Finite Group Databasebeta

← #11713706

Subgroups — #11713706

Order 768.

593 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 593 conjugacy classes of subgroups — too many to draw. Only its 65 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#85289C211yesmaximal, characteristic
3#73053C214yesmaximal, complemented
4C4×A4⋊C8C211yesmaximal, characteristic, complemented
5#25903—31nomaximal, Sylow
6C2×Q8×A4—21no—
7C2×C4⋊S4—21no—
8C2×C4⋊S4—21no—
9C2×C4⋊S4C2212yes—
10A4×C4⋊C4—21no—
11A4×C4⋊C4C2212yes—
12C42×A4C2211yescharacteristic
13C2×A4⋊C8C2212yes—
14#5088—41nomaximal
15C22×C4⋊Q8D311yescharacteristic, complemented
16C2×#486—31no—
17C2×#486—31no—
18#2856—31no—
19#2023—34no—
20#1990—31no—
21#1990—31no—
22#1874—34no—
23C4×C22⋊C8—31no—
24C22×S4—22no—
25Q8×A4—42no—
26C2×C4×A4—22no—
27C2×C4×A4D411yescharacteristic
28C2×C4×A4D411yescharacteristic
29C2×C4×A4C2311yescharacteristic
30C4⋊S4—24no—
31C4⋊S4—42no—
32C4⋊S4—42no—
33C3×C4⋊Q8—41no—
34#874—41no—
35A4⋊C8—22no—
36#809—44no—
37C4×C3⋊C8—41no—
38C23×Q8—21no—
39C2×C4⋊Q8—31no—
40C2×C4⋊Q8—31no—
41C2×C4⋊Q8—32no—
42C2×C4⋊Q8—61no—
43C2×C4⋊Q8—61no—
44C2×#127—31no—
45C2×#121—32no—
46C2×#121—61no—
47C2×#121—61no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×D312yes—
50C22×C42C2×D311yescharacteristic
51#486—62no—
52#486—62no—
53#486—62no—
54#486—61no—
55#486—61no—
56C2×#102—34no—
57C2×#102—34no—
58C2×C22⋊C8—32no—
59C2×C4×C8—31no—
60C2×C4×C8—31no—
61#390—32no—
62#390—61no—
63#390—61no—
64#384—31no—
65#340—34no—
66#340—34no—
67#336—32no—
68C22×A4C2×D411yescharacteristic
69C2×S4—42no—
70C2×S4—42no—
71C2×S4—42no—
72C2×C3×Q8—81no—
73C2×D12—42no—
74C2×D12—81no—
75C2×D12—81no—
76C4×A4—42no—
77C4×A4D812yes—
78C4×A4SD1612yes—
79C4×A4#4112yes—
80C3×C4⋊C4—42no—
81C3×C4⋊C4—81no—
82C3×C42—41no—
83C2×C3⋊C8—42no—
84C22×Q8—42no—
85C22×Q8—61no—
86C22×Q8—61no—
87C22×Q8—62no—
88C22×Q8—122no—
89C22×D4—32no—
90C22×D4—61no—
91C22×D4—61no—
92C22×D4—62no—
93C23×C4—22no—
94C23×C4#6711yescharacteristic
95C23×C4#6711yescharacteristic
96C23×C4C22×D311yescharacteristic
97C22×C8—32no—
98C22×C8—32no—
99C4⋊Q8—32no—
100C4⋊Q8—32no—
101C4⋊Q8—32no—
102C4⋊Q8—31no—
103C4⋊Q8—61no—
104C4⋊Q8—61no—
105C4⋊Q8—61no—
106C4⋊Q8—64no—
107C4⋊Q8S411yescharacteristic, complemented
108#127—61no—
109#127—61no—
110#127—61no—
111#127—61no—
112#121—64no—
113#121—64no—
114#121—122no—
115#121—122no—
116#121—122no—
117#121—122no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—32no—
120C2×C4⋊C4—32no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4—62no—
123C2×C4⋊C4—62no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4—61no—
126C2×C4⋊C4—61no—
127C2×C22⋊C4—62no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4—62no—
130C2×C42—31no—
131C2×C42—31no—
132C2×C42—31no—
133C2×C42—31no—
134#102—68no—
135#102—64no—
136#102—64no—
137C22⋊C8—62no—
138C22⋊C8—62no—
139C4×C8—61no—
