Finite Group Databasebeta

← #10706110

Subgroups — #10706110

Order 768.

438 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 91 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×#2591C211yesmaximal, characteristic
2#71562C211yesmaximal, characteristic, complemented
3C2×#4528C211yesmaximal, characteristic, complemented
4#67652C212yesmaximal, complemented
5#67647C212yesmaximal, complemented
6#11852—31nomaximal, Sylow
7C2×C3×#121C411yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C3×#400C411yescharacteristic
10C3×#400C2211yescharacteristic
11C3×#394C411yescharacteristic
12C3×#394C2211yescharacteristic
13C3×#388—21no—
14C3×#388—21no—
15C3×#386—21no—
16C3×#386—21no—
17C3×C4×C22⋊C4C411yescharacteristic
18#5207—21no—
19C2×#830—21no—
20#4530—21no—
21#4530C2212yes—
22#4528—21no—
23#4528C2212yes—
24#2591D311yescharacteristic
25#1821—31no—
26C2×#324—31no—
27#1353—32no—
28#1345—32no—
29C22×C3×D4—21no—
30C22×C3×D4C2×C411yescharacteristic
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C3×#121—41no—
35C3×#121—41no—
36C2×C3×C4⋊C4—41no—
37C2×C3×C4⋊C4—41no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C42—21no—
50C2×#208—22no—
51C22×C3⋊C8—41no—
52C3×#95—21no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95C2×C411yescharacteristic
57C3×#95C2×C411yescharacteristic
58#830—22no—
59#830—41no—
60#830—41no—
61#830—41no—
62C2×#121Dic311yescharacteristic
63C22×C4⋊C4C2×D311yescharacteristic
64#407—61no—
65C2×C22⋊C8—61no—
66#400Dic311yescharacteristic
67#400C2×D311yescharacteristic
68#394Dic311yescharacteristic
69#394C2×D311yescharacteristic
70#388—21no—
71#388—21no—
72#386—21no—
73#386—21no—
74C4×C22⋊C4Dic311yescharacteristic
75#343—32no—
76#343—61no—
77#324—32no—
78#324—61no—
79C24×C3—21no—
80C24×C3C22⋊C411yescharacteristic
81C2×C3×D4—41no—
82C2×C3×D4—42no—
83C2×C3×D4—42no—
84C2×C3×D4—41no—
85C2×C3×D4—41no—
86C2×C3×D4—81no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—41no—
98C22×C3×C4—81no—
99C22×C3×C4—81no—
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C22⋊C4—42no—
112C3×C22⋊C4—42no—
113C3×C22⋊C4—42no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—81no—
116C3×C22⋊C4—81no—
117C3×C42—81no—
118#208—42no—
119#208—42no—
120C2×C3⋊C8—42no—
121C2×C3⋊C8—42no—
122C2×C3⋊C8—82no—
123C22×D4—21no—
124C22×D4C2×Dic311yescharacteristic
125C23×C4C2×Dic311yescharacteristic
126C23×C4#6711yescharacteristic
127C23×C4#6711yescharacteristic
128C2×M16—62no—
129C22×C8—121no—
130#121—41no—
131#121—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4C2×Dic311yescharacteristic
135C2×C4⋊C4C2×Dic311yescharacteristic
136C2×C4⋊C4#6711yescharacteristic
137C2×C4⋊C4#6711yescharacteristic
138C2×C4⋊C4C22×D311yescharacteristic
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C42—21no—
146C22⋊C8—62no—
147C22⋊C8—121no—
148C22⋊C8—121no—
149C22⋊C8—121no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95C2×Dic311yescharacteristic
155#95C2×Dic311yescharacteristic
156C23×C3—22no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3—41no—
164C23×C3—41no—
165C23×C3—41no—
166C23×C3—41no—
167C23×C3#9911yescharacteristic
168C23×C3#9911yescharacteristic
169C23×C3C2×C22⋊C411yescharacteristic
170C3×D4—82no—
171C3×D4—82no—
172C3×D4—82no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
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—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—81no—
199C2×C3×C4—81no—
200C2×C3×C4—81no—
201C2×C3×C4#13512yes—
202C3⋊C8—82no—
203C3⋊C8—82no—
204C24—21no—
205C24#21711yescharacteristic
206C2×D4—41no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—81no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4#21711yescharacteristic
226C22×C4#21711yescharacteristic
227C22×C4#21711yescharacteristic
228C22×C4C22×Dic311yescharacteristic
229C22×C4C2×#6711yescharacteristic
230C22×C4C2×#6711yescharacteristic
231M16—122no—
232M16—122no—
233C2×C8—122no—
234C2×C8—122no—
235C2×C8—242no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—81no—
246C22⋊C4—81no—
247C42—81no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—81no—
258C22×C3—81no—
259C22×C3—81no—
260C22×C3—81no—
261C22×C3—81no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3#35512yes—
265C22×C3C2×#9911yescharacteristic
266C22×C3#41611yescharacteristic
267C22×C3#42111yescharacteristic
268C3×C4—22no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—81no—
275C3×C4—81no—
276C3×C4—81no—
277C23—21no—
278C23—21no—
279C23—22no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#83411yescharacteristic
289C23#83411yescharacteristic
290C23C2×#21711yescharacteristic
291D4—82no—
292D4—82no—
293D4—82no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4#95012yes—
323C8—242no—
324C8—242no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—22no—
328C2×C3—41no—
329C2×C3—161no—
330C2×C3#151611yescharacteristic
331C2×C3#212111yescharacteristic
332C2×C3C2×#35511yescharacteristic
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—42no—
341C22—42no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22#454212yes—
350C22C2×#83411yescharacteristic
351C22#523311yescharacteristic
352C22#523611yescharacteristic
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C3#1185211yescharacteristic, complemented, Sylow
363C2—21no—
364C2—21no—
365C2—22no—
366C2—41no—
367C2—161no—
368C2#7171011yescharacteristic
369C2C2×#454211yescharacteristic
370C2#7184711yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2591C2
socleC22×C3C2×#99
prime radicalC2×C3×#121C4
cosocleC2×C3×C4⋊C4C23
hypercenter#2591D3
hypocenterC3#11852
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
2. #71562C2semidirect
3. C2×#4528C2semidirect
4. #67652C2semidirect
5. #67647C2semidirect
362. C3#11852semidirect

Maximal subgroups

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

subgroupindex
C3×#25912
#715622
C2×#45282
#676522
#676472
#118523