Finite Group Databasebeta

← #10692559

Subgroups — #10692559

Order 768.

594 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83259C211yesmaximal, characteristic
2#79286C211yesmaximal, characteristic
3C3×#1549C211yesmaximal, characteristic, complemented
4#71775C211yesmaximal, characteristic, complemented
5#70527C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#9395—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#946—22no—
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×#823—21no—
15#4993C414yes—
16#4988—21no—
17#4975—21no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4925C2211yescharacteristic
20#4924C2211yescharacteristic
21#4834—22no—
22D3×C22⋊C8—22no—
23C8×D12—22no—
24C3×#332—22no—
25#4554—21no—
26#4547—22no—
27#2972—31no—
28#2542—31no—
29#1800—31no—
30#1549—31no—
31#1549D311yescharacteristic, complemented
32#1303—31no—
33#1303—31no—
34C22×C3×Q8—21no—
35C22×D12D411yescharacteristic
36C22×C4×D3D412yes—
37C2×C3×M16—21no—
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×C412yes—
44C2×C3×C42C2311yescharacteristic
45#946—48no—
46C2×(C2×D3)⋊C4—24no—
47C2×(C2×D3)⋊C4C2×C412yes—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4—24no—
51C2×C4×Dic3—21no—
52C2×#208—21no—
53C2×D3×C8—24no—
54C4×D12—21no—
55C4×D12—21no—
56C4×D12—21no—
57C4×D12—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8D412yes—
61C3×C22⋊C8—21no—
62C3×C4×C8—21no—
63C3⋊#95C2×C412yes—
64#830—21no—
65#823—42no—
66(C2×D3)⋊C8—24no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)D412yes—
70C4×C3⋊C8—21no—
71C2×C4⋊Q8C2×D311yescharacteristic
72C2×C22⋊Q8—62no—
73C2×C4×D4—31no—
74C8×D4—62no—
75C8×D4—62no—
76#431—31no—
77#431C2×D311yescharacteristic
78#423—31no—
79#423C2×D311yescharacteristic
80C2×#100—61no—
81C2×C22⋊C8—62no—
82#399—34no—
83#390—61no—
84#384—61no—
85C4×C4⋊C4—31no—
86#343—61no—
87#332—22no—
88#332—62no—
89C23×D3C22⋊C412yes—
90C2×C3×Q8—44no—
91C2×C3×Q8—41no—
92C2×C3×Q8—41no—
93C22×C3×C4—22no—
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—22no—
98C22×Dic3C22⋊C412yes—
99C2×D12—21no—
100C2×D12—21no—
101C2×D12—21no—
102C2×D12—21no—
103C2×D12SD1612yes—
104C2×D12Q1612yes—
105C2×C4×D3—22no—
106C2×C4×D3—22no—
107C2×C4×D3—44no—
108C3×M16—41no—
109C3×M16—41no—
110C2×C3×C8—22no—
111C2×C3×C8—41no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C42C2×D411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42#4111yescharacteristic
119C3×C42#4111yescharacteristic
120(C2×D3)⋊C4—22no—
121(C2×D3)⋊C4—22no—
122(C2×D3)⋊C4—48no—
123C4⋊Dic3—24no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126Dic3⋊C4—48no—
127C4×Dic3—41no—
128C4×Dic3—41no—
129#208—41no—
130#208—41no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—41no—
133D3×C8—84no—
134C22×Q8—21no—
135C22×D4—31no—
136C23×C4—32no—
137C2×M16—21no—
138C2×M16—61no—
139C22×C8—64no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C22⋊Q8—128no—
145C4×D4—61no—
146C4×D4—61no—
147C4×D4—61no—
148C4×D4—61no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—64no—
153C2×C4⋊C4C4×D312yes—
154C2×C22⋊C4—32no—
155C2×C22⋊C4—64no—
156C2×C42—61no—
157C2×C42C22×D311yescharacteristic
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—32no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163C4⋊C8#6712yes—
164#100—122no—
165C22⋊C8—21no—
166C22⋊C8—64no—
167C22⋊C8—61no—
168C4×C8—21no—
169C4×C8—61no—
170#95—32no—
171C23×C3C2×C22⋊C411yescharacteristic
172C22×D3—22no—
173C22×D3—22no—
174C22×D3—42no—
175C22×D3—42no—
176C22×D3#10314yes—
177C3×Q8—82no—
178C3×Q8—82no—
179C2×C3×C4—22no—
180C2×C3×C4—41no—
181C2×C3×C4—44no—
182C2×C3×C4—42no—
183C2×C3×C4C4×D411yescharacteristic
184C2×C3×C4C4×D411yescharacteristic
185C2×C3×C4#12011yescharacteristic
