Finite Group Databasebeta

← #10690699

Subgroups — #10690699

Order 768.

538 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82455C211yesmaximal, characteristic
2#79288C211yesmaximal, characteristic
3C3×#1479C211yesmaximal, characteristic, complemented
4#71621C211yesmaximal, characteristic, complemented
5#70783C211yesmaximal, characteristic
6#67737C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#9103—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#947C411yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5229C411yescharacteristic
15#4989—22no—
16#4988C411yescharacteristic
17#4975C411yescharacteristic
18#4940—22no—
19#4932C2211yescharacteristic
20#4925C2211yescharacteristic
21#4925C2211yescharacteristic
22#4838—22no—
23#4726—22no—
24#4688—22no—
25C3×Q8⋊C8—22no—
26#4535—22no—
27#4468—22no—
28#2975—31no—
29#2326—31no—
30#1479—31no—
31#1479D311yescharacteristic, complemented
32#1460—31no—
33#1274—31no—
34#1274—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×D12C2×C411yescharacteristic
37C22×C4×D3D412yes—
38C3×C4×Q8—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46#947—22no—
47#947—22no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4D412yes—
51C2×C4⋊Dic3C2×C411yescharacteristic
52C2×Dic3⋊C4—22no—
53C2×C4×Dic3C2×C411yescharacteristic
54C2×C4×Dic3C2×C411yescharacteristic
55C2×ΓL(1,𝔽25)—24no—
56C4×D12—21no—
57C4×D12—21no—
58C4×D12—21no—
59C4×D12—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×C4×C8—21no—
65C3×C4×C8D412yes—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—22no—
69#830—21no—
70#830—21no—
71(C2×D3)⋊C8—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D412yes—
76C2×#124—31no—
77C2×C4×Q8C2×D311yescharacteristic
78C2×C4×D4—31no—
79#436—62no—
80#436—62no—
81#431—31no—
82#431—31no—
83#431C2×D311yescharacteristic
84#431C2×D311yescharacteristic
85#407—62no—
86#391—31no—
87#390—31no—
88#388—62no—
89#385—62no—
90#384—31no—
91#376—31no—
92Q8⋊C8—22no—
93Q8⋊C8—62no—
94#323—62no—
95C23×D3C22⋊C412yes—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8M1612yescomplemented
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3C22⋊C412yes—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×C4×D3—22no—
117C2×C4×D3—22no—
118C2×C4×D3—82no—
119C2×C3×C8—22no—
120C2×C3×C8—41no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C42—41no—
126C3×C42—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131(C2×D3)⋊C4—28no—
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—82no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137Dic3⋊C4—82no—
138C4×Dic3—21no—
139C4×Dic3—21no—
140C4×Dic3—21no—
141C4×Dic3M1612yescomplemented
142C2×C3⋊C8—22no—
143C2×C3⋊C8—41no—
144ΓL(1,𝔽25)—84no—
145C22×Q8C4×D311yescharacteristic
146C22×D4—31no—
147C23×C4—32no—
148C2×M16—64no—
149#124—62no—
150#124—62no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×D4—61no—
156C4×D4—61no—
157C4×D4—61no—
158C4×D4—61no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—62no—
162C2×C4⋊C4C4×D311yescharacteristic
163C2×C22⋊C4—32no—
164C2×C22⋊C4—62no—
165C2×C22⋊C4—62no—
166C2×C42—21no—
167C2×C42—31no—
168C2×C42—31no—
169C2×C42C22×D311yescharacteristic
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174C22⋊C8—21no—
175C22⋊C8—21no—
176C22⋊C8—64no—
177C22⋊C8—61no—
178C22⋊C8—61no—
179C4×C8—21no—
180C4×C8—32no—
181C4×C8—61no—
182C4×C8#6712yes—
183#95—62no—
184#95—61no—
185#95—61no—
186C23×C3C2×C22⋊C411yescharacteristic
187C22×D3—22no—
188C22×D3—22no—
189C22×D3—22no—
190C22×D3—42no—
191C22×D3—42no—
192C3×Q8—22no—
193C3×Q8—22no—
194C3×Q8—41no—
195C3×Q8—41no—
196C2×C3×C4—22no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
