Finite Group Databasebeta

← #10692554

Subgroups — #10692554

Order 768.

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

normal subgroups only

This group has 506 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#83239C211yesmaximal, characteristic
2#79263C211yesmaximal, characteristic
3C3×#1546C211yesmaximal, characteristic
4#71773C211yesmaximal, characteristic
5#70528C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic
7#67715C211yesmaximal, characteristic
8#9394—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#944—21no—
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5229—21no—
15C2×#824—21no—
16#4983C414yes—
17#4972—22no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4925C2211yescharacteristic
20#4924C2211yescharacteristic
21#4830—22no—
22#4718—22no—
23C8×Dic6—22no—
24C3×#332—22no—
25#4554—21no—
26#4547—22no—
27#2982—31no—
28#2546—31no—
29#1801—31no—
30#1546—31no—
31#1546D311yescharacteristic
32#1303—31no—
33#1303—31no—
34C22×C3×Q8—21no—
35C22×Dic6D411yescharacteristic
36C2×C3×M16—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2×C412yes—
43C2×C3×C42C2311yescharacteristic
44#944—42no—
45#944—42no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3D411yescharacteristic
48C2×Dic3⋊C4—24no—
49C2×Dic3⋊C4C2×C412yes—
50C2×C4×Dic3—21no—
51C2×C4×Dic3D412yes—
52C2×#208—21no—
53C4×Dic6—21no—
54C4×Dic6—21no—
55C4×Dic6—21no—
56C4×Dic6—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8D412yes—
60C3×C22⋊C8—21no—
61C3×C4×C8—21no—
62C3⋊#95—24no—
63C3⋊#95C2×C412yes—
64#830—21no—
65#824—42no—
66Dic3⋊C8—24no—
67C8×Dic3—24no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)D412yes—
71C4×C3⋊C8—21no—
72C2×C4⋊Q8—61no—
73C2×C4⋊Q8C2×D311yescharacteristic
74C2×C4×Q8—31no—
75C8×Q8—62no—
76C8×Q8—62no—
77#431—31no—
78#431—62no—
79#431C2×D311yescharacteristic
80#423—31no—
81#423C2×D311yescharacteristic
82C2×#101—61no—
83#398—34no—
84#391—61no—
85#389—62no—
86C4×C4⋊C4—31no—
87#343—61no—
88#332—22no—
89#332—62no—
90C2×C3×Q8—41no—
91C2×C3×Q8—41no—
92C2×C3×Q8—41no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C22×C3×C4—22no—
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—22no—
101C22×Dic3C22⋊C412yes—
102C22×Dic3C22⋊C412yes—
103C2×Dic6—21no—
104C2×Dic6—21no—
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×Dic6D812yes—
108C2×Dic6SD1612yes—
109C3×M16—41no—
110C3×M16—41no—
111C2×C3×C8—22no—
112C2×C3×C8—41no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42#4111yescharacteristic
120C3×C42#4111yescharacteristic
121C4⋊Dic3—24no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124Dic3⋊C4—22no—
125Dic3⋊C4—22no—
126Dic3⋊C4—48no—
127C4×Dic3—22no—
128C4×Dic3—22no—
129C4×Dic3—41no—
130C4×Dic3—41no—
131#208—41no—
132#208—41no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—41no—
135C22×Q8—21no—
136C22×Q8—31no—
137C2×M16—21no—
138C2×M16—61no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—122no—
144C4⋊Q8—122no—
145C4×Q8—61no—
146C4×Q8—61no—
147C4×Q8—61no—
148C4×Q8—61no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—32no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—64no—
154C2×C4⋊C4C4×D312yes—
155C2×C42—32no—
156C2×C42—61no—
157C2×C42C22×D311yescharacteristic
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—32no—
161C4⋊C8—64no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8#6712yes—
165#101—122no—
166C22⋊C8—21no—
167C22⋊C8—61no—
168C4×C8—21no—
