Finite Group Databasebeta

← #10690423

Subgroups — #10690423

Order 768.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78522C211yesmaximal, characteristic
2#78468C211yesmaximal, characteristic
3#70789C211yesmaximal, characteristic
4#70768C211yesmaximal, characteristic, complemented
5#69868C211yesmaximal, characteristic, complemented
6C3×#1278C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#9055—31nomaximal, Sylow
9C2×#907—21no—
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15#5129—21no—
16#5114—22no—
17C4×C4⋊Dic3C2211yescharacteristic
18C2×C4⋊(C3⋊C8)C2211yescharacteristic
19#4706—22no—
20#4694C411yescharacteristic, complemented
21#4693C411yescharacteristic, complemented
22C4×#204—22no—
23#4688—21no—
24C8×D12—21no—
25#4681C411yescharacteristic, complemented
26#4680C411yescharacteristic, complemented
27#4555—21no—
28#4555—21no—
29#4470—22no—
30#4456—21no—
31#4456—21no—
32#3167—31no—
33#2935—31no—
34#1986—31no—
35#1502—31no—
36#1481—31no—
37#1279—31no—
38#1278D311yescharacteristic, complemented
39C2×#235—21no—
40C2×C3×M16D411yescharacteristic
41C2×C3×M16D411yescharacteristic
42C22×C3×C8—21no—
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930D412yes—
46#926—21no—
47C2×C4⋊Dic3—21no—
48#924—21no—
49#924—21no—
50C2×C4×Dic3—21no—
51C22×C3⋊C8—21no—
52#907—42no—
53#907—42no—
54C2×#204—22no—
55C2×ΓL(1,𝔽25)—22no—
56C2×D3×C8—22no—
57#875—21no—
58#874C2×C411yescharacteristic
59C4×D12—21no—
60C4×D12D412yes—
61#870—21no—
62#869C2×C411yescharacteristic
63#869C2×C411yescharacteristic
64#869C2×C411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#97—21no—
70C3×C4×C8—21no—
71C3×C4×C8—21no—
72C3×C4×C8—21no—
73C3×C4×C8C2×C411yescharacteristic
74C3×C4×C8C2×C411yescharacteristic
75C3⋊#95—21no—
76#821—22no—
77(C2×D3)⋊C8—22no—
78(C2×D3)⋊C8—22no—
79#817—22no—
80#816—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D411yescharacteristic
83C4⋊(C3⋊C8)D411yescharacteristic
84C2×#131—61no—
85#533—31no—
86#532—31no—
87#498—31no—
88#492—31no—
89#487—31no—
90#486—31no—
91#465—62no—
92C4×SD16—62no—
93#436—61no—
94#435—62no—
95C8×D4—61no—
96#431C2×D311yescharacteristic
97C2×C4⋊C8—31no—
98#408—61no—
99C4×M16C2×D311yescharacteristic
100C2×C4×C8C2×D311yescharacteristic
101C4×C4⋊C4—31no—
102#339—61no—
103#339—61no—
104#327—62no—
105Q8⋊C8—61no—
106Q8⋊C8—61no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×#67—22no—
111C22×Dic3—41no—
112#235—41no—
113#235—41no—
114#235—41no—
115#235—41no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12C22⋊C412yes—
119C2×C4×D3—22no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C3×M16—21no—
124C3×M16—21no—
125C3×M16—21no—
126C3×M16SD1612yescomplemented
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—42no—
132C2×C3×C8—41no—
133C3×C42C22×C411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137#217—41no—
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—22no—
140C4⋊Dic3—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—41no—
147C4⋊Dic3C22⋊C412yes—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C4×Dic3—41no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157#204—44no—
158ΓL(1,𝔽25)—44no—
159D3×C8—44no—
160C2×#41—61no—
161C2×SD16—62no—
162#131—122no—
163#131—122no—
164C2×M16—62no—
165C2×M16D1211yescharacteristic
166C2×M16D1211yescharacteristic
167C22×C8—21no—
168C22×C8—62no—
169C22×C8—61no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172C4⋊Q8—31no—
173#127—31no—
174#125—61no—
175#124—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178#121—32no—
179#121—61no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×D4—32no—
183C4×D4—61no—
184#117—61no—
185C2×C4⋊C4—61no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—31no—
191C4⋊C8—31no—
192C4⋊C8—61no—
193#103—62no—
194#102—62no—
195C22⋊C8—21no—
196C22⋊C8—62no—
197C22⋊C8—62no—
198#97—21no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8C4×D311yescharacteristic
203C4×C8C4×D311yescharacteristic
204#95—61no—
205C23×C3#12111yescharacteristic
206C22×D3—22no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—41no—
