Finite Group Databasebeta

← #10765396

Subgroups — #10765396

Order 768.

666 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79298C211yesmaximal, characteristic
2#79044C211yesmaximal, characteristic, complemented
3C2×#4838C211yesmaximal, characteristic
4C3×#1570C211yesmaximal, characteristic, complemented
5#70740C211yesmaximal, characteristic, complemented
6#70496C211yesmaximal, characteristic
7#70429C211yesmaximal, characteristic, complemented
8#68523C214yesmaximal, complemented
9#68520C214yesmaximal, complemented
10#17930—31nomaximal, Sylow
11C22×ΓL(1,𝔽25)—21no—
12C2×C4×D12C2211yescharacteristic
13#5470C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×#423C2211yescharacteristic
16C2×C3×C4⋊C8C2211yescharacteristic
17#5122—21no—
18#5119—21no—
19C2×(C2×D3)⋊C8—21no—
20#4925C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22#4842C412yes—
23#4841C412yes—
24#4838—21no—
25#4838—21no—
26#4838C2212yes—
27#4838C2212yes—
28#4831—21no—
29#4831C2212yescomplemented
30#4830—21no—
31#4830C2212yescomplemented
32#4731C412yes—
33#4721C412yes—
34C4×#204C2214yes—
35C3×#334C2214yescomplemented
36#4484C2214yescomplemented
37#4483C2214yes—
38#4454C2212yescomplemented
39C8×C3⋊C8C2212yescomplemented
40#3101—31no—
41#2960—31no—
42C2×#436—31no—
43#1590—34no—
44#1583—34no—
45#1570D311yescharacteristic, complemented
46#1486—31no—
47#1481—31no—
48#1450—31no—
49C22×D12C2×C411yescharacteristic
50C22×C4×D3—21no—
51C2×C3×M16—21no—
52C22×C3×C8—21no—
53C2×C3×C42C2311yescharacteristic
54C2×(C2×D3)⋊C4—21no—
55#926—21no—
56C2×C4⋊Dic3C2×C411yescharacteristic
57#924—21no—
58#924C2×C412yes—
59C22×C3⋊C8D411yescharacteristic
60C22×C3⋊C8D411yescharacteristic
61C2×#204C2×C414yes—
62C2×ΓL(1,𝔽25)—22no—
63C2×ΓL(1,𝔽25)—22no—
64C2×ΓL(1,𝔽25)—44no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12C2×C412yes—
68C4×D12C2312yes—
69#870—21no—
70#869C2×C411yescharacteristic
71#869C2×C411yescharacteristic
72C4×Dic6—21no—
73C4×Dic6C2312yes—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8C2×C412yes—
77C3×C4⋊C8C2312yes—
78C3×C4⋊C8C2312yes—
79C3×C22⋊C8C2×C412yes—
80C3×C4×C8C2312yes—
81#830—21no—
82#821C2×C414yes—
83(C2×D3)⋊C8—22no—
84(C2×D3)⋊C8—22no—
85#817C2×C414yes—
86#816C2×C414yes—
87#815—22no—
88Dic3⋊C8—22no—
89Dic3⋊C8—22no—
90C8×Dic3—22no—
91C4⋊(C3⋊C8)—21no—
92C4×C3⋊C8D411yescharacteristic
93C4×C3⋊C8D411yescharacteristic
94C4×C3⋊C8C2312yes—
95C4×C3⋊C8C2311yescharacteristic
96C4×C3⋊C8C2311yescharacteristic
97C22×M16—61no—
98#533—31no—
99C2×C4×D4—31no—
100#483—32no—
101#476—32no—
102#461—32no—
103#450—32no—
104#446—32no—
105#446—61no—
106C8×Q8—32no—
107C8×Q8—61no—
108C4×SD16—34no—
109#436—32no—
110#436—32no—
111#436—61no—
112#436—61no—
113#431—31no—
114#431C2×D311yescharacteristic
115#424—61no—
116#423C2×D311yescharacteristic
117C2×C4⋊C8C2×D311yescharacteristic