169C4×C8—64no—
170C4×C8—61no—
171#95—32no—
172#95—64no—
173C23×C3C2×C22⋊C411yescharacteristic
174C3×Q8—81no—
175C3×Q8—81no—
176C3×Q8—81no—
177C3×Q8—81no—
178C2×C3×C4—22no—
179C2×C3×C4—41no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
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×D811yescharacteristic
190C2×C3×C4C2×SD1611yescharacteristic
191C2×C3×C4#13511yescharacteristic
192C2×C3×C4#13511yescharacteristic
193C2×Dic3—22no—
194C2×Dic3—22no—
195C2×Dic3—24no—
196C2×Dic3—22no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3#10214yes—
201Dic6—22no—
202Dic6—22no—
203Dic6—22no—
204Dic6—22no—
205C3×C8—42no—
206C3×C8—81no—
207C3⋊C8—42no—
208C3⋊C8—81no—
209C2×Q8—32no—
210C2×Q8—32no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×Q8—61no—
221C22×C4—22no—
222C22×C4—32no—
223C22×C4—32no—
224C22×C4—62no—
225C22×C4C2×C4×D311yescharacteristic
226C22×C4D3×D411yescharacteristic
227C22×C4D3×D411yescharacteristic
228M16—41no—
229M16—41no—
230M16—121no—
231M16—121no—
232C2×C8—22no—
233C2×C8—41no—
234C2×C8—62no—
235C2×C8—121no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—64no—
241C4⋊C4—62no—
242C4⋊C4—61no—
243C4⋊C4—62no—
244C4⋊C4—61no—
245C4⋊C4—128no—
246C42—62no—
247C42—62no—
248C42—121no—
249C42—121no—
250C42#23511yescharacteristic
251C42D3×D411yescharacteristic
252C42#23711yescharacteristic
253C42C2×#6711yescharacteristic
254C22×C3—21no—
255C22×C3—21no—
256C22×C3#38611yescharacteristic
257C22×C3C2×#10211yescharacteristic
258C22×C3#41611yescharacteristic
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—82no—
263C3×C4#45612yes—
264C3×C4#46511yescharacteristic
265C3×C4#46511yescharacteristic
266C3×C4#46611yescharacteristic
267C3×C4#46611yescharacteristic
268Dic3—24no—
269Dic3—44no—
270Dic3—42no—
271C23D3×C22⋊C411yescharacteristic
272Q8—62no—
273Q8—62no—
274Q8—62no—
275Q8—62no—
276Q8—81no—
277Q8—81no—
278Q8—81no—
279Q8—81no—
280C2×C4—22no—
281C2×C4—34no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—62no—
287C2×C4—62no—
288C2×C4—64no—
289C2×C4—62no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4#67⋊C411yescharacteristic
294C2×C4#88311yescharacteristic
295C2×C4#88411yescharacteristic
296C2×C4D3×D811yescharacteristic
297C2×C4#91211yescharacteristic
298C2×C4D3×SD1611yescharacteristic
299C2×C4#91611yescharacteristic
300C2×C4C4×#6711yescharacteristic
301C2×C4#93711yescharacteristic
302C2×C4#93811yescharacteristic
303C8—42no—
304C8—81no—
305C8—122no—
306C8—241no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3#188611yescharacteristic
310C2×C3#189311yescharacteristic
311C2×C3#191511yescharacteristic
312C22—21no—
313C22—21no—
314C22D3×#10211yescharacteristic
315C22#477211yescharacteristic
316C22#495611yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—64no—
321C4—82no—
322C4—124no—
323C4—122no—
324C4#477511yescharacteristic
325C4#477811yescharacteristic
326C4#515711yescharacteristic
327C4#517111yescharacteristic
328C4#520512yes—
329C3#939411yescharacteristic, complemented, Sylow
330C2—21no—
331C2—21no—
332C2#7012411yescharacteristic
333C2#7020211yescharacteristic
334C2#7034511yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1546D3
hypocenterC3#9394
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
329. C3#9394semidirect

Maximal subgroups

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

subgroupindex
#832392
#792632
C3×#15462
#717732
#705282
#677462
#677152
#93943