Finite Group Databasebeta

← #10691187

Subgroups — #10691187

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#83262C211yesmaximal, characteristic
2#79048C211yesmaximal, characteristic
3C3×#1548C211yesmaximal, characteristic, complemented
4#71774C211yesmaximal, characteristic, complemented
5#70782C211yesmaximal, characteristic
6#67738C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9191—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×D3×M16—21no—
11C2×#872C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5231C412yes—
15#5230C412yes—
16#5118—21no—
17#4993—21no—
18#4987—21no—
19#4985—21no—
20#4972—21no—
21#4936C2211yescharacteristic
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4839—22no—
25#4839—22no—
26C3×#333—22no—
27#4546—22no—
28#4469—21no—
29#4468—21no—
30#2905—31no—
31#2628—31no—
32#1548—31no—
33#1548D311yescharacteristic, complemented
34#1458—31no—
35#1286—31no—
36#1286—31no—
37C22×C3×Q8C2×C412yes—
38C22×C4×D3D411yescharacteristic
39C2×C3×M16—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4C2×C411yescharacteristic
49C2×(C2×D3)⋊C4D411yescharacteristic
50C2×C4⋊Dic3—21no—
51C2×Dic3⋊C4—22no—
52C2×Dic3⋊C4C2×C411yescharacteristic
53C2×Dic3⋊C4D411yescharacteristic
54C2×C4×Dic3D411yescharacteristic
55C2×#208—21no—
56D3×M16—42no—
57D3×M16—42no—
58C2×ΓL(1,𝔽25)—22no—
59C2×D3×C8—22no—
60#872—21no—
61#872—21no—
62#872—21no—
63#872—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D412yes—
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69C3⋊#95—21no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95C2×C412yes—
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⋊Q8C2×D311yescharacteristic
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—61no—
90#399—61no—
91#395—32no—
92#393—32no—
93#389—61no—
94#382—31no—
95#380—61no—
96#333—22no—
97#333—62no—
98#324—61no—
99#323—61no—
100C23×D3C22⋊C411yescharacteristic
101C2×C3×Q8—22no—
102C2×C3×Q8—22no—
103C2×C3×Q8—82no—
104C22×C3×C4—22no—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—22no—
109C22×Dic3C22⋊C411yescharacteristic
110C22×Dic3C22⋊C411yescharacteristic
111C22×Dic3C22⋊C411yescharacteristic
112C2×C4×D3—21no—
113C2×C4×D3—21no—
114C2×C4×D3—21no—
115C2×C4×D3—21no—
116C2×C4×D3SD1612yes—
117C2×C4×D3Q1612yes—
118C3×M16—41no—
119C3×M16—41no—
120C2×C3×C8—22no—
121C2×C3×C8—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129C3×C42#4111yescharacteristic
130(C2×D3)⋊C4—21no—
131(C2×D3)⋊C4—21no—
132(C2×D3)⋊C4—21no—
133(C2×D3)⋊C4—21no—
134(C2×D3)⋊C4—82no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137Dic3⋊C4—21no—
138Dic3⋊C4—21no—
139Dic3⋊C4—21no—
140Dic3⋊C4—21no—
141Dic3⋊C4—82no—
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×Q8C4×D312yes—
151C23×C4—31no—
152C2×M16—21no—
153C2×M16—62no—
154C2×M16—61no—
155C2×M16—122no—
156C2×M16—122no—
157C22×C8—62no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162#117—61no—
163#117—61no—
164#117—61no—
165#117—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
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⋊C8#6712yes—
184C22⋊C8—21no—
185C22⋊C8—61no—
186C4×C8—21no—
187C4×C8—61no—
188#95—32no—
189#95—62no—
190#95—62no—
191#95—61no—
192C23×C3C2×C22⋊C411yescharacteristic
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—41no—
196C22×D3—41no—
197C22×D3#10312yes—
198C3×Q8—82no—
199C3×Q8—82no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12311yescharacteristic
211C2×C3×C4C2×SD1611yescharacteristic
212C2×C3×C4C2×Q1611yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214C2×C3×C4#13711yescharacteristic
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3#10312yes—
225C4×D3—22no—
226C4×D3—22no—
227C4×D3—22no—
228C4×D3—22no—
229C3×C8—42no—
230C3×C8—81no—
231C3⋊C8—42no—
232C3⋊C8—81no—
233C24—31no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—82no—
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—41no—
264C4⋊C4—41no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
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—
279C42#23511yescharacteristic
280C42D3×D411yescharacteristic
281C42#23711yescharacteristic
282C42C2×#6711yescharacteristic
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—82no—
313Q8—82no—
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×C4#67⋊C411yescharacteristic
334C2×C4#88311yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4D3×SD1611yescharacteristic
337C2×C4#91411yescharacteristic
338C2×C4D3×Q1611yescharacteristic
339C2×C4#91811yescharacteristic
340C2×C4C4×#6711yescharacteristic
341C2×C4#93711yescharacteristic
342C2×C4#93811yescharacteristic
343C8—42no—
344C8—81no—
345C8—122no—
346C8—241no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3#187211yescharacteristic
350C2×C3#190911yescharacteristic
351C2×C3#191411yescharacteristic
352D3—42no—
353C22—21no—
354C22—21no—
355C22—62no—
356C22—122no—
357C22—121no—
358C22—121no—
359C22—241no—
360C22D3×#10311yescharacteristic
361C22#480311yescharacteristic
362C22#495611yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—62no—
368C4—82no—
369C4—121no—
370C4—121no—
371C4—242no—
372C4#480511yescharacteristic
373C4#481111yescharacteristic
374C4#516911yescharacteristic
375C4#518611yescharacteristic
376C3#919111yescharacteristic, complemented, Sylow
377C2—21no—
378C2—21no—
379C2—122no—
380C2#7008911yescharacteristic
381C2#7026811yescharacteristic
382C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#386
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1548D3
hypocenterC3#9191
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. C3×#1548C2semidirect
4. #71774C2semidirect
6. #67738C2semidirect
7. #67713C2semidirect
33. #1548D3semidirect
376. C3#9191semidirect

Maximal subgroups

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

subgroupindex
#832622
#790482
C3×#15482
#717742
#707822
#677382
#677132
#91913