Finite Group Databasebeta

← #10692604

Subgroups — #10692604

Order 768.

450 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 109 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#83244C211yesmaximal, characteristic
2#79264C211yesmaximal, characteristic
3C3×#1549C211yesmaximal, characteristic
4#71775C211yesmaximal, characteristic
5#70528C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic
7#67715C211yesmaximal, characteristic
8#9404—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5229—22no—
14C2×#824—21no—
15#4985C414yes—
16#4977—21no—
17#4973—21no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4925C2211yescharacteristic
20#4924C2211yescharacteristic
21#4831—22no—
22#4719—22no—
23#4679—22no—
24C3×#332—22no—
25#4554—21no—
26#4547—22no—
27#2983—31no—
28#2568—31no—
29#1801—31no—
30#1549—31no—
31#1549D311yescharacteristic
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
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3D411yescharacteristic
46C2×Dic3⋊C4C2×C412yes—
47C2×C4×Dic3—21no—
48C2×C4×Dic3D412yes—
49C2×#208—21no—
50C4×Dic6—21no—
51C4×Dic6—21no—
52C4×Dic6—21no—
53C4×Dic6—21no—
54C3×C4⋊C8—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8D412yes—
57C3×C22⋊C8—21no—
58C3×C4×C8—21no—
59C3⋊#95—24no—
60C3⋊#95—24no—
61C3⋊#95C2×C412yes—
62#830—21no—
63#824—42no—
64#815—24no—
65Dic3⋊C8—24no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)D412yes—
69C4×C3⋊C8—21no—
70C2×C4⋊Q8C2×D311yescharacteristic
71C2×C4×Q8—31no—
72#446—62no—
73#446—62no—
74#432—62no—
75#431—31no—
76#431C2×D311yescharacteristic
77#423—31no—
78#423C2×D311yescharacteristic
79C2×#101—61no—
80#400—34no—
81#391—62no—
82#391—61no—
83#382—61no—
84C4×C4⋊C4—31no—
85#343—61no—
86#332—22no—
87#332—62no—
88C2×C3×Q8—44no—
89C2×C3×Q8—41no—
90C2×C3×Q8—41no—
91C22×C3×C4—22no—
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×Dic3—22no—
96C22×Dic3C22⋊C412yes—
97C22×Dic3C22⋊C412yes—
98C2×Dic6—21no—
99C2×Dic6—21no—
100C2×Dic6—21no—
101C2×Dic6—21no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C3×M16—41no—
105C3×M16—41no—
106C2×C3×C8—22no—
107C2×C3×C8—41no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C42C2×D411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42#4111yescharacteristic
115C3×C42#4111yescharacteristic
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—42no—
119Dic3⋊C4—22no—
120Dic3⋊C4—22no—
121C4×Dic3—22no—
122C4×Dic3—22no—
123C4×Dic3—41no—
124C4×Dic3—41no—
125#208—41no—
126#208—41no—
127C2×C3⋊C8—22no—
128C2×C3⋊C8—41no—
129C22×Q8—21no—
130C22×Q8—31no—
131C2×M16—21no—
132C2×M16—61no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4×Q8—61no—
138C4×Q8—61no—
139C4×Q8—61no—
140C4×Q8—61no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—32no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4C4×D312yes—
146C2×C42—32no—
147C2×C42—61no—
148C2×C42C22×D311yescharacteristic
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—32no—
152C4⋊C8—64no—
153C4⋊C8—61no—
154C4⋊C8—61no—
155C4⋊C8#6712yes—
156#101—122no—
157C22⋊C8—21no—
158C22⋊C8—61no—
159#97—64no—
160C4×C8—21no—
161C4×C8—61no—
162#95—32no—
163#95—64no—
164#95—64no—
165C23×C3C2×C22⋊C411yescharacteristic
166C3×Q8—82no—
167C3×Q8—82no—
168C2×C3×C4—22no—
169C2×C3×C4—41no—
170C2×C3×C4—44no—
171C2×C3×C4—42no—
172C2×C3×C4C4×D411yescharacteristic