118C2×C22⋊C8—61no—
119#405—61no—
120C2×C4×C8—31no—
121#334C2×D314yescomplemented
122Q8⋊C8—34no—
123D4⋊C8—34no—
124#322—32no—
125C82—32no—
126C23×D3—21no—
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×Dic3—21no—
131C2×D12—21no—
132C2×D12—42no—
133C2×D12C22×C412yes—
134C2×C4×D3—22no—
135C2×C4×D3—22no—
136C2×C4×D3—42no—
137C2×C4×D3—42no—
138C2×Dic6—21no—
139C2×Dic6C22×C412yes—
140C3×M16—42no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—42no—
144C2×C3×C8C22×C414yes—
145C3×C42C22×C411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148C3×C42C2411yescharacteristic
149#217—22no—
150(C2×D3)⋊C4—22no—
151(C2×D3)⋊C4—22no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3C22×C412yes—
157C4⋊Dic3C22×C412yes—
158Dic3⋊C4—22no—
159Dic3⋊C4—22no—
160C4×Dic3—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8SD1614yescomplemented
164C2×C3⋊C8C2×D411yescharacteristic
165C2×C3⋊C8C2×D411yescharacteristic
166C2×C3⋊C8C2×D411yescharacteristic
167C2×C3⋊C8C2×D411yescharacteristic
168#204—44no—
169ΓL(1,𝔽25)—44no—
170ΓL(1,𝔽25)—44no—
171C22×D4—31no—
172C23×C4—61no—
173C2×SD16—34no—
174C2×M16—21no—
175C2×M16—62no—
176C2×M16—62no—
177C2×M16—124no—
178C22×C8—21no—
179C22×C8—31no—
180C22×C8—31no—
181C4⋊Q8—31no—
182C4⋊Q8—31no—
183#125—61no—
184C22⋊Q8—32no—
185C22⋊Q8—61no—
186C4×Q8—32no—
187C4×Q8—61no—
188C4×D4—32no—
189C4×D4—32no—
190C4×D4—61no—
191C4×D4—61no—
192#117—61no—
193C2×C4⋊C4—31no—
194C2×C22⋊C4—61no—
195C2×C42C22×D311yescharacteristic
196#106—34no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—62no—
200C4⋊C8—62no—
201C4⋊C8—61no—
202C4⋊C8C4×D312yes—
203C4⋊C8C22×D312yes—
204C4⋊C8C22×D312yes—
205#103—34no—
206#102—34no—
207C22⋊C8—62no—
208C22⋊C8—62no—
209C22⋊C8—61no—
210C22⋊C8C4×D312yes—
211#97—62no—
212C4×C8—31no—
213C4×C8—31no—
214C4×C8—31no—
215C4×C8—31no—
216C4×C8—32no—
217C4×C8—62no—
218C4×C8C22×D312yes—
219C23×C3C4×D411yescharacteristic
220C22×D3—22no—
221C22×D3—42no—
222C22×D3—42no—
223C22×D3—42no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4C2×SD1612yes—
230C2×C3×C4#13512yes—
231C2×C3×C4C23×C411yescharacteristic
232C2×C3×C4C22×D411yescharacteristic
233C2×C3×C4C2×#4111yescharacteristic
234C2×Dic3—22no—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—42no—
238D12—22no—
239D12—22no—
240D12—42no—
241C4×D3—42no—
242C4×D3—42no—
243C4×D3—44no—
244Dic6—24no—
245Dic6—42no—
246C3×C8—24no—
247C3×C8—42no—
248C3×C8—42no—
249C3⋊C8—21no—
250C3⋊C8—21no—
251C3⋊C8—42no—
252C3⋊C8C2×SD1612yes—
253C3⋊C8C2×SD1612yes—
254C24—61no—
255C2×Q8—32no—
256C2×Q8—61no—
257C2×D4—32no—
258C2×D4—61no—
259C2×D4—122no—
260C22×C4—61no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—122no—
264C22×C4—122no—
265C22×C4C2×C4×D311yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4#23911yescharacteristic
268SD16—124no—
269M16—42no—