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×M1611yescharacteristic
210C2×C3×C4C2×M1611yescharacteristic
211C2×C3×C4#13111yescharacteristic
212C2×C3×C4#13111yescharacteristic
213C2×Dic3—22no—
214C2×Dic3—22no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221D12—41no—
222D12—41no—
223D12—41no—
224D12—41no—
225C4×D3—44no—
226C4×D3—82no—
227C3×C8—42no—
228C3×C8—81no—
229C3⋊C8—42no—
230C3⋊C8—81no—
231C24—32no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8ΓL(1,𝔽25)12yescomplemented
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—32no—
247C22×C4—61no—
248C22×C4—62no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4—242no—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255M16—244no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—62no—
259C2×C8—121no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—242no—
267C22⋊C4—68no—
268C22⋊C4—62no—
269C22⋊C4—62no—
270C22⋊C4—242no—
271C42—32no—
272C42—41no—
273C42—41no—
274C42—61no—
275C42—61no—
276C42—61no—
277C42#23511yescharacteristic
278C42D3×D411yescharacteristic
279C42#23711yescharacteristic
280C42C2×#6711yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3#38611yescharacteristic
284C22×C3#40711yescharacteristic
285C22×C3#40811yescharacteristic
286C2×D3—44no—
287C2×D3—42no—
288C2×D3—42no—
289C2×D3—42no—
290C2×D3—82no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—42no—
296C3×C4—81no—
297C3×C4#43511yescharacteristic
298C3×C4#43511yescharacteristic
299C3×C4#43611yescharacteristic
300C3×C4#43611yescharacteristic
301Dic3—42no—
302Dic3—42no—
303Dic3—81no—
304C23—62no—
305C23—62no—
306C23—62no—
307C23—122no—
308C23—122no—
309C23D3×C22⋊C411yescharacteristic
310Q8—22no—
311Q8—22no—
312Q8—41no—
313Q8—41no—
314D4—121no—
315D4—121no—
316D4—121no—
317D4—121no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—62no—
326C2×C4—62no—
327C2×C4—124no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—242no—
335C2×C4#67⋊C411yescharacteristic
336C2×C4#88311yescharacteristic
337C2×C4#88411yescharacteristic
338C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
339C2×C4#90111yescharacteristic
340C2×C4D3×M1611yescharacteristic
341C2×C4#90711yescharacteristic
342C2×C4C4×#6711yescharacteristic
343C2×C4#93711yescharacteristic
344C2×C4#93811yescharacteristic
345C8—42no—
346C8—81no—
347C8—122no—
348C8—241no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3#187011yescharacteristic
352C2×C3#190911yescharacteristic
353C2×C3#191111yescharacteristic
354D3—82no—
355C22—21no—
356C22—21no—
357C22—124no—
358C22—122no—
359C22—122no—
360C22—122no—
361C22—242no—
362C22#472611yescharacteristic
363C22#472711yescharacteristic
364C22#495611yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—81no—
371C4—122no—
372C4—122no—
373C4—241no—
374C4#472911yescharacteristic
375C4#473011yescharacteristic
376C4#511111yescharacteristic
377C4#512811yescharacteristic
378C3#910311yescharacteristic, complemented, Sylow
379C2—21no—
380C2—21no—
381C2—242no—
382C2#6995811yescharacteristic
383C2#7035011yescharacteristic
384C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1479C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1479D3
hypocenterC3#9103
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
3. C3×#1479C2semidirect
4. #71621C2semidirect
6. #67737C2semidirect
7. #67737C2semidirect
31. #1479D3semidirect
101. C2×C3×Q8M16semidirect
141. C4×Dic3M16semidirect
237. C2×Q8ΓL(1,𝔽25)semidirect
378. C3#9103semidirect

Maximal subgroups

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

subgroupindex
#824552
#792882
C3×#14792
#716212
#707832
#677372
#91033