Finite Group Databasebeta

← #10692532

Subgroups — #10692532

Order 768.

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

normal subgroups only

This group has 496 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
1C3×#2921C211yesmaximal, characteristic, complemented
2#82260C211yesmaximal, characteristic
3#71572C211yesmaximal, characteristic, complemented
4#69860C212yesmaximal, complemented
5#67665C212yesmaximal, complemented
6#9389—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#875C2211yescharacteristic
9C3×#433—24no—
10C3×#423—21no—
11C3×#407—21no—
12C3×C4×M16C2212yes—
13#5213—21no—
14#4991C411yescharacteristic
15#4990—21no—
16#4979C411yescharacteristic
17#4960C411yescharacteristic
18#4957—21no—
19#4947C411yescharacteristic
20#4940—21no—
21#4932C2211yescharacteristic
22#4926C2212yes—
23#4528—21no—
24#4525—21no—
25#4464—24no—
26#2921D311yescharacteristic, complemented
27#2610—31no—
28#1527—32no—
29#1464—31no—
30#1300—32no—
31C23×C3×C4D411yescharacteristic
32C22×#67—21no—
33C22×D12C2×C411yescharacteristic
34C22×Dic6C2×C411yescharacteristic
35C2×C3×M16—22no—
36C3×#117—22no—
37C3×#117—22no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C22⋊C4C2×C411yescharacteristic
41C2×C3×C22⋊C4D411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×C3×C42C2311yescharacteristic
44C2×#217—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52#875—22no—
53#875—22no—
54C3×C4⋊C8—24no—
55C3×C22⋊C8—24no—
56C3×#97D414yescomplemented
57C3×C4×C8—22no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95C2×C411yescharacteristic
62C3⋊#95C2×C411yescharacteristic
63#830—22no—
64C4⋊(C3⋊C8)—22no—
65#803—22no—
66C2×#124—31no—
67C2×#117C2×D311yescharacteristic
68#433—24no—
69#432—32no—
70#423—21no—
71#407—21no—
72C4×M16C2×D312yes—
73#395—31no—
74#394—31no—
75#393—31no—
76#392—31no—
77#390—61no—
78#388—61no—
79#388—61no—
80#386—61no—
81#376—31no—
82#329—64no—
83#324—61no—
84#323—61no—
85C24×C3C22⋊C411yescharacteristic
86C23×D3—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C2×#67—42no—
98C2×#67—42no—
99C2×#67—81no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C2×D12—22no—
104C2×D12—81no—
105C2×Dic6—22no—
106C2×Dic6—81no—
107C3×M16—42no—
108C3×M16—42no—
109C2×C3×C8—24no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—22no—
114C3×C42—21no—
115C3×C42—21no—
116C3×C42C2×D412yes—
117C3×C42#4112yes—
118#217—81no—
119(C2×D3)⋊C4—42no—
120(C2×D3)⋊C4—42no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—42no—
123(C2×D3)⋊C4—41no—
124(C2×D3)⋊C4—81no—
125Dic3⋊C4—81no—
126C4×Dic3—42no—
127C2×C3⋊C8—42no—
128C22×Q8—31no—
129C22×D4—31no—
130C22×D4—61no—
131C23×C4#6711yescharacteristic
132C2×M16—22no—
133#124—62no—
134#124—62no—
135#117—22no—
136#117—22no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4C4×D311yescharacteristic
139C2×C4⋊C4D1211yescharacteristic
140C2×C22⋊C4—61no—
141C2×C22⋊C4—61no—
142C2×C22⋊C4—61no—
143C2×C22⋊C4—61no—
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4C4×D311yescharacteristic
147C2×C22⋊C4D1211yescharacteristic
148C2×C42—61no—
149C2×C42#6711yescharacteristic
150C2×C42C22×D311yescharacteristic
151C4⋊C8—24no—
152C4⋊C8—62no—
153C22⋊C8—24no—
154C22⋊C8—62no—
155#97—62no—
156#97#6714yescomplemented
157C4×C8—22no—
158#95—31no—
159#95—31no—
160#95—61no—
161#95—61no—
162#95—61no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3C2×C22⋊C411yescharacteristic
169C22×D3—41no—
170C22×D3—41no—
171C22×D3—41no—
172C22×D3—41no—
