Finite Group Databasebeta

← #10692966

Subgroups — #10692966

Order 768.

716 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2973C211yesmaximal, characteristic
2#82349C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic
4#69963C212yesmaximal, complemented
5#67676C212yesmaximal, complemented
6#9471—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C2×#924—21no—
10C3×#435—24no—
11C3×#432—21no—
12C3×#407—21no—
13C2×C3×C22⋊C8C2212yes—
14#5220—21no—
15C2×#834—21no—
16#5212C2212yes—
17#5212C2211yescharacteristic
18C2×#816—24no—
19#4979C412yescomplemented
20#4947C412yescomplemented
21#4946—21no—
22#4929—21no—
23#4526—21no—
24#2973D311yescharacteristic
25#2409—31no—
26#1799—31no—
27#1787—32no—
28#1289—32no—
29C22×C3×D4D411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C23×Dic3—21no—
33C22×Dic6—22no—
34C22×Dic6—21no—
35C22×Dic6—22no—
36C22×Dic6—44no—
37C22×Dic6C2×C412yes—
38C2×C3×M16—22no—
39C22×C3×C8D414yes—
40C3×C4×D4—22no—
41C3×C4×D4—22no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×#217—22no—
47C2×#217—21no—
48C2×#217—22no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—22no—
51#924—44no—
52C2×Dic3⋊C4—22no—
53C3×C4⋊C8—22no—
54C3×C22⋊C8—22no—
55C3×C22⋊C8—22no—
56C3×C22⋊C8—22no—
57C3×#97—22no—
58#834—42no—
59C3⋊#95—22no—
60C3⋊#95—22no—
61C3⋊#95—22no—
62C3⋊#95C2×C412yes—
63#816—48no—
64C23×Q8—31no—
65C2×C22⋊Q8—61no—
66C2×C4×D4C2×D311yescharacteristic
67#435—24no—
68#432—21no—
69C2×#103—64no—
70C2×#99—61no—
71#407—21no—
72C2×C22⋊C8C2×D312yes—
73#395—32no—
74#393—32no—
75#391—61no—
76#386—61no—
77#385—61no—
78#380—31no—
79#380—32no—
80#342—61no—
81C24×C3C22⋊C411yescharacteristic
82C24×C3C2×D411yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—21no—
85C2×C3×D4—21no—
86C2×C3×D4—22no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—42no—
90C22×C3×C4—42no—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D412yes—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4#4112yes—
98C22×Dic3—22no—
99C22×Dic3—22no—
100C22×Dic3—44no—
101C22×Dic3—42no—
102C22×Dic3—42no—
103C2×Dic6—28no—
104C2×Dic6—24no—
105C2×Dic6—24no—
106C2×Dic6—44no—
107C2×Dic6—48no—
108C2×Dic6—44no—
109C2×Dic6—42no—
110C2×Dic6—41no—
111C2×Dic6—41no—
112C3×M16—82no—
113C2×C3×C8—24no—
114C2×C3×C8—44no—
115C3×C4⋊C4—22no—
116C3×C22⋊C4—22no—
117C3×C22⋊C4—22no—
118C3×C42—21no—
119C3×C42—21no—
120#217—44no—
121#217—41no—
122#217—41no—
123#217—42no—
124#217—42no—
125C4⋊Dic3—42no—
126C4⋊Dic3—44no—
127Dic3⋊C4—44no—
128C22×Q8—32no—
129C22×Q8—61no—
130C22×Q8—62no—
131C22×Q8—62no—
132C22×Q8—124no—
133C22×D4D1211yescharacteristic
134C23×C4—61no—
135C23×C4#6711yescharacteristic
136C23×C4C22×D311yescharacteristic
137C2×M16—22no—
138C22×C8#6714yes—
139C22⋊Q8—124no—
140C4×D4—22no—
141C4×D4—22no—
142C2×C4⋊C4—62no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—62no—
145C2×C4⋊C4C4×D311yescharacteristic
146C2×C22⋊C4—62no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—62no—
149C2×C22⋊C4C4×D311yescharacteristic
150C2×C22⋊C4D1211yescharacteristic
151C2×C42#6711yescharacteristic
152C4⋊C8—22no—
153#103—128no—
154#99—122no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157C22⋊C8—22no—
158#97—22no—
159#95—32no—
160#95—62no—
161#95—62no—
162#95—62no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3C2×C22⋊C411yescharacteristic
171C23×C3#12112yes—
172C23×C3AGL(1,ℤ/8)11yescharacteristic
173C23×C3#13711yescharacteristic
174C23×C3#13712yes—
175C3×D4—42no—
176C3×D4—42no—
