Finite Group Databasebeta

← #10690256

Subgroups — #10690256

Order 768.

534 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2973C211yesmaximal, characteristic
2#82348C211yesmaximal, characteristic
3C4×#833C211yesmaximal, characteristic
4#70802C212yesmaximal, complemented
5#67748C212yesmaximal, complemented
6#9029—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×C4×Dic6C2211yescharacteristic
9C3×#435—24no—
10C3×#432C2212yes—
11C3×#407—21no—
12C2×C3×C22⋊C8—21no—
13#5222—21no—
14C2×#833—21no—
15C4×#217—21no—
16C4×#217—21no—
17#4985—21no—
18#4953C411yescharacteristic
19#4953C411yescharacteristic
20#4947C411yescharacteristic
21#4947C411yescharacteristic
22#4932C2211yescharacteristic
23C4×#208C2212yes—
24#4530—21no—
25#4484—24no—
26#2973D311yescharacteristic
27#2398—31no—
28C4×#100—31no—
29#1491—32no—
30#1300—32no—
31C22×C3×D4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C22×Dic6C2×C411yescharacteristic
35C2×C3×M16—22no—
36C22×C3×C8—22no—
37C3×C4×D4—22no—
38C3×C4×D4—22no—
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C22⋊C4C2×C411yescharacteristic
41C2×C3×C22⋊C4C2×C411yescharacteristic
42C2×C3×C42C2311yescharacteristic
43C2×#217—21no—
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—21no—
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×Dic3⋊C4—21no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3—21no—
51C2×#208—22no—
52C4×Dic6—22no—
53C4×Dic6—22no—
54C3×C4⋊C8D414yes—
55C3×C22⋊C8—22no—
56C3×C22⋊C8—22no—
57C3×C22⋊C8—22no—
58C3×#97—22no—
59#833—24no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95C2×C411yescharacteristic
66C3⋊#95C2×C411yescharacteristic
67#803—22no—
68C4×C3⋊C8—22no—
69C2×C4×Q8—31no—
70C2×C4×D4C2×D311yescharacteristic
71#435—24no—
72#432C2×D312yes—
73C2×#100—61no—
74#407—21no—
75C2×C22⋊C8—21no—
76C4×M16—32no—
77#400—61no—
78#397—31no—
79#397—31no—
80#394—61no—
81#393—31no—
82#393—31no—
83C4×C22⋊C4—61no—
84C4×C22⋊C4—61no—
85#376—31no—
86#343—61no—
87Q8⋊C8—64no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C22⋊C411yescharacteristic
90C2×C3×D4—22no—
91C2×C3×D4—22no—
92C2×C3×D4—41no—
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×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C2×Dic6—22no—
109C2×Dic6—42no—
110C3×M16—82no—
111C2×C3×C8—24no—
112C2×C3×C8—82no—
113C3×C4⋊C4—22no—
114C3×C22⋊C4—22no—
115C3×C22⋊C4—22no—
116C3×C42C2×D412yes—
117C3×C42#4112yes—
118#217—42no—
119#217—42no—
120#217—42no—
121#217—42no—
122C4⋊Dic3—22no—
123Dic3⋊C4—42no—
124C4×Dic3—42no—
125C4×Dic3—41no—
126C4×Dic3—41no—
127#208—24no—
128#208—42no—
129C2×C3⋊C8—42no—
130C22×Q8—31no—
131C22×D4D1211yescharacteristic
132C23×C4#6711yescharacteristic
133C23×C4#6711yescharacteristic
134C2×M16—22no—
135C2×M16—62no—
136C22×C8—22no—
137C4×Q8—62no—
138C4×Q8—62no—
139C4×D4—22no—
140C4×D4—22no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4D1211yescharacteristic
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4C4×D311yescharacteristic
149C2×C22⋊C4C4×D311yescharacteristic
150C2×C42—61no—
151C2×C42—61no—
152C2×C42C22×D311yescharacteristic
153C4⋊C8#6714yes—
154#100—64no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157C22⋊C8—22no—
158#97—22no—
159#97—62no—
160C4×C8—62no—
161#95—31no—
162#95—31no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95—61no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3#10412yes—
178C23×C3C2×C22⋊C411yescharacteristic
179C3×D4—44no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—81no—
191C2×C3×C4—81no—
192C2×C3×C4C4×D412yes—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12112yes—
195C2×C3×C4#12311yescharacteristic
196C2×Dic3—22no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208Dic6—24no—
209Dic6—42no—
210C3×C8—44no—
211C3⋊C8—44no—
212C24(C2×D3)⋊C411yescharacteristic
213C24(C2×D3)⋊C411yescharacteristic
214C2×Q8—62no—
215C2×Q8—122no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4(C2×D3)⋊C411yescharacteristic
230C22×C4(C2×D3)⋊C411yescharacteristic
231C22×C4C2×C4×D311yescharacteristic
232C22×C4C2×D1211yescharacteristic
233C22×C4C2×#6711yescharacteristic
234M16—64no—
235M16—82no—
236M16—122no—
237C2×C8—24no—
238C2×C8—82no—
239C2×C8—122no—
240C4⋊C4—22no—
241C4⋊C4—62no—
242C4⋊C4—122no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—122no—
246C22⋊C4—122no—
247C22⋊C4—122no—
248C22⋊C4—122no—
249C42—122no—
250C42—121no—
251C42—121no—
252C42#23512yes—
253C42C2×#6712yes—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—42no—
258C22×C3—42no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—81no—
265C22×C3—81no—
266C22×C3#38611yescharacteristic
267C22×C3C2×#10411yescharacteristic
268C22×C3#42111yescharacteristic
269C3×C4—22no—
270C3×C4—21no—
271C3×C4—22no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4#45512yes—
275Dic3—42no—
276Dic3—81no—
277Dic3—81no—
278Dic3—81no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#82512yes—
289C23C2×(C2×D3)⋊C411yescharacteristic
290Q8—64no—
291Q8—122no—
292D4—44no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—62no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4C4×#6712yes—
318C2×C4#92911yescharacteristic
319C2×C4#93012yes—
320C2×C4#95311yescharacteristic
321C8—44no—
322C8—124no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—42no—
326C2×C3—81no—
327C2×C3#187111yescharacteristic
328C2×C3#188511yescharacteristic
329C2×C3#189111yescharacteristic
330C22—21no—
331C22—21no—
332C22—22no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22C2×#82511yescharacteristic
343C22#513911yescharacteristic
344C22#521311yescharacteristic
345C4—22no—
346C4—22no—
347C4—21no—
348C4—41no—
349C4—41no—
350C4—122no—
351C4—241no—
352C4—241no—
353C4—241no—
354C4#524212yes—
355C3#902911yescharacteristic, complemented, Sylow
356C2—21no—
357C2—21no—
358C2—42no—
359C2—81no—
360C2#7164911yescharacteristic
361C2#7198411yescharacteristic
362C2#7202911yescharacteristic
subgroupG/H
centerC2×C4#953
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2973C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#2973D3
hypocenterC3#9029
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #70802C2semidirect
5. #67748C2semidirect
355. C3#9029semidirect

Maximal subgroups

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

subgroupindex
C3×#29732
#823482
C4×#8332
#708022
#677482
#90293