Finite Group Databasebeta

← #10690419

Subgroups — #10690419

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#83343C211yesmaximal, characteristic
2#78522C211yesmaximal, characteristic
3C3×#1986C211yesmaximal, characteristic, complemented
4#71824C211yesmaximal, characteristic, complemented
5#69469C211yesmaximal, characteristic
6#67667C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#9055—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#907—21no—
11#5477C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5129—21no—
15#5088C411yescharacteristic, complemented
16#5085C411yescharacteristic, complemented
17#5080C411yescharacteristic, complemented
18#5079C411yescharacteristic, complemented
19#5048—22no—
20C4×#215—22no—
21#4925C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4706—22no—
25#4688—21no—
26C8×D12—21no—
27C3×#339—21no—
28C3×#339—21no—
29#4535—21no—
30#4535—21no—
31#4451—22no—
32#3167—31no—
33#2935—31no—
34#1986D311yescharacteristic, complemented
35#1502—31no—
36#1481—31no—
37#1279—31no—
38#1278—31no—
39C2×#235—21no—
40C2×C3×M16D411yescharacteristic
41C2×C3×M16D411yescharacteristic
42C3×C4⋊Q8C2×C411yescharacteristic
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#215—22no—
54#930—21no—
55#930D412yes—
56C22×C3⋊C8—21no—
57C22×C3⋊C8—21no—
58#907—42no—
59#907—42no—
60C2×ΓL(1,𝔽25)—22no—
61C2×D3×C8—22no—
62#875—21no—
63#874C2×C411yescharacteristic
64C4×D12—21no—
65C4×D12D412yes—
66#869C2×C411yescharacteristic
67C3×#97—21no—
68C3×C4×C8—21no—
69C3×#95—21no—
70#835—22no—
71#830—21no—
72(C2×D3)⋊C8—22no—
73(C2×D3)⋊C8—22no—
74#809—22no—
75#808—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D411yescharacteristic
79C4⋊(C3⋊C8)D411yescharacteristic
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8C2×C411yescharacteristic
83C4×C3⋊C8C2×C411yescharacteristic
84C2×#131—61no—
85#533C2×D311yescharacteristic
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#431—31no—
97C2×C4⋊C8—31no—
98#408—61no—
99C4×M16C2×D311yescharacteristic
100C2×C4×C8—31no—
101C4×C4⋊C4C2×D311yescharacteristic
102#339—21no—
103#339—21no—
104#327—62no—
105Q8⋊C8—61no—
106Q8⋊C8—61no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C2×#67—22no—
114#235—41no—
115#235—41no—
116#235—41no—
117#235—41no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12C22⋊C412yes—
121C2×C4×D3—22no—
122C2×Dic6—21no—
123C3×M16—21no—
124C3×M16—21no—
125C3×M16—21no—
126C3×M16SD1612yescomplemented
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42—41no—
139C3×C42—41no—
140C3×C42—41no—
141C3×C42C22×C411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145#215—44no—
146(C2×D3)⋊C4—22no—
147(C2×D3)⋊C4—22no—
148C4⋊Dic3—21no—
149C4⋊Dic3C22⋊C412yes—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—42no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158ΓL(1,𝔽25)—44no—
159D3×C8—44no—
160C2×#41—61no—
161C2×SD16—62no—
162#131—122no—
163#131—122no—
164C2×M16—62no—
165C2×M16#6711yescharacteristic
166C2×M16#6711yescharacteristic
167C22×C8—62no—
168C22×C8—61no—
169C22×C8—61no—
170C4⋊Q8—31no—
171C4⋊Q8C4×D311yescharacteristic
172C4⋊Q8C4×D311yescharacteristic
173#127—31no—
174#125—21no—
175#124—61no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178#121—32no—
179#121—61no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×D4—32no—
183C4×D4—61no—
184#117—21no—
185C2×C4⋊C4—21no—
186C2×C42—21no—
187C2×C42C22×D311yescharacteristic
188#106—62no—
189C4⋊C8—31no—
190C4⋊C8—31no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#103—62no—
194#102—62no—
195C22⋊C8—62no—
196C22⋊C8—62no—
197C22⋊C8—61no—
198#97—21no—
199C4×C8—21no—
200C4×C8—31no—
201C4×C8—31no—
