Finite Group Databasebeta

← #10691223

Subgroups — #10691223

Order 768.

578 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 111 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#82286C211yesmaximal, characteristic
2#82209C211yesmaximal, characteristic, complemented
3C3×#1458C211yesmaximal, characteristic, complemented
4#70774C212yesmaximal, complemented
5#67738C212yesmaximal, complemented
6#9198—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#930—21no—
9C2×#874C2211yescharacteristic
10C3×#431C2212yes—
11#5207—21no—
12#5000—24no—
13#4991—21no—
14#4991C412yes—
15#4960—21no—
16#4960C412yes—
17#4939—21no—
18#4936C2211yescharacteristic
19#4924C2212yes—
20C4×#208—21no—
21C3×#324—21no—
22C3×#323—21no—
23#4485—24no—
24#2921—31no—
25#2619—31no—
26#1545—32no—
27#1458D311yescharacteristic, complemented
28#1286—32no—
29C23×C3×C4D411yescharacteristic
30C22×#67—22no—
31C22×D12—21no—
32C22×D12—21no—
33C22×D12C2×C412yes—
34C3×#117—22no—
35C3×#117—22no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C22⋊C4C2×C411yescharacteristic
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42#930—44no—
43C2×(C2×D3)⋊C4—22no—
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—22no—
46C2×C4⋊Dic3—21no—
47C2×#208—22no—
48#874—22no—
49#874—22no—
50C3×C4⋊C8—22no—
51C3×C22⋊C8—22no—
52C3×C4×C8—22no—
53C3⋊#95—21no—
54C3⋊#95C2×C412yes—
55#830—24no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)D414yescomplemented
58#803—22no—
59C4×C3⋊C8—22no—
60C2×#127—31no—
61C2×#121—61no—
62C2×#117C2×D311yescharacteristic
63#433—64no—
64#431C2×D312yes—
65#423—32no—
66#407—61no—
67C4×M16—61no—
68#394—32no—
69#394—61no—
70#392—32no—
71#392—61no—
72#390—61no—
73#382—31no—
74#331—64no—
75#324—21no—
76#323—21no—
77C24×C3C22⋊C411yescharacteristic
78C23×D3—22no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C2×#67—44no—
90C2×#67—82no—
91C2×#67—82no—
92C22×Dic3—22no—
93C2×D12—24no—
94C2×D12—41no—
95C2×D12—42no—
96C2×D12—41no—
97C2×D12—42no—
98C2×D12—42no—
99C2×D12—82no—
100C2×D12—82no—
101C2×C3×C8—42no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—22no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—22no—
106C3×C42—21no—
107C3×C42—21no—
108C3×C42C2×D412yes—
109C3×C42#4112yes—
110(C2×D3)⋊C4—44no—
111(C2×D3)⋊C4—82no—
112(C2×D3)⋊C4—82no—
113C4⋊Dic3—42no—
114#208—42no—
115#208—42no—
116C2×C3⋊C8—24no—
117C22×D4—32no—
118C22×D4—62no—
119C22×D4—61no—
120C22×D4—61no—
121C23×C4D1211yescharacteristic
122C2×M16—62no—
123#127—62no—
124#127—62no—
125#121—124no—
126#117—22no—
127#117—22no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4C4×D311yescharacteristic
130C2×C4⋊C4#6711yescharacteristic
131C2×C22⋊C4—62no—
132C2×C22⋊C4—62no—
133C2×C22⋊C4—62no—
134C2×C22⋊C4C4×D311yescharacteristic
135C2×C22⋊C4#6711yescharacteristic
136C2×C42D1211yescharacteristic
137C2×C42C22×D311yescharacteristic
138C4⋊C8—22no—
139C4⋊C8—34no—
140C4⋊C8—62no—
141C22⋊C8—22no—
142C22⋊C8—64no—
143#97—62no—
144C4×C8—22no—
145C4×C8—62no—
146#95—32no—
147#95—61no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—21no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3C2×C22⋊C411yescharacteristic
154C22×D3—42no—
155C22×D3—42no—
156C22×D3—42no—
157C22×D3—42no—
158C22×D3—42no—
159C22×D3—42no—
160C22×D3—42no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4C4×D412yes—
