Finite Group Databasebeta

← #11713708

Subgroups — #11713708

Order 768.

512 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 53 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
1A4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#85290C211yesmaximal, characteristic
3#73053C212yesmaximal
4#73052C212yesmaximal, complemented
5#73044C211yesmaximal, characteristic, complemented
6#25908—31nomaximal, Sylow
7C2×Q8×A4—21no—
8C2×C4⋊S4C2211yescharacteristic
9C2×A4⋊Q8C2211yescharacteristic
10A4×C4⋊C4—21no—
11A4×C4⋊C4C2212yes—
12C42×A4C2211yescharacteristic
13(C2×S4)⋊C4—21no—
14(C2×S4)⋊C4—21no—
15C2×A4⋊C8C2212yes—
16#5089—41nomaximal
17C22×C4⋊Q8D311yescharacteristic, complemented
18C2×#489—31no—
19C2×#489—31no—
20#2850—31no—
21#2024—32no—
22#2023—32no—
23#1993—31no—
24#1993—31no—
25#1874—32no—
26#1872—32no—
27#1749—31no—
28C22×S4—21no—
29Q8×A4—42no—
30C2×C4×A4—22no—
31C2×C4×A4D411yescharacteristic
32C2×C4×A4D411yescharacteristic
33C2×C4×A4C2311yescharacteristic
34C2×SL(2,ℤ/4)—21no—
35C4⋊S4—22no—
36A4⋊Q8—22no—
37C3×C4⋊Q8—41no—
38#875—41no—
39A4⋊C8—22no—
40#810—42no—
41#809—42no—
42#803—41no—
43C23×Q8—21no—
44C2×C4⋊Q8—31no—
45C2×C4⋊Q8—31no—
46C2×C4⋊Q8—32no—
47C2×C4⋊Q8—61no—
48C2×C4⋊Q8—61no—
49C2×#124—31no—
50C2×C22⋊Q8—31no—
51C2×#121—31no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2×D312yes—
54C22×C42C2×D311yescharacteristic
55#489—62no—
56#489—62no—
57#489—62no—
58#489—61no—
59#489—61no—
60C2×#103—32no—
61C2×#103—32no—
62C2×#102—32no—
63C2×#102—32no—
64C2×C22⋊C8—32no—
65C2×#97—31no—
66C2×#97—31no—
67#391—31no—
68#390—31no—
69#388—61no—
70#388—61no—
71#386—61no—
72#386—61no—
73#382—31no—
74#340—32no—
75#340—32no—
76#340—32no—
77#340—32no—
78#336—32no—
79C22×A4C2×D411yescharacteristic
80C2×S4—41no—
81C2×S4—41no—
82C2×S4—41no—
83C2×C3×Q8—81no—
84C2×D12—41no—
85C2×Dic6—41no—
86C4×A4—21no—
87C4×A4—21no—
88C4×A4—42no—
89C4×A4#4112yes—
90SL(2,ℤ/4)—41no—
91C3×C4⋊C4—42no—
92C3×C4⋊C4—81no—
93C3×C42—41no—
94(C2×D3)⋊C4—81no—
95(C2×D3)⋊C4—81no—
96C2×C3⋊C8—42no—
97C22×Q8—31no—
98C22×Q8—42no—
99C22×Q8—61no—
100C22×Q8—62no—
101C22×Q8—61no—
102C22×Q8—122no—
103C22×D4—31no—
104C22×D4—61no—
105C23×C4—22no—
106C23×C4#6711yescharacteristic
107C23×C4#6711yescharacteristic
108C23×C4C22×D311yescharacteristic
109C22×C8—32no—
110C22×C8—32no—
111C4⋊Q8—32no—
112C4⋊Q8—32no—
113C4⋊Q8—32no—
114C4⋊Q8—31no—
115C4⋊Q8—61no—
116C4⋊Q8—61no—
117C4⋊Q8—61no—
118C4⋊Q8—64no—
119C4⋊Q8S411yescharacteristic, complemented
120#124—61no—
121#124—61no—
122#124—61no—
123#124—61no—
124C22⋊Q8—62no—
125C22⋊Q8—62no—
126#121—62no—
127#121—62no—
128C2×C4⋊C4—32no—
129C2×C4⋊C4—32no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—62no—
134C2×C4⋊C4—62no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C22⋊C4—61no—
138C2×C22⋊C4—61no—
139C2×C22⋊C4—61no—
140C2×C22⋊C4—61no—
141C2×C22⋊C4—61no—
142C2×C22⋊C4—61no—
143C2×C42—31no—
144C2×C42—31no—
145C2×C42—31no—
146C2×C42—31no—
147#103—64no—
148#103—62no—
149#103—62no—
150#102—64no—
151#102—62no—
152#102—62no—
153C22⋊C8—62no—
154C22⋊C8—62no—
155#97—61no—
156#97—61no—
157#97—61no—
158#97—61no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—61no—
163C22×D3—81no—
164C2×A4#12411yescharacteristic
165C2×A4AGL(1,ℤ/8)11yescharacteristic
166C2×A4#13711yescharacteristic
167S4—81no—
168C3×Q8—162no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—82no—
