Finite Group Databasebeta

← #10693072

Subgroups — #10693072

Order 768.

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

normal subgroups only

This group has 600 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
1C3×#2598C211yesmaximal, characteristic
2#82202C211yesmaximal, characteristic, complemented
3#71599C212yesmaximal, complemented
4#71572C211yesmaximal, characteristic, complemented
5#67633C212yesmaximal, complemented
6#9492—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C22×#208—21no—
10C3×#400C411yescharacteristic
11C3×#399C411yescharacteristic
12C3×#394C411yescharacteristic
13C3×#392C411yescharacteristic
14C3×#388—21no—
15C3×#386—21no—
16C3×#386—21no—
17C3×#385—21no—
18C3×#383C2211yescharacteristic
19#4997—24no—
20#4926C2212yes—
21C4×#208—21no—
22C4×#208C2212yes—
23D4⋊(C3⋊C8)—24no—
24#4528—21no—
25#4525—21no—
26#2873—31no—
27#2598D311yescharacteristic
28#1490—32no—
29#1464—31no—
30#1288—32no—
31C22×C3×D4—21no—
32C22×C3×D4C2×C411yescharacteristic
33C23×C3×C4—21no—
34C23×C3×C4D411yescharacteristic
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C3×#117—22no—
38C3×#117—22no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4D411yescharacteristic
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⋊C4C2×C411yescharacteristic
49C2×C3×C22⋊C4D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52C2×#208—22no—
53C2×#208—24no—
54C2×#208—24no—
55C2×#208—22no—
56C22×C3⋊C8—22no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95C2×C411yescharacteristic
62C3×#95C2×C411yescharacteristic
63#830—22no—
64C4⋊(C3⋊C8)—22no—
65#803—22no—
66#803—22no—
67#803—22no—
68C4×C3⋊C8—22no—
69C4×C3⋊C8D414yescomplemented
70C22×M16—61no—
71C2×C4×D4C2×D311yescharacteristic
72C2×#117C2×D311yescharacteristic
73#432—32no—
74#405—64no—
75C4×M16—32no—
76C4×M16—61no—
77#400Dic311yescharacteristic
78#399Dic311yescharacteristic
79#394Dic311yescharacteristic
80#392Dic311yescharacteristic
81#388—21no—
82#386—21no—
83#386—21no—
84#385—21no—
85#383C2×D311yescharacteristic
86D4⋊C8—64no—
87#324—61no—
88#323—61no—
89C24×C3—21no—
90C24×C3C22⋊C411yescharacteristic
91C2×C3×D4—22no—
92C2×C3×D4—42no—
93C2×C3×D4—42no—
94C2×C3×D4—42no—
95C2×C3×D4—81no—
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—42no—
104C22×C3×C4—81no—
105C22×C3×C4—81no—
106C22×C3×C4C22⋊C411yescharacteristic
107C22×C3×C4C22⋊C411yescharacteristic
108C22×C3×C4C22⋊C411yescharacteristic
109C22×C3×C4C22×C411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C3×C4⋊C4—22no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—81no—
116C3×C22⋊C4—22no—
117C3×C22⋊C4—22no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C22⋊C4—81no—
123C3×C22⋊C4—81no—
124C3×C42—21no—
125C3×C42—21no—
126C3×C42C2×D412yes—
127C3×C42#4112yes—
128#208—28no—
129#208—24no—
130#208—24no—
131C2×C3⋊C8—28no—
132C2×C3⋊C8—24no—
133C2×C3⋊C8—42no—
134C22×D4—21no—
135C22×D4C2×Dic311yescharacteristic
136C23×C4—21no—
137C23×C4#6711yescharacteristic
138C2×M16—62no—
139C2×M16—64no—
140C2×M16—64no—
141C2×M16—62no—
142C22×C8—62no—
143C4×D4—22no—
144C4×D4—22no—
145#117—22no—
146#117—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4C2×Dic311yescharacteristic
149C2×C4⋊C4C2×Dic311yescharacteristic
150C2×C4⋊C4#6711yescharacteristic
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4C2×Dic311yescharacteristic
157C2×C22⋊C4#6711yescharacteristic
158C2×C42#6711yescharacteristic
159C2×C42C22×D311yescharacteristic
160C4⋊C8—62no—
161C22⋊C8—62no—
162#97—62no—
163#97—62no—
