Finite Group Databasebeta

← #10690184

Subgroups — #10690184

Order 768.

800 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#4979C211yesmaximal, characteristic
2#82156C211yesmaximal, characteristic
3C2×C3×#324C211yesmaximal, characteristic
4#69963C212yesmaximal
5#67675C212yesmaximal
6#9015—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C2×#926—21no—
10C22×Dic3⋊C4—21no—
11C2×C3×C22⋊C8C2212yes—
12#5212—21no—
13#5212C2212yes—
14C2×C3⋊#95—21no—
15C2×C3⋊#95C2211yescharacteristic
16C2×#816—24no—
17#4984—24no—
18#4979—22no—
19#4979C414yescomplemented
20C3×#324—22no—
21C2×#395—31no—
22#2316—31no—
23#1787—32no—
24C2×#324D311yescharacteristic
25#1294—32no—
26C23×C3×C4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×C3×C4C2311yescharacteristic
29C23×Dic3—21no—
30C23×Dic3—21no—
31C22×Dic6—22no—
32C22×Dic6—22no—
33C22×Dic6—21no—
34C22×Dic6—44no—
35C22×Dic6C2×C412yes—
36C22×C3×C8—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4C2×C412yes—
40C2×C3×C4⋊C4D412yes—
41C2×#217—22no—
42#926—42no—
43#926—42no—
44C2×C4⋊Dic3—22no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3D414yes—
47C2×Dic3⋊C4—24no—
48C2×Dic3⋊C4—44no—
49C2×C4×Dic3—22no—
50C3×C22⋊C8—22no—
51C3×C22⋊C8—22no—
52C3⋊#95—22no—
53C3⋊#95—24no—
54C3⋊#95—21no—
55C3⋊#95—22no—
56C3⋊#95C2×C412yes—
57#816—48no—
58C23×Q8—31no—
59C2×#117—61no—
60C22×C4⋊C4—61no—
61C22×C4⋊C4C2×D311yescharacteristic
62C2×#103—64no—
63C2×C22⋊C8C2×D312yes—
64#395—34no—
65#395—62no—
66#389—64no—
67#380—32no—
68#380—61no—
69C2×#95—31no—
70C2×#95—61no—
71#324—22no—
72C24×C3C2×D411yescharacteristic
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—42no—
80C22×C3×C4—42no—
81C22×C3×C4C22⋊C412yes—
82C22×C3×C4C22⋊C412yes—
83C22×C3×C4SD1612yes—
84C22×C3×C4Q1612yes—
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C2×D412yes—
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4#4112yes—
89C22×Dic3—24no—
90C22×Dic3—24no—
91C22×Dic3—22no—
92C22×Dic3—22no—
93C22×Dic3—24no—
94C22×Dic3—42no—
95C22×Dic3—42no—
96C22×Dic3—41no—
97C22×Dic3—41no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C2×Dic6—28no—
101C2×Dic6—24no—
102C2×Dic6—44no—
103C2×Dic6—48no—
104C2×Dic6—44no—
105C2×Dic6—44no—
106C2×Dic6—42no—
107C2×C3×C8—44no—
108C2×C3×C8—42no—
109C3×C4⋊C4—24no—
110C3×C4⋊C4—24no—
111#217—42no—
112#217—42no—
113C4⋊Dic3—44no—
114C4⋊Dic3—44no—
115C4⋊Dic3—82no—
116Dic3⋊C4—48no—
117C4×Dic3—44no—
118C22×Q8—32no—
119C22×Q8—61no—
120C22×Q8—62no—
121C22×Q8—62no—
122C22×Q8—124no—
123C23×C4—61no—
124C23×C4—61no—
125C23×C4D1211yescharacteristic
126C23×C4D1211yescharacteristic
127C23×C4C22×D311yescharacteristic
128C22×C8—22no—
129#117—122no—
130#117—122no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—34no—
134C2×C4⋊C4—62no—
135C2×C4⋊C4—64no—
136C2×C4⋊C4—62no—
137C2×C4⋊C4—124no—
138C2×C4⋊C4C4×D312yes—
139C2×C4⋊C4#6712yes—
140C2×C22⋊C4—62no—
141C2×C42—62no—
142#103—128no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145#95—32no—
146#95—62no—
147#95—64no—
148#95—62no—
149#95—61no—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3C2×C22⋊C411yescharacteristic
153C23×C3#12011yescharacteristic
154C23×C3#12311yescharacteristic
155C23×C3C2×SD1611yescharacteristic
156C23×C3C2×Q1611yescharacteristic
157C23×C3AGL(1,ℤ/8)11yescharacteristic
158C23×C3#13711yescharacteristic
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—24no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—81no—
