Finite Group Databasebeta

← #10691233

Subgroups — #10691233

Order 768.

560 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 127 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#82183C211yesmaximal, characteristic
2#82156C211yesmaximal, characteristic
3C2×C3×#324C211yesmaximal, characteristic
4#69946C212yesmaximal
5#67676C212yesmaximal
6#9200—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#926—21no—
9C2×#924C2211yescharacteristic
10C2×C3×C22⋊C8C2212yes—
11#5212—21no—
12#5212C2211yescharacteristic
13#5212C2212yes—
14#5211—21no—
15#5210—21no—
16#4986C412yescomplemented
17#4984—24no—
18#4982—22no—
19#4979C412yescomplemented
20C3×#324—22no—
21#4467—24no—
22#2627—31no—
23#2316—31no—
24#1520—32no—
25C2×#324D311yescharacteristic
26#1289—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×Dic6C2×C411yescharacteristic
31C22×C3×C8—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4C2×C412yes—
35C2×C3×C4⋊C4D412yes—
36C2×#217—21no—
37C2×#217—21no—
38C2×#217—22no—
39#926—42no—
40#926—42no—
41C2×C4⋊Dic3—22no—
42C2×C4⋊Dic3—22no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—22no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46#924—22no—
47#924—22no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—22no—
50C2×C4×Dic3—22no—
51C3×C22⋊C8—22no—
52C3×C22⋊C8—22no—
53C3⋊#95—22no—
54C3⋊#95—22no—
55C3⋊#95—22no—
56C3⋊#95—22no—
57C3⋊#95C2×C412yes—
58C3⋊#95D414yes—
59C2×C22⋊Q8—31no—
60C2×#117—61no—
61C22×C4⋊C4C2×D311yescharacteristic
62C2×C22⋊C8C2×D312yes—
63#400—32no—
64#398—62no—
65#395—32no—
66#389—64no—
67#385—61no—
68#381—61no—
69#380—31no—
70#380—32no—
71#380—61no—
72#328—64no—
73#324—22no—
74C24×C3C2×D411yescharacteristic
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—81no—
82C22×C3×C4—81no—
83C22×C3×C4C22⋊C412yes—
84C22×C3×C4C22⋊C412yes—
85C22×C3×C4SD1612yes—
86C22×C3×C4Q1612yes—
87C22×C3×C4C22×C411yescharacteristic
88C22×C3×C4C2×D412yes—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4#4112yes—
91C22×Dic3—21no—
92C22×Dic3—22no—
93C22×Dic3—21no—
94C22×Dic3—24no—
95C2×Dic6—22no—
96C2×Dic6—81no—
97C2×C3×C8—42no—
98C2×C3×C8—82no—
99C3×C4⋊C4—24no—
100C3×C4⋊C4—24no—
101#217—41no—
102#217—41no—
103#217—42no—
104#217—42no—
105#217—41no—
106#217—41no—
107C4⋊Dic3—22no—
108C4⋊Dic3—44no—
109C4⋊Dic3—81no—
110C4⋊Dic3—82no—
111C4⋊Dic3—82no—
112Dic3⋊C4—42no—
113Dic3⋊C4—82no—
114C4×Dic3—44no—
115C22×Q8—31no—
116C23×C4D1211yescharacteristic
117C23×C4D1211yescharacteristic
118C23×C4C22×D311yescharacteristic
119C22×C8—22no—
120C22⋊Q8—62no—
121C22⋊Q8—62no—
122#117—122no—
123#117—122no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4—62no—
130C2×C4⋊C4—62no—
131C2×C4⋊C4—62no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4C4×D312yes—
134C2×C4⋊C4#6712yes—
135C2×C22⋊C4—62no—
136C2×C22⋊C4—61no—
137C2×C22⋊C4—61no—
138C2×C42—62no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141#95—32no—
142#95—34no—
143#95—62no—
144#95—62no—
145#95—62no—
146#95—62no—
147C23×C3—41no—
148C23×C3—41no—
149C23×C3C2×C22⋊C411yescharacteristic
150C23×C3#12011yescharacteristic
151C23×C3#12311yescharacteristic
152C23×C3C2×SD1611yescharacteristic
153C23×C3C2×Q1611yescharacteristic
