Finite Group Databasebeta

← #10692527

Subgroups — #10692527

Order 768.

532 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#2873C211yesmaximal, characteristic
2#82261C211yesmaximal, characteristic
3#71572C211yesmaximal, characteristic
4#69863C212yesmaximal, complemented
5#67665C212yesmaximal, complemented
6#9388—31nomaximal, Sylow
7C22×C3×M16—21no—
8C2×C3×#117C2211yescharacteristic
9C2×#870C2211yescharacteristic
10C3×#405—24no—
11C3×C4×M16—21no—
12C3×C4×M16C2212yes—
13#5210—21no—
14#4986C411yescharacteristic
15#4981C411yescharacteristic
16#4976—21no—
17#4953C411yescharacteristic
18#4949C411yescharacteristic
19#4945—21no—
20#4933—21no—
21#4932C2211yescharacteristic
22#4926C2212yes—
23#4528—21no—
24#4525—21no—
25#4465—24no—
26#2873D311yescharacteristic
27#2611—31no—
28#1528—32no—
29#1464—31no—
30#1300—32no—
31C23×C3×C4D411yescharacteristic
32C2×C3×M16—24no—
33C2×C3×M16—24no—
34C2×C3×M16—22no—
35C2×C3×M16—22no—
36C22×C3×C8—22no—
37C3×#117—22no—
38C3×#117—22no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C22⋊C4C2×C411yescharacteristic
42C2×C3×C22⋊C4D411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×C3×C42C2311yescharacteristic
45C2×#217—21no—
46C2×#217—21no—
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3—21no—
54#870—22no—
55#870—22no—
56C3×#97—22no—
57C3×#97D414yes—
58C3×C4×C8—22no—
59C3×C4×C8—22no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95C2×C411yescharacteristic
66C3⋊#95C2×C411yescharacteristic
67#830—22no—
68C4⋊(C3⋊C8)—22no—
69#803—22no—
70C22×M16—21no—
71C2×#125—31no—
72C2×#117C2×D311yescharacteristic
73#432—32no—
74#405—24no—
75C4×M16—21no—
76C4×M16C2×D312yes—
77#400—31no—
78#399—31no—
79#398—31no—
80#397—31no—
81#389—61no—
82#388—61no—
83#387—61no—
84#385—61no—
85#376—31no—
86#330—64no—
87#324—61no—
88#323—61no—
89C24×C3C22⋊C411yescharacteristic
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C3×M16—28no—
105C3×M16—24no—
106C3×M16—24no—
107C2×C3×C8—28no—
108C2×C3×C8—24no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C42—21no—
114C3×C42—21no—
115C3×C42C2×D412yes—
116C3×C42#4112yes—
117#217—42no—
118#217—42no—
119C4⋊Dic3—22no—
120C4⋊Dic3—22no—
121Dic3⋊C4—42no—
122Dic3⋊C4—42no—
123Dic3⋊C4—41no—
124Dic3⋊C4—42no—
125Dic3⋊C4—41no—
126C4×Dic3—42no—
127C2×C3⋊C8—42no—
128C23×C4#6711yescharacteristic
129C2×M16—22no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—22no—
133C22×C8—22no—
134#125—62no—
135#125—62no—
136#117—22no—
137#117—22no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4C4×D311yescharacteristic
145C2×C4⋊C4D1211yescharacteristic
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4C4×D311yescharacteristic
149C2×C22⋊C4D1211yescharacteristic
150C2×C42—61no—
151C2×C42#6711yescharacteristic
152C2×C42C22×D311yescharacteristic
153C4⋊C8—62no—
154C22⋊C8—62no—
155#97—22no—
156#97—62no—
157#97#6714yes—
158C4×C8—22no—
159C4×C8—22no—
160#95—31no—
161#95—31no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3#10412yes—
172C23×C3C2×C22⋊C411yescharacteristic
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—24no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4#10412yes—
187C2×C3×C4C4×D412yes—
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12112yes—
190C2×C3×C4#12311yescharacteristic
191C2×Dic3—22no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C3×C8—28no—
204C3⋊C8—82no—
205C24(C2×D3)⋊C411yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4(C2×D3)⋊C411yescharacteristic
215C22×C4(C2×D3)⋊C411yescharacteristic
216C22×C4(C2×D3)⋊C411yescharacteristic
217C22×C4C2×C4×D311yescharacteristic
218C22×C4C2×D1211yescharacteristic
219C22×C4C2×#6711yescharacteristic
220M16—28no—
221M16—24no—
222M16—24no—
223C2×C8—28no—
224C2×C8—24no—
225C2×C8—122no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—62no—
229C4⋊C4—62no—
230C4⋊C4—122no—
231C4⋊C4—122no—
232C4⋊C4—121no—
233C4⋊C4—122no—
234C4⋊C4—121no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—122no—
238C22⋊C4—122no—
239C42—21no—
240C42—21no—
241C42—122no—
242C42#23512yes—
243C42C2×#6712yes—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—22no—
247C22×C3—42no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—81no—
251C22×C3#38611yescharacteristic
252C22×C3C2×#10411yescharacteristic
253C22×C3C2×#10411yescharacteristic
254C3×C4—22no—
255C3×C4—21no—
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—22no—
259C3×C4—41no—
260C3×C4—41no—
261Dic3—42no—
262Dic3—81no—
263Dic3—81no—
264Dic3—81no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23#80512yes—
270C23C2×(C2×D3)⋊C411yescharacteristic
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—62no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4#80512yes—
297C2×C4C4×#6712yes—
298C2×C4#92911yescharacteristic
299C2×C4#93012yes—
300C2×C4#95311yescharacteristic
301C8—28no—
302C8—242no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—42no—
306C2×C3#188411yescharacteristic
307C2×C3#189611yescharacteristic
308C2×C3#191511yescharacteristic
309C22—21no—
310C22—21no—
311C22—22no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22C2×#80511yescharacteristic
317C22C2×#80511yescharacteristic
318C22#521311yescharacteristic
319C4—22no—
320C4—22no—
321C4—21no—
322C4—22no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—122no—
327C4—241no—
328C4—241no—
329C4—241no—
330C3#938811yescharacteristic, complemented, Sylow
331C2—21no—
332C2—21no—
333C2—42no—
334C2#7153211yescharacteristic
335C2#7202111yescharacteristic
336C2#7203111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2873C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2873D3
hypocenterC3#9388
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2873

Complements

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

NH ≅ G/Ntype
4. #69863C2semidirect
5. #67665C2semidirect
330. C3#9388semidirect

Maximal subgroups

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

subgroupindex
C3×#28732
#822612
#715722
#698632
#676652
#93883