173C2×C3×C4#10314yescomplemented
174C2×C3×C4C4×D412yes—
175C2×C3×C4#12112yes—
176C2×Dic3—24no—
177C2×Dic3—24no—
178C2×Dic3—42no—
179C2×Dic3—42no—
180C2×Dic3—44no—
181C2×Dic3—44no—
182C2×Dic3—44no—
183C2×Dic3—42no—
184C2×Dic3—42no—
185C2×Dic3—44no—
186C2×Dic3—44no—
187C2×Dic3—44no—
188C2×Dic3—44no—
189Dic6—28no—
190Dic6—48no—
191Dic6—44no—
192C3×C8—82no—
193C24C2×D1211yescharacteristic
194C2×Q8—68no—
195C2×Q8—64no—
196C2×Q8—124no—
197C2×Q8—124no—
198C2×Q8—124no—
199C2×Q8—128no—
200C2×Q8—122no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—64no—
210C22×C4—64no—
211C22×C4—62no—
212C22×C4—62no—
213C22×C4—64no—
214C22×C4—122no—
215C22×C4—122no—
216C22×C4—121no—
217C22×C4—121no—
218C22×C4—121no—
219C22×C4—121no—
220C22×C4#20412yes—
221C22×C4Dic1212yes—
222C22×C4(C2×D3)⋊C412yes—
223C22×C4(C2×D3)⋊C412yes—
224C22×C4C2×C4×D311yescharacteristic
225C22×C4D3×D412yes—
226C22×C4#23712yes—
227C22×C4C2×#6711yescharacteristic
228C2×C8—44no—
229C2×C8—42no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—124no—
233C4⋊C4—124no—
234C4⋊C4—128no—
235C4⋊C4—242no—
236C22⋊C4—122no—
237C22⋊C4—122no—
238C42—124no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3#38611yescharacteristic
248C22×C3C2×#10311yescharacteristic
249C22×C3#41711yescharacteristic
250C22×C3#45011yescharacteristic
251C22×C3#45111yescharacteristic
252C22×C3#45612yes—
253C22×C3B(4,𝔽2)12yes—
254C22×C3#48311yescharacteristic
255C22×C3#48411yescharacteristic
256C3×C4—24no—
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—42no—
260Dic3—44no—
261Dic3—44no—
262Dic3—84no—
263C23—22no—
264C23—22no—
265C23#88211yescharacteristic
266C23#88611yescharacteristic
267C23C2×#20411yescharacteristic
268C23C2×Dic1211yescharacteristic
269C23#90811yescharacteristic
270C23#90911yescharacteristic
271C23C2×(C2×D3)⋊C411yescharacteristic
272Q8—68no—
273Q8—128no—
274Q8—124no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—64no—
289C2×C4—64no—
290C2×C4—81no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—124no—
294C2×C4—124no—
295C2×C4—124no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4—124no—
299C2×C4—124no—
300C2×C4—124no—
301C2×C4—124no—
302C2×C4#81614yescomplemented
303C2×C4#67⋊C412yes—
304C2×C4#93912yes—
305C8—82no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3#160412yes—
311C2×C3#160512yes—
312C2×C3#187211yescharacteristic
313C2×C3#188611yescharacteristic
314C2×C3#188911yescharacteristic
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22#472111yescharacteristic
324C22#472211yescharacteristic
325C22#473111yescharacteristic
326C22#473911yescharacteristic
327C22#474112yes—
328C22#475212yes—
329C22#495811yescharacteristic
330C22C2×#81611yescharacteristic
331C22#513311yescharacteristic
332C4—24no—
333C4—22no—
334C4—22no—
335C4—42no—
336C4—124no—
337C4—124no—
338C4—244no—
339C3#901511yescharacteristic, complemented, Sylow
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#6813312yes—
345C2#6813912yes—
346C2#6993011yescharacteristic
347C2#7002211yescharacteristic
348C2#7010311yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#324D3
hypocenterC3#9015
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
19. #4979C4semidirect
173. C2×C3×C4#103semidirect
302. C2×C4#816semidirect
339. C3#9015semidirect

Maximal subgroups

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

subgroupindex
C2×#49792
#821562
C2×C3×#3242
#699632
#676752
#90153