202C4×C8—61no—
203C4×C8—61no—
204#95—21no—
205C23×C3#12111yescharacteristic
206C22×D3—22no—
207C3×Q8—42no—
208C3×Q8—41no—
209C3×Q8—41no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4C2×C22⋊C411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12311yescharacteristic
224C2×C3×C4C2×SD1611yescharacteristic
225C2×C3×C4#13511yescharacteristic
226C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
227C2×C3×C4#13711yescharacteristic
228#67—42no—
229#67—42no—
230C2×Dic3—22no—
231D12—42no—
232D12—41no—
233D12—41no—
234D12—41no—
235D12—41no—
236C4×D3—42no—
237C4×D3—42no—
238Dic6—41no—
239Dic6—41no—
240C3×C8—22no—
241C3×C8—41no—
242C3⋊C8—22no—
243C3⋊C8—41no—
244C3⋊C8—41no—
245C3⋊C8—41no—
246C3⋊C8—81no—
247#41—121no—
248#41—121no—
249#41—121no—
250#41—121no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—61no—
254C2×D4—32no—
255C2×D4—62no—
256C2×D4—61no—
257C2×D4—61no—
258C22×C4—41no—
259C22×C4—62no—
260C22×C4D3×D411yescharacteristic
261C22×C4#23711yescharacteristic
262C22×C4C2×#6711yescharacteristic
263SD16—124no—
264M16—21no—
265M16—21no—
266M16—21no—
267M16—124no—
268M16#21512yescomplemented
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—61no—
272C2×C8—61no—
273C2×C8—61no—
274C2×C8—61no—
275C2×C8—124no—
276C2×C8—122no—
277C2×C8—121no—
278C2×C8—121no—
279C2×C8—121no—
280C4⋊C4—22no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—32no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—61no—
290C22⋊C4—41no—
291C22⋊C4—62no—
292C22⋊C4—62no—
293C42—41no—
294C42—41no—
295C42—41no—
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×C4—82no—
312C3×C4—81no—
313C3×C4—81no—
314C3×C4C4×SD1611yescharacteristic
315C3×C4#44011yescharacteristic
316C3×C4#44311yescharacteristic
317C3×C4#44411yescharacteristic
318C3×C4#45011yescharacteristic
319C3×C4#45211yescharacteristic
320Dic3—42no—
321C23—62no—
322C23#93811yescharacteristic
323Q8—42no—
324Q8—41no—
325Q8—41no—
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—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—62no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4D3×C22⋊C411yescharacteristic
347C2×C4#67⋊C411yescharacteristic
348C2×C4#88311yescharacteristic
349C2×C4#88411yescharacteristic
350C2×C4C4×#6711yescharacteristic
351C2×C4#93711yescharacteristic
352C2×C4C2×#21511yescharacteristic
353C2×C4#94211yescharacteristic
354C2×C4#94911yescharacteristic
355C2×C4#95011yescharacteristic
356C8—22no—
357C8—41no—
358C8—62no—
359C8—121no—
360C8—121no—
361C8—121no—
362C8—241no—
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#495611yescharacteristic
375C22#505711yescharacteristic
376C22#508311yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—82no—
381C4—81no—
382C4—81no—
383C4—122no—
384C4#482311yescharacteristic
385C4#482411yescharacteristic
386C4C4×#21511yescharacteristic
387C4#502611yescharacteristic
388C4#504611yescharacteristic
389C4#504711yescharacteristic
390C3#905511yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—242no—
394C2#6968711yescharacteristic
395C2#7039011yescharacteristic
396C2#7041711yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1986D3
hypocenterC3#9055
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
3. C3×#1986C2semidirect
4. #71824C2semidirect
6. #67667C2semidirect
7. #67619C2semidirect
15. #5088C4semidirect
16. #5085C4semidirect
17. #5080C4semidirect
18. #5079C4semidirect
34. #1986D3semidirect
126. C3×M16SD16semidirect
268. M16#215semidirect
390. C3#9055semidirect

Maximal subgroups

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

subgroupindex
#833432
#785222
C3×#19862
#718242
#694692
#676672
#676192
#90553