210C2×C3×C4C2×C22⋊C411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12311yescharacteristic
216C2×C3×C4C2×SD1611yescharacteristic
217C2×C3×C4#13511yescharacteristic
218C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
219C2×C3×C4#13711yescharacteristic
220#67—42no—
221#67—42no—
222C2×Dic3—22no—
223C2×Dic3—42no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228D12—42no—
229D12—41no—
230D12—41no—
231D12—41no—
232D12—41no—
233C4×D3—42no—
234C4×D3—42no—
235Dic6—41no—
236Dic6—41no—
237Dic6—42no—
238Dic6—41no—
239Dic6—41no—
240C3×C8—22no—
241C3×C8—22no—
242C3×C8—41no—
243C3×C8—41no—
244C3×C8—81no—
245C3⋊C8—41no—
246C3⋊C8—41no—
247#41—121no—
248#41—121no—
249#41—121no—
250#41—121no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×D4—32no—
255C2×D4—62no—
256C2×D4—61no—
257C2×D4—61no—
258C22×C4—62no—
259C22×C4—121no—
260C22×C4C2×D1211yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4#23911yescharacteristic
263SD16—124no—
264M16—21no—
265M16—21no—
266M16—21no—
267M16—124no—
268M16#20412yescomplemented
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—61no—
276C2×C8—61no—
277C2×C8—124no—
278C2×C8—121no—
279C2×C8—121no—
280C4⋊C4—32no—
281C4⋊C4—62no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C4⋊C4—121no—
290C22⋊C4—62no—
291C22⋊C4—62no—
292C22⋊C4—121no—
293C42—121no—
294C42—121no—
295C42—121no—
296C42C2×C4×D311yescharacteristic
297C42#23511yescharacteristic
298C42D3×D411yescharacteristic
299C42D3×D411yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3#38611yescharacteristic
303C22×C3#46111yescharacteristic
304C22×C3#46311yescharacteristic
305C2×D3—42no—
306C2×D3—42no—
307C2×D3—42no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4C4×SD1611yescharacteristic
312C3×C4#44011yescharacteristic
313C3×C4#44311yescharacteristic
314C3×C4#44411yescharacteristic
315C3×C4#45011yescharacteristic
316C3×C4#45211yescharacteristic
317Dic3—42no—
318Dic3—82no—
319Dic3—81no—
320Dic3—81no—
321C23—62no—
322C23#89511yescharacteristic
323Q8—122no—
324Q8—121no—
325Q8—121no—
326Q8—121no—
327Q8—121no—
328D4—122no—
329D4—122no—
330D4—122no—
331D4—121no—
332D4—121no—
333D4—121no—
334D4—121no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—62no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—241no—
345C2×C4—241no—
346C2×C4C4×D1211yescharacteristic
347C2×C4D3×C22⋊C411yescharacteristic
348C2×C4#88211yescharacteristic
349C2×C4#88411yescharacteristic
350C2×C4#89311yescharacteristic
351C2×C4#89411yescharacteristic
352C2×C4C2×#20411yescharacteristic
353C2×C4#90411yescharacteristic
354C2×C4#90811yescharacteristic
355C2×C4#90911yescharacteristic
356C8—22no—
357C8—22no—
358C8—41no—
359C8—41no—
360C8—81no—
361C8—121no—
362C8—121no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3#187611yescharacteristic
366C2×C3#187711yescharacteristic
367C2×C3#188411yescharacteristic
368D3—82no—
369C22—21no—
370C22—21no—
371C22—122no—
372C22—122no—
373C22—122no—
374C22#466511yescharacteristic
375C22#484111yescharacteristic
376C22#484411yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—122no—
381C4—242no—
382C4—241no—
383C4—241no—
384C4C4×#20411yescharacteristic
385C4#471011yescharacteristic
386C4#473111yescharacteristic
387C4#473311yescharacteristic
388C4#489411yescharacteristic
389C4#489511yescharacteristic
390C3#905511yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—242no—
394C2#6867211yescharacteristic
395C2#6877711yescharacteristic
396C2#6894611yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1278C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1278D3
hypocenterC3#9055
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
4. #70768C2semidirect
5. #69868C2semidirect
6. C3×#1278C2semidirect
7. #67655C2semidirect
20. #4694C4semidirect
21. #4693C4semidirect
25. #4681C4semidirect
26. #4680C4semidirect
38. #1278D3semidirect
126. C3×M16SD16semidirect
268. M16#204semidirect
390. C3#9055semidirect

Maximal subgroups

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

subgroupindex
#785222
#784682
#707892
#707682
#698682
C3×#12782
#676552
#90553