173C22×D3—41no—
174C22×D3—41no—
175C22×D3—41no—
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—41no—
184C2×C3×C4—41no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4C4×D412yes—
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12112yes—
192C2×C3×C4#12311yescharacteristic
193#67—84no—
194C2×Dic3—22no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203D12—82no—
204Dic6—82no—
205C3×C8—44no—
206C3⋊C8—82no—
207C24—61no—
208C24(C2×D3)⋊C411yescharacteristic
209C2×Q8—62no—
210C2×Q8—241no—
211C2×D4—62no—
212C2×D4—122no—
213C2×D4—122no—
214C2×D4—241no—
215C2×D4—241no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4(C2×D3)⋊C411yescharacteristic
224C22×C4(C2×D3)⋊C411yescharacteristic
225C22×C4(C2×D3)⋊C411yescharacteristic
226C22×C4C2×C4×D311yescharacteristic
227C22×C4C2×D1211yescharacteristic
228C22×C4C2×#6711yescharacteristic
229M16—42no—
230M16—42no—
231C2×C8—24no—
232C2×C8—122no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—241no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—122no—
239C22⋊C4—122no—
240C22⋊C4—121no—
241C22⋊C4—122no—
242C22⋊C4—121no—
243C22⋊C4—241no—
244C22⋊C4—241no—
245C42—21no—
246C42—21no—
247C42—122no—
248C42#23512yes—
249C42C2×#6712yes—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—42no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—81no—
257C22×C3#38611yescharacteristic
258C22×C3#42111yescharacteristic
259C22×C3#42111yescharacteristic
260C2×D3—81no—
261C2×D3—81no—
262C2×D3—81no—
263C2×D3—81no—
264C2×D3—81no—
265C2×D3—81no—
266C2×D3—81no—
267C3×C4—22no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—22no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274Dic3—42no—
275Dic3—81no—
276Dic3—81no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—121no—
283C23—121no—
284C23—121no—
285C23—121no—
286C23—121no—
287C23—121no—
288C23—121no—
289C23C2×(C2×D3)⋊C411yescharacteristic
290Q8—242no—
291D4—242no—
292D4—244no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—62no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4C4×#6712yes—
316C2×C4#92911yescharacteristic
317C2×C4#93012yes—
318C2×C4#95311yescharacteristic
319C8—44no—
320C8—242no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—81no—
324C2×C3#188411yescharacteristic
325C2×C3#189711yescharacteristic
326C2×C3#191411yescharacteristic
327D3—161no—
328C22—21no—
329C22—21no—
330C22—42no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—241no—
336C22—241no—
337C22—241no—
338C22—241no—
339C22—241no—
340C22—241no—
341C22—241no—
342C22#500511yescharacteristic
343C22#500511yescharacteristic
344C22#521311yescharacteristic
345C4—22no—
346C4—22no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—122no—
353C4—241no—
354C4—241no—
355C3#938911yescharacteristic, complemented, Sylow
356C2—21no—
357C2—21no—
358C2—81no—
359C2—481no—
360C2#7159111yescharacteristic
361C2#7202111yescharacteristic
362C2#7203011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2921C2
socleC22×C3#386
prime radical#4960C4
cosocleC2×C3×C42C23
hypercenter#2921D3
hypocenterC3#9389
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
1. C3×#2921C2semidirect
3. #71572C2semidirect
4. #69860C2semidirect
5. #67665C2semidirect
26. #2921D3semidirect
56. C3×#97D4semidirect
156. #97#67semidirect
355. C3#9389semidirect

Maximal subgroups

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

subgroupindex
C3×#29212
#822602
#715722
#698602
#676652
#93893