Finite Group Databasebeta

← #10691199

Subgroups — #10691199

Order 768.

504 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79048C211yesmaximal, characteristic
2#74194C211yesmaximal, characteristic
3#70782C211yesmaximal, characteristic
4#70780C211yesmaximal, characteristic, complemented
5#70508C211yesmaximal, characteristic, complemented
6C3×#1286C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9193—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×#872C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×#382C2211yescharacteristic
15#5118—21no—
16#4987—21no—
17#4984—21no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20#4839—22no—
21#4839—22no—
22#4677C412yes—
23#4673—21no—
24#4659C412yes—
25#4656—21no—
26#4486—22no—
27#4482—22no—
28#4469—21no—
29#4468—21no—
30#2905—31no—
31#2628—31no—
32#1549—31no—
33#1549—31no—
34#1458—31no—
35#1286—31no—
36#1286D311yescharacteristic, complemented
37C22×C4×D3D411yescharacteristic
38C22×Dic6—21no—
39C2×C3×M16—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C42C2311yescharacteristic
42C2×(C2×D3)⋊C4—22no—
43C2×(C2×D3)⋊C4C2×C411yescharacteristic
44C2×(C2×D3)⋊C4D411yescharacteristic
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—22no—
47C2×C4⋊Dic3C2×C412yes—
48C2×Dic3⋊C4C2×C411yescharacteristic
49C2×Dic3⋊C4D411yescharacteristic
50C2×C4×Dic3D411yescharacteristic
51C2×#208—21no—
52D3×M16—42no—
53D3×M16—42no—
54C2×ΓL(1,𝔽25)—22no—
55C2×D3×C8—22no—
56#872—21no—
57#872—21no—
58#872—21no—
59#872—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D412yes—
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69C3×#95—21no—
70C3×#95C2×C412yes—
71C3⋊#95—22no—
72C3⋊#95—22no—
73#830—21no—
74Dic3⋊C8—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yes—
78C4×C3⋊C8—21no—
79C22×M16—61no—
80C2×C4⋊Q8—31no—
81C2×#117—31no—
82#431—31no—
83#431C2×D311yescharacteristic
84#424—62no—
85#424—62no—
86#423—31no—
87#423—61no—
88#423C2×D311yescharacteristic
89#400—32no—
90#399—32no—
91#395—61no—
92#393—61no—
93#389—61no—
94#382C2×D311yescharacteristic
95#380—61no—
96#332—62no—
97#332—62no—
98#324—61no—
99#323—61no—
100C23×D3C22⋊C411yescharacteristic
101C22×C3×C4—22no—
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—22no—
106C22×Dic3C22⋊C411yescharacteristic
107C22×Dic3C22⋊C411yescharacteristic
108C22×Dic3C22⋊C411yescharacteristic
109C2×C4×D3—21no—
110C2×C4×D3—21no—
111C2×C4×D3—21no—
112C2×C4×D3—21no—
113C2×C4×D3SD1612yes—
114C2×C4×D3Q1612yes—
115C2×Dic6—41no—
116C2×Dic6—41no—
117C2×Dic6—82no—
118C3×M16—41no—
119C3×M16—41no—
120C2×C3×C8—22no—
121C2×C3×C8—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127C3×C42#4111yescharacteristic
128(C2×D3)⋊C4—21no—
129(C2×D3)⋊C4—21no—
130(C2×D3)⋊C4—21no—
131(C2×D3)⋊C4—21no—
132(C2×D3)⋊C4—82no—
133C4⋊Dic3—22no—
134C4⋊Dic3—22no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C4⋊Dic3—82no—
138Dic3⋊C4—21no—
139Dic3⋊C4—21no—
140Dic3⋊C4—21no—
141Dic3⋊C4—21no—
142C4×Dic3—21no—
143C4×Dic3—21no—
144#208—41no—
145#208—41no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148ΓL(1,𝔽25)—44no—
149D3×C8—44no—
150C22×Q8—61no—
151C23×C4—31no—
152C2×M16—21no—
153C2×M16—62no—
154C2×M16—61no—
155C2×M16—122no—
156C2×M16—122no—
157C22×C8—62no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162#117—61no—
163#117—61no—
164#117—61no—
165#117—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—32no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—62no—
172C2×C22⋊C4—31no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—62no—
175C2×C42—31no—
176C2×C42C22×D311yescharacteristic
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—32no—
180C4⋊C8—64no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183C4⋊C8D1212yes—
184C22⋊C8—21no—
185C22⋊C8—61no—
