C4#46111yescharacteristic
291C3×C4#46311yescharacteristic
292C3×C4#47111yescharacteristic
293C3×C4#47211yescharacteristic
294Dic3—42no—
295Dic3—82no—
296C23—62no—
297C23C4×D1211yescharacteristic
298Q8—121no—
299Q8—121no—
300Q8—242no—
301Q8—241no—
302Q8—241no—
303D4—122no—
304D4—122no—
305D4—122no—
306D4—121no—
307D4—121no—
308D4—121no—
309D4—121no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—62no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4—242no—
318C2×C4D3×C22⋊C411yescharacteristic
319C2×C4#88211yescharacteristic
320C2×C4#88411yescharacteristic
321C2×C4#89311yescharacteristic
322C2×C4#89411yescharacteristic
323C2×C4#89511yescharacteristic
324C2×C4C2×#20411yescharacteristic
325C2×C4#90411yescharacteristic
326C2×C4#90811yescharacteristic
327C2×C4#90911yescharacteristic
328C8—22no—
329C8—42no—
330C8—41no—
331C8—81no—
332C8—121no—
333C8—121no—
334C8—241no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3#187711yescharacteristic
338C2×C3#190711yescharacteristic
339C2×C3#191511yescharacteristic
340D3—82no—
341C22—21no—
342C22—21no—
343C22—122no—
344C22—122no—
345C22—122no—
346C22#466511yescharacteristic
347C22C4×#20411yescharacteristic
348C22#471011yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—122no—
353C4—242no—
354C4#473111yescharacteristic
355C4#473311yescharacteristic
356C4#474911yescharacteristic
357C4#475411yescharacteristic
358C4#484111yescharacteristic
359C4#484411yescharacteristic
360C3#926911yescharacteristic, complemented, Sylow
361C2—21no—
362C2—21no—
363C2—242no—
364C2#6867211yescharacteristic
365C2#6900611yescharacteristic
366C2#6904111yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1283D3
hypocenterC3#9269
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
3. C2×#4482C2semidirect
5. #69879C2semidirect
6. C3×#1283C2semidirect
7. #67745C2semidirect
37. #1283D3semidirect
360. C3#9269semidirect

Maximal subgroups

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

subgroupindex
#792692
#785222
C2×#44822
#704322
#698792
C3×#12832
#677452
#92693