270M16—124no—
271M16—124no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—34no—
275C2×C8—31no—
276C2×C8—31no—
277C2×C8—31no—
278C2×C8—31no—
279C2×C8—42no—
280C2×C8—62no—
281C2×C8—62no—
282C2×C8C2×C4×D314yes—
283C4⋊C4—32no—
284C4⋊C4—32no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—62no—
289C4⋊C4—61no—
290C4⋊C4—62no—
291C22⋊C4—62no—
292C22⋊C4—62no—
293C22⋊C4—62no—
294C42—62no—
295C42C2×C4×D311yescharacteristic
296C42D3×D411yescharacteristic
297C42#23911yescharacteristic
298C42C23×D311yescharacteristic
299C22×C3—21no—
300C22×C3—21no—
301C22×C3C4×SD1612yes—
302C22×C3C2×C4×D411yescharacteristic
303C2×D3—42no—
304C2×D3—42no—
305C2×D3—42no—
306C2×D3—84no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4C4×SD1612yes—
311C3×C4#56812yes—
312C3×C4C22×SD1611yescharacteristic
313C3×C4C2×#13511yescharacteristic
314Dic3—42no—
315Dic3—42no—
316Dic3—42no—
317C23—62no—
318C23—122no—
319C23—122no—
320C23—122no—
321C23#89311yescharacteristic
322Q8—64no—
323Q8—122no—
324D4—62no—
325D4—62no—
326D4—122no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—124no—
336C2×C4#89311yescharacteristic
337C2×C4#89311yescharacteristic
338C2×C4#89311yescharacteristic
339C2×C4D3×SD1612yes—
340C2×C4#91612yes—
341C2×C4C22×C4×D311yescharacteristic
342C2×C4C2×D3×D411yescharacteristic
343C2×C4C2×#23911yescharacteristic
344C8—24no—
345C8—32no—
346C8—32no—
347C8—42no—
348C8—42no—
349C8—61no—
350C8—61no—
351C8—122no—
352C2×C3—22no—
353C2×C3#293011yescharacteristic
354C2×C3C2×C4×SD1611yescharacteristic
355C2×C3#295511yescharacteristic
356D3—82no—
357C22—21no—
358C22—21no—
359C22—122no—
360C22—122no—
361C22—122no—
362C22—244no—
363C22#485212yes—
364C22C2×#89311yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—122no—
369C4—122no—
370C4—122no—
371C4#485212yes—
372C4#575012yes—
373C4C2×D3×SD1611yescharacteristic
374C4C2×#91611yescharacteristic
375C3#1793011yescharacteristic, complemented, Sylow
376C2—22no—
377C2—242no—
378C2#7907411yescharacteristic
379C2C2×#485211yescharacteristic
380C2#7925711yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1570C2
socleC22×C3C2×C4×D4
prime radicalC22×D12C2×C4
cosocleC3×C42C24
hypercenter#1570D3
hypocenterC3#17930
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
2. #79044C2semidirect
4. C3×#1570C2semidirect
5. #70740C2semidirect
7. #70429C2semidirect
8. #68523C2semidirect
9. #68520C2semidirect
29. #4831C22semidirect
31. #4830C22semidirect
35. C3×#334C22semidirect
36. #4484C22semidirect
38. #4454C22semidirect
39. C8×C3⋊C8C22semidirect
45. #1570D3semidirect
121. #334C2×D3semidirect
163. C2×C3⋊C8SD16semidirect
375. C3#17930semidirect

Maximal subgroups

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

subgroupindex
#792982
#790442
C2×#48382
C3×#15702
#707402
#704962
#704292
#685232
#685202
#179303