173C2×C3×C4#12011yescharacteristic
174C2×C3×C4#12112yes—
175C2×C3×C4#12311yescharacteristic
176C2×C3×C4#13512yes—
177C2×C3×C4AGL(1,ℤ/8)12yes—
178#67—88no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181C2×Dic3—42no—
182D12—84no—
183D12—84no—
184D12—82no—
185D12—82no—
186C3×C8—82no—
187C3⋊C8—44no—
188C24—62no—
189C24(C2×D3)⋊C411yescharacteristic
190C2×D4—64no—
191C2×D4—124no—
192C2×D4—121no—
193C2×D4—121no—
194C2×D4—122no—
195C2×D4—122no—
196C2×D4—122no—
197C2×D4—242no—
198C2×D4—242no—
199C2×D4—242no—
200C2×D4—242no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—62no—
206C22×C4(C2×D3)⋊C411yescharacteristic
207C22×C4(C2×D3)⋊C411yescharacteristic
208C22×C4(C2×D3)⋊C411yescharacteristic
209C22×C4C2×C4×D311yescharacteristic
210C22×C4C2×D1211yescharacteristic
211C22×C4C2×#6711yescharacteristic
212M16—122no—
213M16—122no—
214C2×C8—42no—
215C2×C8—64no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—122no—
219C22⋊C4—22no—
220C22⋊C4—22no—
221C22⋊C4—124no—
222C22⋊C4—242no—
223C22⋊C4—242no—
224C42—21no—
225C42—21no—
226C42D3×D412yes—
227C42#23712yes—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—42no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—81no—
235C22×C3#38611yescharacteristic
236C22×C3#41611yescharacteristic
237C22×C3#41811yescharacteristic
238C2×D3—82no—
239C2×D3—82no—
240C2×D3—82no—
241C2×D3—82no—
242C2×D3—82no—
243C2×D3—82no—
244C2×D3—82no—
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—41no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4#44912yes—
251C3×C4#48212yes—
252Dic3—82no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—122no—
259C23—122no—
260C23—122no—
261C23—122no—
262C23—122no—
263C23—122no—
264C23—122no—
265C23C2×(C2×D3)⋊C411yescharacteristic
266D4—244no—
267D4—244no—
268D4—248no—
269D4—242no—
270D4—242no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4#67⋊C412yes—
286C2×C4#88211yescharacteristic
287C2×C4#88611yescharacteristic
288C2×C4#90412yes—
289C2×C4#90812yes—
290C2×C4#93912yes—
291C8—82no—
292C8—124no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—81no—
296C2×C3#187511yescharacteristic
297C2×C3#191011yescharacteristic
298C2×C3#191411yescharacteristic
299D3—162no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22—242no—
308C22—242no—
309C22—242no—
310C22—242no—
311C22—242no—
312C22—242no—
313C22—242no—
314C22#495811yescharacteristic
315C22#511311yescharacteristic
316C22#513111yescharacteristic
317C4—22no—
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—242no—
323C4#473412yes—
324C4#473712yes—
325C3#919811yescharacteristic, complemented, Sylow
326C2—21no—
327C2—21no—
328C2—81no—
329C2—482no—
330C2#6993311yescharacteristic
331C2#7004711yescharacteristic
332C2#7013711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1458C2
socleC22×C3#386
prime radical#82286C2
cosocleC2×C3×C42C23
hypercenter#1458D3
hypocenterC3#9198
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #82209C2semidirect
3. C3×#1458C2semidirect
4. #70774C2semidirect
5. #67738C2semidirect
27. #1458D3semidirect
57. C4⋊(C3⋊C8)D4semidirect
325. C3#9198semidirect

Maximal subgroups

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

subgroupindex
#822862
#822092
C3×#14582
#707742
#677382
#91983