173C2×C3×C4C4×D411yescharacteristic
174C2×C3×C4#12011yescharacteristic
175C2×C3×C4#12111yescharacteristic
176C2×C3×C4#12111yescharacteristic
177C2×C3×C4#12311yescharacteristic
178C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
179C2×C3×C4#13711yescharacteristic
180C2×C3×C4#13711yescharacteristic
181C2×C3×C4#13711yescharacteristic
182C2×Dic3—22no—
183C2×Dic3—22no—
184C2×Dic3—22no—
185C2×Dic3—22no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188C2×Dic3—42no—
189C2×Dic3—42no—
190Dic6—41no—
191Dic6—41no—
192Dic6—41no—
193Dic6—41no—
194C3×C8—42no—
195C3×C8—81no—
196C3⋊C8—42no—
197C3⋊C8—81no—
198C2×Q8—44no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—61no—
202C2×Q8—61no—
203C2×Q8—61no—
204C2×Q8—61no—
205C2×Q8—61no—
206C2×Q8—61no—
207C22×C4—22no—
208C22×C4—32no—
209C22×C4—32no—
210C22×C4—62no—
211C22×C4C2×C4×D311yescharacteristic
212C22×C4D3×D411yescharacteristic
213C22×C4D3×D411yescharacteristic
214M16—41no—
215M16—41no—
216M16—121no—
217M16—121no—
218C2×C8—22no—
219C2×C8—41no—
220C2×C8—62no—
221C2×C8—121no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—62no—
227C4⋊C4—61no—
228C4⋊C4—62no—
229C4⋊C4—61no—
230C4⋊C4—122no—
231C42—62no—
232C42—62no—
233C42—121no—
234C42—121no—
235C42#23511yescharacteristic
236C42D3×D411yescharacteristic
237C42#23711yescharacteristic
238C42C2×#6711yescharacteristic
239C22×C3—21no—
240C22×C3—21no—
241C22×C3#38611yescharacteristic
242C22×C3#41711yescharacteristic
243C22×C3#41911yescharacteristic
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—82no—
248C3×C4#45412yes—
249C3×C4#46811yescharacteristic
250C3×C4#46811yescharacteristic
251C3×C4#47011yescharacteristic
252C3×C4#47011yescharacteristic
253Dic3—42no—
254Dic3—42no—
255Dic3—82no—
256C23D3×C22⋊C411yescharacteristic
257Q8—82no—
258Q8—82no—
259Q8—121no—
260Q8—121no—
261Q8—121no—
262Q8—121no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—122no—
272C2×C4—122no—
273C2×C4—122no—
274C2×C4—122no—
275C2×C4#67⋊C411yescharacteristic
276C2×C4#88311yescharacteristic
277C2×C4#88411yescharacteristic
278C2×C4#91411yescharacteristic
279C2×C4#91511yescharacteristic
280C2×C4#91811yescharacteristic
281C2×C4#91811yescharacteristic
282C2×C4C4×#6711yescharacteristic
283C2×C4#93711yescharacteristic
284C2×C4#93811yescharacteristic
285C8—42no—
286C8—81no—
287C8—122no—
288C8—241no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3#188811yescharacteristic
292C2×C3#189411yescharacteristic
293C2×C3#191511yescharacteristic
294C22—21no—
295C22—21no—
296C22#480211yescharacteristic
297C22#480311yescharacteristic
298C22#495611yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—82no—
303C4—122no—
304C4—122no—
305C4—242no—
306C4#481211yescharacteristic
307C4#481611yescharacteristic
308C4#517411yescharacteristic
309C4#518911yescharacteristic
310C4#520112yes—
311C3#940411yescharacteristic, complemented, Sylow
312C2—21no—
313C2—21no—
314C2#7012411yescharacteristic
315C2#7027111yescharacteristic
316C2#7032511yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1549C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1549D3
hypocenterC3#9404
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
311. C3#9404semidirect

Maximal subgroups

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

subgroupindex
#832442
#792642
C3×#15492
#717752
#705282
#677462
#677152
#94043