186C2×C3×C4#12111yescharacteristic
187C2×C3×C4#12111yescharacteristic
188C2×C3×C4#12311yescharacteristic
189C2×C3×C4C2×SD1611yescharacteristic
190C2×C3×C4C2×Q1611yescharacteristic
191C2×C3×C4#13511yescharacteristic
192C2×C3×C4#13511yescharacteristic
193C2×Dic3—24no—
194C2×Dic3—22no—
195C2×Dic3—42no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198D12—22no—
199D12—22no—
200D12—22no—
201D12—22no—
202C4×D3—44no—
203C4×D3—82no—
204C3×C8—42no—
205C3×C8—81no—
206C3⋊C8—42no—
207C3⋊C8—81no—
208C24—32no—
209C2×Q8—44no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×D4—32no—
213C2×D4—32no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—61no—
217C2×D4—61no—
218C22×C4—22no—
219C22×C4—32no—
220C22×C4—62no—
221C22×C4—62no—
222C22×C4—62no—
223C22×C4—124no—
224C22×C4C2×C4×D311yescharacteristic
225C22×C4D3×D411yescharacteristic
226C22×C4D3×D411yescharacteristic
227M16—41no—
228M16—41no—
229M16—121no—
230M16—121no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—62no—
234C2×C8—121no—
235C2×C8—244no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—64no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—128no—
244C22⋊C4—62no—
245C22⋊C4—62no—
246C22⋊C4—128no—
247C42—121no—
248C42—121no—
249C42#23511yescharacteristic
250C42D3×D411yescharacteristic
251C42#23711yescharacteristic
252C42C2×#6711yescharacteristic
253C22×C3—21no—
254C22×C3—21no—
255C22×C3#38611yescharacteristic
256C22×C3C2×#10311yescharacteristic
257C22×C3#41611yescharacteristic
258C2×D3—24no—
259C2×D3—42no—
260C2×D3—42no—
261C2×D3—44no—
262C2×D3—82no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—82no—
267C3×C4#45312yes—
268C3×C4#46511yescharacteristic
269C3×C4#46511yescharacteristic
270C3×C4#46711yescharacteristic
271C3×C4#46711yescharacteristic
272Dic3—44no—
273Dic3—42no—
274C23—34no—
275C23—62no—
276C23—62no—
277C23—122no—
278C23—122no—
279C23D3×C22⋊C411yescharacteristic
280Q8—82no—
281Q8—82no—
282D4—62no—
283D4—62no—
284D4—62no—
285D4—62no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—64no—
291C2×C4—62no—
292C2×C4—122no—
293C2×C4—122no—
294C2×C4—124no—
295C2×C4—122no—
296C2×C4—242no—
297C2×C4#67⋊C411yescharacteristic
298C2×C4#88311yescharacteristic
299C2×C4#88411yescharacteristic
300C2×C4D3×SD1611yescharacteristic
301C2×C4#91611yescharacteristic
302C2×C4D3×Q1611yescharacteristic
303C2×C4#91911yescharacteristic
304C2×C4C4×#6711yescharacteristic
305C2×C4#93711yescharacteristic
306C2×C4#93811yescharacteristic
307C8—42no—
308C8—81no—
309C8—122no—
310C8—241no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3#188711yescharacteristic
314C2×C3#189211yescharacteristic
315C2×C3#191511yescharacteristic
316D3—44no—
317C22—21no—
318C22—21no—
319C22—64no—
320C22—124no—
321C22—122no—
322C22—122no—
323C22—242no—
324C22D3×#10311yescharacteristic
325C22#480411yescharacteristic
326C22#495611yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—82no—
331C4—124no—
332C4—122no—
333C4#480711yescharacteristic
334C4#481311yescharacteristic
335C4#517111yescharacteristic
336C4#518811yescharacteristic
337C4#519812yes—
338C3#939511yescharacteristic, complemented, Sylow
339C2—21no—
340C2—21no—
341C2—124no—
342C2#7012411yescharacteristic
343C2#7027011yescharacteristic
344C2#7032311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1549C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1549D3
hypocenterC3#9395
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. C3×#1549C2semidirect
4. #71775C2semidirect
6. #67746C2semidirect
7. #67715C2semidirect
31. #1549D3semidirect
338. C3#9395semidirect

Maximal subgroups

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

subgroupindex
#832592
#792862
C3×#15492
#717752
#705272
#677462
#677152
#93953