164#97—62no—
165C4×C8—34no—
166C4×C8—62no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95C2×Dic311yescharacteristic
172#95C2×Dic311yescharacteristic
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—22no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—41no—
183C23×C3—41no—
184C23×C3#10412yes—
185C23×C3C2×C22⋊C411yescharacteristic
186C3×D4—24no—
187C3×D4—42no—
188C3×D4—84no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—24no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—84no—
212C2×C3×C4#10412yes—
213C2×C3×C4C4×D412yes—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12112yes—
216C2×C3×C4#12311yescharacteristic
217C3⋊C8—28no—
218C3⋊C8—82no—
219C24—21no—
220C24#21711yescharacteristic
221C2×D4—22no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—81no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4#21711yescharacteristic
237C22×C4#21711yescharacteristic
238C22×C4#21711yescharacteristic
239C22×C4C22×Dic311yescharacteristic
240C22×C4C2×#6711yescharacteristic
241C22×C4C2×#6711yescharacteristic
242M16—68no—
243M16—64no—
244M16—64no—
245C2×C8—68no—
246C2×C8—64no—
247C2×C8—122no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—81no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—81no—
259C22⋊C4—81no—
260C42—21no—
261C42—21no—
262C42D3×D412yes—
263C42#23712yes—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—22no—
267C22×C3—42no—
268C22×C3—42no—
269C22×C3—42no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—42no—
273C22×C3—81no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3#38611yescharacteristic
279C22×C3C2×#10411yescharacteristic
280C22×C3C2×#10411yescharacteristic
281C3×C4—22no—
282C3×C4—21no—
283C3×C4—22no—
284C3×C4—21no—
285C3×C4—22no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—81no—
289C3×C4—81no—
290C3×C4—81no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23#83712yes—
303C23C2×#21711yescharacteristic
304D4—24no—
305D4—42no—
306D4—84no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—84no—
330C2×C4#83712yes—
331C2×C4D4×Dic312yes—
332C2×C4#93511yescharacteristic
333C2×C4#93711yescharacteristic
334C2×C4#93912yes—
335C8—68no—
336C8—242no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—42no—
340C2×C3—82no—
341C2×C3#189111yescharacteristic
342C2×C3#189611yescharacteristic
343C2×C3#191011yescharacteristic
344C22—21no—
345C22—21no—
346C22—22no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—41no—
351C22—41no—
352C22—42no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22#522011yescharacteristic
359C22C2×#83711yescharacteristic
360C22C2×#83711yescharacteristic
361C4—21no—
362C4—22no—
363C4—21no—
364C4—22no—
365C4—22no—
366C4—41no—
367C4—41no—
368C4—81no—
369C4—81no—
370C4—81no—
371C3#949211yescharacteristic, complemented, Sylow
372C2—21no—
373C2—21no—
374C2—42no—
375C2—82no—
376C2#7171111yescharacteristic
377C2#7171411yescharacteristic
378C2#7185411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2598C2
socleC22×C3#386
prime radicalC3×#392C4
cosocleC2×C3×C42C23
hypercenter#2598D3
hypocenterC3#9492
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2598

Complements

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

NH ≅ G/Ntype
2. #82202C2semidirect
3. #71599C2semidirect
4. #71572C2semidirect
5. #67633C2semidirect
69. C4×C3⋊C8D4semidirect
371. C3#9492semidirect

Maximal subgroups

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

subgroupindex
C3×#25982
#822022
#715992
#715722
#676332
#94923