Finite Group Databasebeta

← #10705757

Subgroups — #10705757

Order 768.

652 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2541C211yesmaximal, characteristic
2C4×#834C211yesmaximal, characteristic, complemented
3#71602C211yesmaximal, characteristic, complemented
4#67645C214yesmaximal, complemented
5#11776—31nomaximal, Sylow
6C2×C3×#127—21no—
7C2×C3×#127C411yescharacteristic, complemented
8C2×C3×#121—21no—
9C2×C3×#121—21no—
10C2×C3×C4×D4C2211yescharacteristic
11C3×#392C412yescomplemented
12C3×#392C2212yes—
13C3×#390—21no—
14C3×C4×C4⋊C4C411yescharacteristic, complemented
15C2×#834—21no—
16C4×#217—21no—
17#5207—21no—
18#4528—21no—
19#4526C2212yes—
20#4525C2212yes—
21#2541D311yescharacteristic
22C4×#99—31no—
23#1465—31no—
24#1343—34no—
25C22×C3×D4—22no—
26C22×C3×D4—22no—
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4C2×C412yes—
30C22×C3×D4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C3×#127—41no—
34C3×#127—41no—
35C3×#127—44no—
36C3×#121—44no—
37C3×#121—44no—
38C3×C4×D4—41no—
39C3×C4×D4—41no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4C2311yescharacteristic
46C2×C3×C42—21no—
47C2×C3×C42C2×C411yescharacteristic
48C2×#217—22no—
49C2×C4×Dic3—41no—
50C2×#208—41no—
51C3×#95C2×C412yes—
52#834—24no—
53C3⋊#95—41no—
54#830—22no—
55#830—41no—
56C2×#127—21no—
57C2×#127Dic311yescharacteristic, complemented
58C2×#121—21no—
59C2×#121—21no—
60C2×C4×D4C2×D311yescharacteristic
61C2×#99—61no—
62#407—61no—
63#392Dic312yescomplemented
64#392C2×D312yes—
65#390—21no—
66C4×C4⋊C4Dic311yescharacteristic, complemented
67C4×C22⋊C4—61no—
68#342—32no—
69#324—61no—
70#323—32no—
71C24×C3—22no—
72C24×C3C22⋊C411yescharacteristic
73C24×C3C2×D411yescharacteristic
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—41no—
77C2×C3×D4—42no—
78C2×C3×D4—42no—
79C2×C3×D4—42no—
80C2×C3×D4—44no—
81C2×C3×D4—44no—
82C2×C3×D4—42no—
83C2×C3×D4—42no—
84C2×C3×D4—44no—
85C2×C3×D4—44no—
86C2×C3×D4—41no—
87C2×C3×D4—41no—
88C2×C3×D4—42no—
89C2×C3×D4—82no—
90C2×C3×D4—82no—
91C2×C3×D4—82no—
92C22×C3×C4—22no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×Dic3—42no—
105C3×C4⋊C4—24no—
106C3×C4⋊C4—41no—
107C3×C22⋊C4—41no—
108C3×C22⋊C4—41no—
109C3×C22⋊C4—44no—
110C3×C22⋊C4—44no—
111C3×C42—21no—
112C3×C42—21no—
113C3×C42—42no—
114#217—24no—
115#217—42no—
116C4×Dic3—82no—
117#208—82no—
118C2×C3⋊C8—42no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4C2×Dic312yes—
124C22×D4#6711yescharacteristic
125C23×C4#6711yescharacteristic
126C23×C4#6711yescharacteristic
127C2×M16—121no—
128#127—41no—
129#127—41no—
130#127—44no—
131#121—44no—
132#121—44no—
133C4×D4—41no—
134C4×D4—41no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4#6711yescharacteristic
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—62no—
140C2×C22⋊C4C2×Dic311yescharacteristic
141C2×C22⋊C4C22×D311yescharacteristic
142C2×C42—21no—
143C2×C42—121no—
144C2×C42C2×Dic311yescharacteristic
145#99—64no—
146C22⋊C8—62no—
147C22⋊C8—121no—
148#95—121no—
149#95C2×Dic312yes—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—41no—
156C23×C3—41no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3—42no—
160C23×C3—42no—
161C23×C3—42no—
162C23×C3—42no—
163C23×C3—42no—
164C23×C3—42no—
165C23×C3—42no—
166C23×C3C2×C22⋊C411yescharacteristic
167C23×C3AGL(1,ℤ/8)12yes—