186C4×C8—21no—
187C4×C8—61no—
188#95—21no—
189#95—62no—
190#95—62no—
191#95C4×D312yes—
192C23×C3C2×C22⋊C411yescharacteristic
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—41no—
196C22×D3—41no—
197C22×D3#10312yes—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12311yescharacteristic
209C2×C3×C4C2×SD1611yescharacteristic
210C2×C3×C4C2×Q1611yescharacteristic
211C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
212C2×C3×C4#13711yescharacteristic
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3#10312yes—
223C4×D3—22no—
224C4×D3—22no—
225C4×D3—22no—
226C4×D3—22no—
227Dic6—82no—
228Dic6—82no—
229C3×C8—42no—
230C3×C8—81no—
231C3⋊C8—42no—
232C3⋊C8—81no—
233C24—31no—
234C2×Q8—121no—
235C2×Q8—121no—
236C2×Q8—242no—
237C22×C4—22no—
238C22×C4—31no—
239C22×C4—32no—
240C22×C4—32no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—62no—
248C22×C4C2×C4×D311yescharacteristic
249C22×C4D3×D411yescharacteristic
250C22×C4D3×D411yescharacteristic
251M16—41no—
252M16—41no—
253M16—124no—
254M16—121no—
255M16—121no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—62no—
259C2×C8—124no—
260C2×C8—121no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—62no—
268C4⋊C4—62no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—242no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C22⋊C4—242no—
277C42—61no—
278C42—61no—
279C42C2×D1211yescharacteristic
280C42#23511yescharacteristic
281C42D3×D411yescharacteristic
282C42#23911yescharacteristic
283C22×C3—21no—
284C22×C3—21no—
285C22×C3#38611yescharacteristic
286C22×C3C2×#10311yescharacteristic
287C22×C3#41711yescharacteristic
288C2×D3—22no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—42no—
292C2×D3—81no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—82no—
298C3×C4#45011yescharacteristic
299C3×C4#45111yescharacteristic
300C3×C4#48311yescharacteristic
301C3×C4#48411yescharacteristic
302Dic3—22no—
303Dic3—41no—
304Dic3—41no—
305Dic3—82no—
306C23—32no—
307C23—61no—
308C23—61no—
309C23—121no—
310C23—121no—
311C23D3×C22⋊C411yescharacteristic
312Q8—242no—
313Q8—242no—
314C2×C4—32no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4C4×D1211yescharacteristic
334C2×C4#88211yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4#89311yescharacteristic
337C2×C4#89411yescharacteristic
338C2×C4#89511yescharacteristic
339C2×C4#91111yescharacteristic
340C2×C4D3×SD1611yescharacteristic
341C2×C4#91511yescharacteristic
342C2×C4D3×Q1611yescharacteristic
343C8—42no—
344C8—81no—
345C8—122no—
346C8—241no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3#187211yescharacteristic
350C2×C3#191011yescharacteristic
351C2×C3#191511yescharacteristic
352D3—42no—
353C22—21no—
354C22—21no—
355C22—62no—
356C22—122no—
357C22—121no—
358C22—121no—
359C22—241no—
360C22#466511yescharacteristic
361C22#477111yescharacteristic
362C22D3×#10311yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—62no—
368C4—82no—
369C4—121no—
370C4—121no—
371C4—242no—
372C4#477611yescharacteristic
373C4#480911yescharacteristic
374C4#486211yescharacteristic
375C4#488411yescharacteristic
376C3#919311yescharacteristic, complemented, Sylow
377C2—21no—
378C2—21no—
379C2—122no—
380C2#6871311yescharacteristic
381C2#6887811yescharacteristic
382C2#6904111yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#386
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1286D3
hypocenterC3#9193
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
4. #70780C2semidirect
5. #70508C2semidirect
6. C3×#1286C2semidirect
7. #67631C2semidirect
36. #1286D3semidirect
376. C3#9193semidirect

Maximal subgroups

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

subgroupindex
#790482
#741942
#707822
#707802
#705082
C3×#12862
#676312
#91933