Finite Group Databasebeta

← #10693066

Subgroups — #10693066

Order 768.

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

normal subgroups only

This group has 544 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#82258C211yesmaximal, characteristic
3#71572C211yesmaximal, characteristic
4#69857C212yesmaximal, complemented
5#67666C212yesmaximal, complemented
6#9491—31nomaximal, Sylow
7C22×C3×M16—21no—
8C2×C3×#117C2211yescharacteristic
9C2×C4×Dic6C2211yescharacteristic
10C3×#405—24no—
11C3×C4×M16—21no—
12C3×C4×M16C2212yes—
13#5210—21no—
14#4981C411yescharacteristic
15#4979C411yescharacteristic
16#4978—21no—
17#4953C411yescharacteristic
18#4947C411yescharacteristic
19#4944—21no—
20#4937C2211yescharacteristic
21#4933—21no—
22#4926C2212yes—
23#4528—21no—
24#4525—21no—
25#4456—24no—
26#2873D311yescharacteristic
27#2599—31no—
28#1491—32no—
29#1464—31no—
30#1288—32no—
31C23×C3×C4D411yescharacteristic
32C22×Dic6C2×C411yescharacteristic
33C2×C3×M16—24no—
34C2×C3×M16—24no—
35C2×C3×M16—22no—
36C2×C3×M16—22no—
37C22×C3×C8—22no—
38C3×#117—22no—
39C3×#117—22no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4D411yescharacteristic
44C2×C3×C42D411yescharacteristic
45C2×C3×C42C2311yescharacteristic
46C2×#217—21no—
47C2×#217—21no—
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C4×Dic6—22no—
54C4×Dic6—22no—
55C3×#97—22no—
56C3×#97—22no—
57C3×C4×C8—22no—
58C3×C4×C8D414yes—
59C3⋊#95—21no—
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×C4×Q8—31no—
72C2×#117C2×D311yescharacteristic
73#432—32no—
74#405—24no—
75C4×M16—21no—
76C4×M16C2×D312yes—
77#398—31no—
78#397—31no—
79#395—31no—
80#393—31no—
81#388—61no—
82#387—61no—
83#387—61no—
84#385—61no—
85#383—31no—
86Q8⋊C8—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—
104C2×Dic6—22no—
105C2×Dic6—42no—
106C3×M16—28no—
107C3×M16—24no—
108C3×M16—24no—
109C2×C3×C8—28no—
110C2×C3×C8—24no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—22no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4—22no—
115C3×C42—21no—
116C3×C42—21no—
117C3×C42C2×D412yes—
118C3×C42#4112yes—
119#217—42no—
120#217—42no—
121C4⋊Dic3—22no—
122Dic3⋊C4—42no—
123Dic3⋊C4—42no—
124Dic3⋊C4—41no—
125Dic3⋊C4—41no—
126C4×Dic3—42no—
127C2×C3⋊C8—42no—
128C22×Q8—31no—
129C23×C4#6711yescharacteristic
130C2×M16—22no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—22no—
134C22×C8—22no—
135C4×Q8—62no—
136C4×Q8—62no—
137#117—22no—
138#117—22no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4C4×D311yescharacteristic
144C2×C4⋊C4D1211yescharacteristic
145C2×C22⋊C4—61no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4C4×D311yescharacteristic
148C2×C22⋊C4D1211yescharacteristic
149C2×C42—61no—
150C2×C42#6711yescharacteristic
151C2×C42C22×D311yescharacteristic
152C4⋊C8—62no—
153C22⋊C8—62no—
154#97—22no—
155#97—22no—
156#97—62no—
157C4×C8—22no—
158C4×C8#6714yes—
159#95—31no—
160#95—31no—
161#95—61no—
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—
203Dic6—24no—
204Dic6—42no—
205C3×C8—28no—
206C3⋊C8—82no—
207C24(C2×D3)⋊C411yescharacteristic
208C2×Q8—62no—
209C2×Q8—122no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4(C2×D3)⋊C411yescharacteristic
219C22×C4(C2×D3)⋊C411yescharacteristic
220C22×C4(C2×D3)⋊C411yescharacteristic
221C22×C4C2×C4×D311yescharacteristic
222C22×C4C2×D1211yescharacteristic
223C22×C4C2×#6711yescharacteristic
224M16—28no—
225M16—24no—
226M16—24no—
227C2×C8—28no—
228C2×C8—24no—
229C2×C8—122no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—62no—
233C4⋊C4—122no—
234C4⋊C4—122no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—122no—
240C22⋊C4—122no—
241C42—21no—
242C42—21no—
243C42—122no—
244C42#23512yes—
245C42C2×#6712yes—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—22no—
249C22×C3—42no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—81no—
253C22×C3#38611yescharacteristic
254C22×C3C2×#10411yescharacteristic
255C22×C3C2×#10411yescharacteristic
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—22no—
259C3×C4—21no—
260C3×C4—22no—
261C3×C4—41no—
262C3×C4—41no—
263Dic3—42no—
264Dic3—81no—
265Dic3—81no—
266Dic3—81no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23#80512yes—
272C23C2×(C2×D3)⋊C411yescharacteristic
273Q8—64no—
274Q8—122no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—62no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4#80512yes—
301C2×C4C4×#6712yes—
302C2×C4#92911yescharacteristic
303C2×C4#93012yes—
304C2×C4#95311yescharacteristic
305C8—28no—
306C8—242no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—42no—
310C2×C3#189111yescharacteristic
311C2×C3#189611yescharacteristic
312C2×C3#190911yescharacteristic
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22C2×#80511yescharacteristic
321C22C2×#80511yescharacteristic
322C22#521311yescharacteristic
323C4—22no—
324C4—22no—
325C4—21no—
326C4—22no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—122no—
331C4—241no—
332C4—241no—
333C4—241no—
334C3#949111yescharacteristic, complemented, Sylow
335C2—21no—
336C2—21no—
337C2—42no—
338C2#7153211yescharacteristic
339C2#7202411yescharacteristic
340C2#7202911yescharacteristic
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#9491
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. #69857C2semidirect
5. #67666C2semidirect
334. C3#9491semidirect

Maximal subgroups

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

subgroupindex
C3×#28732
#822582
#715722
#698572
#676662
#94913