Finite Group Databasebeta

← #10848804

Subgroups — #10848804

Order 768.

1052 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5731C211yesmaximal, characteristic
2C2×C3×#494C211yesmaximal, characteristic, complemented
3#83261C211yesmaximal, characteristic
4C2×#5082C211yesmaximal, characteristic, complemented
5#72071C214yesmaximal
6#72065C214yesmaximal, complemented
7#71786C211yesmaximal, characteristic, complemented
8#69778C211yesmaximal, characteristic, complemented
9#69753C211yesmaximal, characteristic
10#25609—31nomaximal, Sylow
11C22×C3×Q16C2214yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946C2212yes—
15C2×#946C2212yes—
16C2×Q8×Dic3C2212yes—
17#5731—24no—
18#5731—24no—
19C2×#898—22no—
20C2×#889—22no—
21C2×#872C2211yescharacteristic
22C2×#870C2211yescharacteristic
23C3×#494—24no—
24C2×C3×C4×C8C2211yescharacteristic
25#5229C2212yes—
26C2×#835C2214yes—
27C2×#835C2214yes—
28C2×(C2×D3)⋊C8C2212yes—
29#5082—24no—
30#4992—22no—
31#4983—22no—
32#4936C2211yescharacteristic
33C2×#803C2211yescharacteristic
34#4551C2214yes—
35#4550C2212yes—
36C2×#541—31no—
37C2×#494D311yescharacteristic, complemented
38C2×#491—31no—
39#2621—31no—
40#2001—34no—
41#1991—31no—
42#1960—31no—
43#1878—34no—
44#1750—31no—
45C22×C3×Q8C2312yes—
46C22×C3×Q8C2312yes—
47C22×C4×D3C2311yescharacteristic
48C2×C3×Q16—28no—
49C2×C3×Q16—28no—
50C2×C3×Q16—44no—
51C22×C3×C8C2312yes—
52C3×C4⋊Q8—22no—
53C3×C4⋊Q8—22no—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8D414yes—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C42C2311yescharacteristic
59#946—24no—
60#946—24no—
61#946—44no—
62Q8×Dic3—24no—
63Q8×Dic3—24no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4D412yes—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3C2312yes—
69C2×C4⋊Dic3C2312yes—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4D412yes—
74C2×C4×Dic3—22no—
75C2×C4×Dic3C2311yescharacteristic
76C22×C3⋊C8C2312yes—
77#898—216no—
78#889—216no—
79#872—22no—
80#872—22no—
81#870—22no—
82#870D414yes—
83C3×C4×C8—22no—
84#835—28no—
85#835—44no—
86C3⋊#95—22no—
87C3⋊#95—24no—
88(C2×D3)⋊C8—24no—
89#803—22no—
90C22×Q16C2×D314yes—
91#541—64no—
92#541—64no—
93C2×C4⋊Q8C2×D311yescharacteristic
94C2×C4⋊Q8C2×D311yescharacteristic
95C2×#126—62no—
96C2×#125—31no—
97C2×#125—62no—
98C2×C22⋊Q8—32no—
99C2×C22⋊Q8—32no—
100C2×C4×Q8—32no—
101C2×#117—31no—
102#494—24no—
103#491—64no—
104C2×#103—34no—
105C2×#103—34no—
106C2×C22⋊C8—32no—
107C2×#97—31no—
108C2×C4×C8C2×D311yescharacteristic
109#398—62no—
110#397—62no—
111#391—32no—
112#382—31no—
113#340—34no—
114#336—32no—
115C23×D3C2×D411yescharacteristic
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—24no—
120C2×C3×Q8—24no—
121C2×C3×Q8—24no—
122C2×C3×Q8—24no—
123C2×C3×Q8C2×D414yes—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2411yescharacteristic
128C22×Dic3—22no—
129C22×Dic3—22no—
130C22×Dic3C2×D412yes—
131C22×Dic3C2×D411yescharacteristic
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×C4×D3—22no—
135C2×C4×D3—22no—
136C3×Q16—416no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8#4114yes—
140C3×C4⋊C4—28no—
141C3×C4⋊C4—44no—
142C3×C42C2×D412yes—
143C3×C42C2×D412yes—
144(C2×D3)⋊C4—28no—
145(C2×D3)⋊C4—24no—
146(C2×D3)⋊C4—44no—
147C4⋊Dic3—28no—
148C4⋊Dic3—22no—
149C4⋊Dic3—24no—
150C4⋊Dic3C2×D414yes—
151Dic3⋊C4—28no—
152Dic3⋊C4—28no—
153Dic3⋊C4—24no—
154Dic3⋊C4—44no—
155C4×Dic3—28no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C22×Q8C22×D312yes—
162C22×Q8C22×D312yes—
163C23×C4—31no—
164C2×Q16—28no—
165C2×Q16—28no—
166C2×Q16—44no—
167C22×C8—32no—
168C22×C8C22×D312yes—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8#6714yes—
173#126—616no—
174#125—34no—
175#125—616no—