173C2×Dic3—81no—
174D12—82no—
175Dic6—82no—
176C3⋊C8—82no—
177C24—61no—
178C24C2×#6711yescharacteristic
179C2×Q8—21no—
180C2×Q8—62no—
181C2×Q8—62no—
182C2×Q8—61no—
183C2×Q8—62no—
184C2×Q8—62no—
185C2×Q8—61no—
186C2×Q8—61no—
187C2×Q8—124no—
188C2×Q8—122no—
189C2×Q8—122no—
190C2×Q8—122no—
191C2×Q8—124no—
192C2×Q8—122no—
193C2×D4—61no—
194C2×D4—62no—
195C2×D4—62no—
196C2×D4—61no—
197C2×D4—121no—
198C2×D4—121no—
199C2×D4—121no—
200C2×D4—121no—
201C2×D4—121no—
202C2×D4—121no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—32no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—42no—
213C22×C4—62no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—62no—
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—122no—
221C22×C4#23912yes—
222C2×C8—62no—
223C2×C8—62no—
224C2×C8—62no—
225C2×C8—62no—
226C2×C8—62no—
227C2×C8—62no—
228C4⋊C4—21no—
229C4⋊C4—34no—
230C4⋊C4—32no—
231C4⋊C4—62no—
232C4⋊C4—62no—
233C4⋊C4—62no—
234C4⋊C4—62no—
235C4⋊C4—62no—
236C4⋊C4—61no—
237C4⋊C4—122no—
238C4⋊C4C2×S412yes—
239C22⋊C4—122no—
240C22⋊C4—122no—
241C22⋊C4—122no—
242C22⋊C4—121no—
243C22⋊C4—121no—
244C22⋊C4—121no—
245C22⋊C4—121no—
246C22⋊C4—121no—
247C22⋊C4—121no—
248C42—31no—
249C42—31no—
250C42—31no—
251C42—61no—
252C42C2×S411yescharacteristic
253C22×C3—41no—
254C2×D3—161no—
255C2×D3—161no—
256C2×D3—161no—
257A4#48911yescharacteristic, complemented
258C3×C4—42no—
259C3×C4—81no—
260C3×C4—81no—
261C3×C4—162no—
262Dic3—161no—
263C23—31no—
264C23—31no—
265C23—31no—
266C23—31no—
267C23—121no—
268C23—121no—
269C23—121no—
270C23—121no—
271C23—121no—
272C23—121no—
273C23—121no—
274C23#93211yescharacteristic
275C23#94211yescharacteristic
276C23#94711yescharacteristic
277Q8—42no—
278Q8—122no—
279Q8—122no—
280Q8—124no—
281Q8—124no—
282Q8—122no—
283D4—122no—
284D4—122no—
285D4—241no—
286D4—241no—
287D4—241no—
288D4—241no—
289C2×C4—22no—
290C2×C4—31no—
291C2×C4—32no—
292C2×C4—32no—
293C2×C4—32no—
294C2×C4—32no—
295C2×C4—32no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—62no—
306C2×C4—122no—
307C2×C4—122no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4GL(2,ℤ/4)11yescharacteristic
320C2×C4GL(2,ℤ/4)11yescharacteristic
321C2×C4C22×S411yescharacteristic
322C8—122no—
323C8—122no—
324C2×C3—41no—
325C2×C3—41no—
326C2×C3—41no—
327D3—321no—
328C22—31no—
329C22—31no—
330C22—31no—
331C22—31no—
332C22—31no—
333C22—31no—
334C22—31no—
335C22—31no—
336C22—31no—
337C22—61no—
338C22—241no—
339C22—241no—
340C22—241no—
341C22—241no—
342C22—241no—
343C22—241no—
344C22—241no—
345C22#508911yescharacteristic, complemented
346C22C2×GL(2,ℤ/4)11yescharacteristic
347C4—21no—
348C4—21no—
349C4—32no—
350C4—42no—
351C4—61no—
352C4—61no—
353C4—122no—
354C4—241no—
355C4—241no—
356C4—241no—
357C4#591912yes—
358C3—41noSylow
359C2—31no—
360C2—31no—
361C2—31no—
362C2—31no—
363C2—481no—
364C2#8548911yescharacteristic
365C2#8549911yescharacteristic
366C2#8550811yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4⋊Q8D3
socleC24C2×#67
prime radicalC2×C4⋊S4C22
hypercenterC4⋊Q8S4
hypocenterA4#489
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. A4×C4⋊Q8C2semidirect
4. #73052C2semidirect
5. #73044C2semidirect
17. C22×C4⋊Q8D3semidirect
119. C4⋊Q8S4semidirect
257. A4#489semidirect
345. C22#5089semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊Q82
#852902
#730532
#730522
#730442
#259083
#50894