177C2×C3×C4—24no—
178C2×C3×C4—22no—
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—42no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—81no—
190C2×C3×C4C4×D412yes—
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12311yescharacteristic
193C2×Dic3—24no—
194C2×Dic3—48no—
195C2×Dic3—42no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—82no—
201C2×Dic3—82no—
202C2×Dic3—82no—
203Dic6—28no—
204Dic6—44no—
205Dic6—44no—
206Dic6—44no—
207C3×C8—44no—
208C24(C2×D3)⋊C411yescharacteristic
209C24C2×D1211yescharacteristic
210C2×Q8—64no—
211C2×Q8—68no—
212C2×Q8—64no—
213C2×Q8—124no—
214C2×Q8—122no—
215C2×Q8—124no—
216C2×Q8—128no—
217C2×Q8—121no—
218C2×Q8—121no—
219C2×D4—21no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4—124no—
230C22×C4—122no—
231C22×C4—122no—
232C22×C4(C2×D3)⋊C411yescharacteristic
233C22×C4(C2×D3)⋊C411yescharacteristic
234C22×C4(C2×D3)⋊C411yescharacteristic
235C22×C4C2×C4×D311yescharacteristic
236C22×C4#23512yes—
237C22×C4C2×#6712yes—
238C22×C4C2×#6711yescharacteristic
239M16—82no—
240C2×C8—24no—
241C2×C8—44no—
242C4⋊C4—22no—
243C4⋊C4—122no—
244C4⋊C4—124no—
245C4⋊C4—124no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—124no—
249C22⋊C4—121no—
250C22⋊C4—121no—
251C22⋊C4—122no—
252C22⋊C4—122no—
253C42—21no—
254C42—21no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—21no—
258C22×C3—22no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—42no—
262C22×C3—42no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—81no—
267C22×C3#38611yescharacteristic
268C22×C3#41711yescharacteristic
269C22×C3#41911yescharacteristic
270C22×C3#46812yes—
271C22×C3#47012yes—
272C3×C4—22no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4B(4,𝔽2)12yes—
278Dic3—44no—
279Dic3—82no—
280Dic3—162no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#90811yescharacteristic
289C23#90912yes—
290C23#90911yescharacteristic
291C23C2×(C2×D3)⋊C411yescharacteristic
292C23#93012yes—
293Q8—68no—
294Q8—124no—
295Q8—124no—
296Q8—124no—
297D4—42no—
298D4—42no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—64no—
312C2×C4—81no—
313C2×C4—122no—
314C2×C4—128no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—242no—
320C2×C4—242no—
321C2×C4—242no—
322C2×C4C4×#6712yes—
323C2×C4#92911yescharacteristic
324C2×C4#95311yescharacteristic
325C8—44no—
326C2×C3—21no—
327C2×C3—22no—
328C2×C3—21no—
329C2×C3—22no—
330C2×C3—81no—
331C2×C3#188911yescharacteristic
332C2×C3#189411yescharacteristic
333C2×C3#190911yescharacteristic
334C22—21no—
335C22—22no—
336C22—21no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22#512011yescharacteristic
347C22#513311yescharacteristic
348C22#513412yes—
349C22#513712yes—
350C22#521311yescharacteristic
351C4—21no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—124no—
357C4—242no—
358C4—482no—
359C4#524412yes—
360C3#947111yescharacteristic, complemented, Sylow
361C2—21no—
362C2—22no—
363C2—21no—
364C2—22no—
365C2—81no—
366C2#7165411yescharacteristic
367C2#7198911yescharacteristic
368C2#7202411yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2973C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2973D3
hypocenterC3#9471
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #69963C2semidirect
5. #67676C2semidirect
19. #4979C4semidirect
20. #4947C4semidirect
360. C3#9471semidirect

Maximal subgroups

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

subgroupindex
C3×#29732
#823492
#716192
#699632
#676762
#94713