154C23×C3AGL(1,ℤ/8)11yescharacteristic
155C23×C3#13711yescharacteristic
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—42no—
164C2×C3×C4—41no—
165C2×C3×C4—42no—
166C2×C3×C4—41no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—81no—
170C2×C3×C4#10314yes—
171C2×C3×C4C4×D412yes—
172C2×C3×C4#12112yes—
173C2×Dic3—42no—
174C2×Dic3—42no—
175C2×Dic3—44no—
176C2×Dic3—42no—
177C2×Dic3—44no—
178C2×Dic3—42no—
179C2×Dic3—41no—
180C2×Dic3—41no—
181C2×Dic3—42no—
182C2×Dic3—44no—
183Dic6—82no—
184C3×C8—82no—
185C24C2×D1211yescharacteristic
186C2×Q8—62no—
187C2×Q8—241no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—64no—
195C22×C4—62no—
196C22×C4—61no—
197C22×C4—61no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4#20412yes—
201C22×C4Dic1212yes—
202C22×C4(C2×D3)⋊C412yes—
203C22×C4(C2×D3)⋊C412yes—
204C22×C4C2×C4×D311yescharacteristic
205C22×C4D3×D412yes—
206C22×C4#23712yes—
207C22×C4C2×#6711yescharacteristic
208C2×C8—42no—
209C2×C8—82no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—62no—
213C4⋊C4—124no—
214C4⋊C4—122no—
215C4⋊C4—242no—
216C4⋊C4—242no—
217C4⋊C4—241no—
218C4⋊C4—242no—
219C22⋊C4—121no—
220C22⋊C4—121no—
221C22⋊C4—122no—
222C22⋊C4—122no—
223C22⋊C4—121no—
224C22⋊C4—121no—
225C42—124no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3#38611yescharacteristic
237C22×C3C2×#10311yescharacteristic
238C22×C3#41711yescharacteristic
239C22×C3#45011yescharacteristic
240C22×C3#45111yescharacteristic
241C22×C3#48311yescharacteristic
242C22×C3#48411yescharacteristic
243C3×C4—24no—
244C3×C4—22no—
245C3×C4—22no—
246C3×C4—42no—
247Dic3—84no—
248Dic3—82no—
249Dic3—81no—
250Dic3—81no—
251C23—41no—
252C23—41no—
253C23#88211yescharacteristic
254C23#88611yescharacteristic
255C23C2×#20411yescharacteristic
256C23C2×Dic1211yescharacteristic
257C23#90811yescharacteristic
258C23#90911yescharacteristic
259C23C2×(C2×D3)⋊C411yescharacteristic
260Q8—242no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—124no—
276C2×C4—122no—
277C2×C4—122no—
278C2×C4—124no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4—124no—
285C2×C4#81614yes—
286C2×C4#67⋊C412yes—
287C2×C4#93912yes—
288C8—82no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3#187211yescharacteristic
296C2×C3#191211yescharacteristic
297C2×C3#191311yescharacteristic
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22#472111yescharacteristic
309C22#472211yescharacteristic
310C22#473111yescharacteristic
311C22#473911yescharacteristic
312C22#495811yescharacteristic
313C22C2×#81611yescharacteristic
314C22#513311yescharacteristic
315C4—24no—
316C4—22no—
317C4—22no—
318C4—42no—
319C4—244no—
320C4—242no—
321C4—241no—
322C4—241no—
323C3#920011yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2#6993011yescharacteristic
331C2#7002411yescharacteristic
332C2#7010511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#324D3
hypocenterC3#9200
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
16. #4986C4semidirect
19. #4979C4semidirect
323. C3#9200semidirect

Maximal subgroups

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

subgroupindex
#821832
#821562
C2×C3×#3242
#699462
#676762
#92003