176#125—62no—
177C22⋊Q8—64no—
178C22⋊Q8—64no—
179C22⋊Q8—124no—
180C4×Q8—64no—
181C4×Q8—64no—
182#117—62no—
183#117—62no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—32no—
187C2×C4⋊C4—32no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4—62no—
193C2×C22⋊C4—32no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—62no—
196C2×C42—31no—
197C2×C42—62no—
198C2×C42C22×D311yescharacteristic
199#103—68no—
200#103—124no—
201C22⋊C8—64no—
202#97—62no—
203C4×C8—22no—
204#95—64no—
205#95—62no—
206C23×C3C22×D411yescharacteristic
207C22×D3—21no—
208C22×D3—21no—
209C22×D3—21no—
210C22×D3—24no—
211C3×Q8—28no—
212C3×Q8—44no—
213C3×Q8—44no—
214C3×Q8—44no—
215C2×C3×C4—24no—
216C2×C3×C4—24no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4#12014yes—
223C2×C3×C4C22×D411yescharacteristic
224C2×C3×C4C22×D411yescharacteristic
225C2×C3×C4C2×#4112yes—
226C2×C3×C4H(5,𝔽2)12yes—
227C2×Dic3—22no—
228C2×Dic3—24no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—24no—
233C2×Dic3—44no—
234C2×Dic3—42no—
235C2×Dic3—44no—
236C4×D3—42no—
237C4×D3—42no—
238C3×C8—24no—
239C3⋊C8—42no—
240C24—31no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8C2×#6714yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—32no—
252C22×C4—31no—
253C22×C4—62no—
254C22×C4—61no—
255C22×C4—62no—
256C22×C4—62no—
257C22×C4—61no—
258C22×C4—62no—
259C22×C4D3×D412yes—
260C22×C4C23×D311yescharacteristic
261Q16—416no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—62no—
265C2×C8—62no—
266C2×C8—62no—
267C2×C8#23714yes—
268C4⋊C4—28no—
269C4⋊C4—34no—
270C4⋊C4—44no—
271C4⋊C4—68no—
272C4⋊C4—68no—
273C4⋊C4—64no—
274C4⋊C4—68no—
275C4⋊C4—62no—
276C4⋊C4—64no—
277C4⋊C4—124no—
278C22⋊C4—68no—
279C22⋊C4—64no—
280C22⋊C4—124no—
281C42—68no—
282C42—61no—
283C42—61no—
284C42D3×D412yes—
285C42C2×#6712yes—
286C22×C3C2×#12011yescharacteristic
287C22×C3#53211yescharacteristic
288C22×C3#53511yescharacteristic
289C22×C3#57814yes—
290C2×D3—42no—
291C2×D3—41no—
292C2×D3—42no—
293C2×D3—42no—
294C3×C4—24no—
295C3×C4—44no—
296C3×C4—82no—
297C3×C4#54612yes—
298C3×C4#55012yes—
299Dic3—41no—
300Dic3—42no—
301Dic3—44no—
302Dic3—82no—
303C23—61no—
304C23—61no—
305C23—64no—
306C23—61no—
307C23C2×D3×D411yescharacteristic
308Q8—28no—
309Q8—44no—
310Q8—44no—
311Q8—44no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—64no—
322C2×C4—62no—
323C2×C4—64no—
324C2×C4—61no—
325C2×C4—124no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—124no—
330C2×C4#93714yes—
331C2×C4C2×D3×D411yescharacteristic
332C2×C4C2×#23712yes—
333C2×C4#100412yes—
334C2×C4C22×#6711yescharacteristic
335C8—24no—
336C8—122no—
337C2×C3#243111yescharacteristic
338C2×C3#301112yes—
339C2×C3#317712yes—
340C2×C3#322312yes—
341D3—81no—
342C22—122no—
343C22—122no—
344C22—122no—
345C22—121no—
346C22#571511yescharacteristic
347C22#571611yescharacteristic
348C22#576614yes—
349C22C2×#93711yescharacteristic
350C4—24no—
351C4—44no—
352C4—82no—
353C4—122no—
354C4—124no—
355C4—121no—
356C4—242no—
357C4#571812yes—
358C4#580612yes—
359C3#2560911yescharacteristic, complemented, Sylow
360C2—241no—
361C2#8179312yes—
362C2#8180412yes—
363C2#8318711yescharacteristic
364C2#8388412yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#494C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#494D3
hypocenterC3#25609
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
2. C2×C3×#494C2semidirect
4. C2×#5082C2semidirect
6. #72065C2semidirect
7. #71786C2semidirect
8. #69778C2semidirect
37. C2×#494D3semidirect
359. C3#25609semidirect

Maximal subgroups

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

subgroupindex
C2×#57312
C2×C3×#4942
#832612
C2×#50822
#720712
#720652
#717862
#697782
#697532
#256093