Finite Group Databasebeta

← #10691173

Subgroups — #10691173

Order 768.

478 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 99 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#82503C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic, complemented
3C3×C4×#101C211yesmaximal, characteristic, complemented
4#69860C212yesmaximal
5#67666C212yesmaximal, complemented
6#9188—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#875C2211yescharacteristic
9C2×C3×#101—21no—
10C3×C4×M16C2212yes—
11#5231—21no—
12#5024—24no—
13#4993—21no—
14#4991C411yescharacteristic
15#4991C411yescharacteristic
16#4979C411yescharacteristic
17#4979C411yescharacteristic
18#4940—21no—
19#4940—21no—
20#4937C2211yescharacteristic
21#4926—21no—
22#4926C2212yes—
23#4925—21no—
24C3×#343—21no—
25#4464—24no—
26#2983—31no—
27#2632—31no—
28C4×#101D311yescharacteristic, complemented
29#1527—32no—
30#1288—32no—
31C22×C3×Q8D411yescharacteristic
32C22×D12C2×C411yescharacteristic
33C22×Dic6C2×C411yescharacteristic
34C2×C3×M16—22no—
35C3×C4×Q8—22no—
36C3×C4×Q8—22no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×C3×C42C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51#875—22no—
52#875—22no—
53C3×#101—24no—
54C3×#97—22no—
55C3×C4×C8—22no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95C2×C411yescharacteristic
61C3⋊#95C2×C411yescharacteristic
62#830—22no—
63C4⋊(C3⋊C8)—22no—
64C4⋊(C3⋊C8)—22no—
65C4⋊(C3⋊C8)—22no—
66#803—22no—
67#803D414yes—
68C4×C3⋊C8—22no—
69C2×#124—31no—
70C2×C4×Q8C2×D311yescharacteristic
71#446—64no—
72#432—32no—
73#432—61no—
74#431—61no—
75C2×#101—21no—
76C4×M16C2×D312yes—
77#399—61no—
78#395—31no—
79#395—31no—
80#394—31no—
81#394—31no—
82#393—61no—
83#388—61no—
84#388—61no—
85#383—31no—
86#343—21no—
87#329—64no—
88C23×D3—21no—
89C2×C3×Q8—22no—
90C2×C3×Q8—22no—
91C2×C3×Q8—41no—
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C2×D12—22no—
103C2×D12—81no—
104C2×Dic6—22no—
105C2×Dic6—81no—
106C3×M16—24no—
107C3×M16—42no—
108C2×C3×C8—42no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—22no—
112C3×C42—21no—
113C3×C42—21no—
114C3×C42—21no—
115C3×C42—21no—
116C3×C42C2×D412yes—
117C3×C42#4112yes—
118(C2×D3)⋊C4—42no—
119(C2×D3)⋊C4—42no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—42no—
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—81no—
124(C2×D3)⋊C4—81no—
125Dic3⋊C4—81no—
126Dic3⋊C4—81no—
127C2×C3⋊C8—24no—
128C22×Q8—31no—
129C22×Q8#6711yescharacteristic
130C22×D4—31no—
131C2×M16—22no—
132#124—62no—
133#124—62no—
134C4×Q8—22no—
135C4×Q8—22no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4C4×D311yescharacteristic
139C2×C4⋊C4C4×D311yescharacteristic
140C2×C4⋊C4#6711yescharacteristic
141C2×C22⋊C4—61no—
142C2×C22⋊C4—61no—
143C2×C22⋊C4—61no—
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4—61no—
147C2×C42D1211yescharacteristic
148C2×C42D1211yescharacteristic
149C2×C42C22×D311yescharacteristic
150C4⋊C8—62no—
151C4⋊C8—62no—
152C4⋊C8—62no—
153#101—24no—
154C22⋊C8—62no—
155#97—22no—
156#97—34no—
157#97—62no—
158C4×C8—22no—
159C4×C8—62no—
160#95—31no—
161#95—31no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166C23×C3C2×C22⋊C411yescharacteristic
167C22×D3—41no—
168C22×D3—41no—
169C22×D3—41no—
170C22×D3—41no—
171C22×D3—41no—
172C22×D3—41no—
173C22×D3—41no—
174C3×Q8—44no—
175C2×C3×C4—21no—
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—21no—
183C2×C3×C4—21no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4#10412yes—
189C2×C3×C4C4×D412yes—
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12112yes—
192C2×C3×C4#12311yescharacteristic
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—
202D12—82no—
203Dic6—82no—
204C3×C8—44no—
205C3⋊C8—44no—
206C24—61no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—41no—
210C2×Q8—62no—
211C2×Q8—241no—
212C2×D4—62no—
213C2×D4—241no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4(C2×D3)⋊C411yescharacteristic
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—24no—
225M16—42no—
226C2×C8—42no—
227C2×C8—64no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—241no—
232C4⋊C4—241no—
233C22⋊C4—122no—
234C22⋊C4—122no—
235C22⋊C4—121no—
236C22⋊C4—121no—
237C22⋊C4—122no—
238C22⋊C4—241no—
239C22⋊C4—241no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42D3×D412yes—
245C42#23712yes—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3#38611yescharacteristic
249C22×C3C2×#10411yescharacteristic
250C22×C3#42111yescharacteristic
251C2×D3—81no—
252C2×D3—81no—
253C2×D3—81no—
254C2×D3—81no—
255C2×D3—81no—
256C2×D3—81no—
257C2×D3—81no—
258C3×C4—22no—
259C3×C4—22no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—22no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—41no—
266Dic3—81no—
267Dic3—81no—
268Dic3—81no—
269C23—121no—
270C23—121no—
271C23—121no—
272C23—121no—
273C23—121no—
274C23—121no—
275C23—121no—
276C23C2×(C2×D3)⋊C411yescharacteristic
277Q8—44no—
278Q8—242no—
279D4—242no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4#80512yes—
303C2×C4#67⋊C412yes—
304C2×C4#88211yescharacteristic
305C2×C4#88611yescharacteristic
306C2×C4#93912yes—
307C8—44no—
308C8—124no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3#187111yescharacteristic
312C2×C3#191211yescharacteristic
313C2×C3#191411yescharacteristic
314D3—161no—
315C22—21no—
316C22—21no—
317C22—241no—
318C22—241no—
319C22—241no—
320C22—241no—
321C22—241no—
322C22—241no—
323C22—241no—
324C22C2×#80511yescharacteristic
325C22#495811yescharacteristic
326C22#500511yescharacteristic
327C4—22no—
328C4—22no—
329C4—21no—
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—241no—
336C4—241no—
337C4—241no—
338C3#918811yescharacteristic, complemented, Sylow
339C2—21no—
340C2—21no—
341C2—481no—
342C2#6945311yescharacteristic
343C2#7010511yescharacteristic
344C2#7013711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×#101C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenterC4×#101D3
hypocenterC3#9188
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. #82436C2semidirect
3. C3×C4×#101C2semidirect
5. #67666C2semidirect
28. C4×#101D3semidirect
338. C3#9188semidirect

Maximal subgroups

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

subgroupindex
#825032
#824362
C3×C4×#1012
#698602
#676662
#91883