168C3×D4—44no—
169C3×D4—82no—
170C3×D4—84no—
171C3×D4—88no—
172C3×D4—88no—
173C3×D4—84no—
174C3×D4—82no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—22no—
178C2×C3×C4—24no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4#9911yescharacteristic
192C2×C3×C4#9911yescharacteristic
193C2×C3×C4#10412yes—
194C2×Dic3—44no—
195C2×Dic3—82no—
196C2×Dic3—82no—
197C3⋊C8—82no—
198C24—22no—
199C24#21711yescharacteristic
200C24C2×#6711yescharacteristic
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—41no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—42no—
216C2×D4—82no—
217C2×D4—82no—
218C2×D4—82no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—122no—
227C22×C4#21711yescharacteristic
228C22×C4#21711yescharacteristic
229C22×C4#21711yescharacteristic
230C22×C4C22×Dic311yescharacteristic
231C22×C4C2×#6711yescharacteristic
232M16—242no—
233C2×C8—122no—
234C4⋊C4—24no—
235C4⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C22⋊C4—64no—
241C22⋊C4—122no—
242C42—21no—
243C42—21no—
244C42—42no—
245C42—242no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—22no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—81no—
257C22×C3—81no—
258C22×C3—82no—
259C22×C3—82no—
260C22×C3—82no—
261C22×C3—82no—
262C22×C3—82no—
263C22×C3—82no—
264C22×C3—82no—
265C22×C3C2×#9911yescharacteristic
266C22×C3#41811yescharacteristic
267C22×C3C2×#10411yescharacteristic
268C3×C4—21no—
269C3×C4—22no—
270C3×C4—44no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4#35312yes—
275Dic3—84no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23#93212yes—
293C23C2×#21711yescharacteristic
294D4—44no—
295D4—82no—
296D4—84no—
297D4—88no—
298D4—88no—
299D4—84no—
300D4—82no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—124no—
318C2×C4—242no—
319C2×C4—242no—
320C2×C4#83411yescharacteristic
321C2×C4#83411yescharacteristic
322C2×C4#83712yes—
323C8—242no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—42no—
327C2×C3—81no—
328C2×C3—162no—
329C2×C3#150711yescharacteristic
330C2×C3#212211yescharacteristic
331C2×C3C2×#35311yescharacteristic
332C22—21no—
333C22—21no—
334C22—22no—
335C22—42no—
336C22—42no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—81no—
344C22—82no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22#521511yescharacteristic
352C22C2×#83411yescharacteristic
353C22C2×#83711yescharacteristic
354C4—21no—
355C4—22no—
356C4—44no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—244no—
361C4#454512yes—
362C3#1177611yescharacteristic, complemented, Sylow
363C2—21no—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—162no—
368C2#7167311yescharacteristic
369C2C2×#454511yescharacteristic
370C2#7185911yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
Fitting, prime radicalC3×#2541C2
socleC22×C3C2×#99
cosocleC2×C3×C22⋊C4C23
hypercenter#2541D3
hypocenterC3#11776
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
2. C4×#834C2semidirect
3. #71602C2semidirect
4. #67645C2semidirect
7. C2×C3×#127C4semidirect
11. C3×#392C4semidirect
14. C3×C4×C4⋊C4C4semidirect
57. C2×#127Dic3semidirect
63. #392Dic3semidirect
66. C4×C4⋊C4Dic3semidirect
362. C3#11776semidirect

Maximal subgroups

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

subgroupindex
C3×#25412
C4×#8342
#716022
#676452
#117763