140C4×C8—61no—
141C4×C8—61no—
142C4×C8—61no—
143C22×D3—82no—
144C2×A4#12411yescharacteristic
145C2×A4C2×D811yescharacteristic
146C2×A4C2×SD1611yescharacteristic
147S4—82no—
148C3×Q8—162no—
149C2×C3×C4—41no—
150C2×C3×C4—41no—
151C2×C3×C4—41no—
152C2×C3×C4—82no—
153D12—84no—
154D12—162no—
155D12—162no—
156C3⋊C8—82no—
157C24—62no—
158C24C2×#6711yescharacteristic
159C2×Q8—21no—
160C2×Q8—62no—
161C2×Q8—62no—
162C2×Q8—61no—
163C2×Q8—122no—
164C2×Q8—122no—
165C2×Q8—122no—
166C2×Q8—124no—
167C2×Q8—122no—
168C2×Q8—124no—
169C2×D4—62no—
170C2×D4—64no—
171C2×D4—64no—
172C2×D4—62no—
173C2×D4—122no—
174C2×D4—122no—
175C2×D4—121no—
176C2×D4—121no—
177C2×D4—122no—
178C2×D4—122no—
179C2×D4—122no—
180C2×D4—122no—
181C2×D4—122no—
182C2×D4—122no—
183C2×D4—122no—
184C2×D4—122no—
185C2×D4—121no—
186C2×D4—121no—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—31no—
190C22×C4—32no—
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—32no—
195C22×C4—32no—
196C22×C4—42no—
197C22×C4—62no—
198C22×C4—62no—
199C22×C4—62no—
200C22×C4—122no—
201C22×C4#21312yes—
202C22×C4#21512yes—
203C22×C4#23912yes—
204C2×C8—62no—
205C2×C8—62no—
206C2×C8—62no—
207C2×C8—62no—
208C2×C8—62no—
209C2×C8—62no—
210C4⋊C4—21no—
211C4⋊C4—34no—
212C4⋊C4—32no—
213C4⋊C4—62no—
214C4⋊C4—62no—
215C4⋊C4—62no—
216C4⋊C4—64no—
217C4⋊C4—61no—
218C4⋊C4—122no—
219C4⋊C4—122no—
220C4⋊C4C2×S412yes—
221C22⋊C4—124no—
222C22⋊C4—124no—
223C22⋊C4—124no—
224C42—31no—
225C42—31no—
226C42—31no—
227C42—61no—
228C42C2×S411yescharacteristic
229C22×C3—41no—
230C2×D3—162no—
231C2×D3—162no—
232C2×D3—162no—
233A4#48611yescharacteristic, complemented
234C3×C4—42no—
235C3×C4—42no—
236C3×C4—42no—
237C3×C4—162no—
238C23—31no—
239C23—31no—
240C23—31no—
241C23—31no—
242C23—122no—
243C23—122no—
244C23—122no—
245C23—122no—
246C23—122no—
247C23—122no—
248C23—122no—
249C23C2×#21311yescharacteristic
250C23C2×#21511yescharacteristic
251C23#94711yescharacteristic
252Q8—42no—
253Q8—124no—
254Q8—124no—
255Q8—122no—
256D4—124no—
257D4—124no—
258D4—242no—
259D4—242no—
260D4—242no—
261D4—242no—
262D4—242no—
263D4—242no—
264D4—242no—
265D4—242no—
266C2×C4—22no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—32no—
271C2×C4—32no—
272C2×C4—32no—
273C2×C4—32no—
274C2×C4—32no—
275C2×C4—32no—
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—61no—
282C2×C4—62no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4—122no—
290C2×C4GL(2,ℤ/4)11yescharacteristic
291C2×C4GL(2,ℤ/4)11yescharacteristic
292C2×C4C22×S411yescharacteristic
293C8—122no—
294C8—122no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298D3—322no—
299C22—31no—
300C22—31no—
301C22—31no—
302C22—31no—
303C22—31no—
304C22—31no—
305C22—31no—
306C22—31no—
307C22—31no—
308C22—61no—
309C22—242no—
310C22—242no—
311C22—242no—
312C22—242no—
313C22—242no—
314C22—242no—
315C22—242no—
316C22#508811yescharacteristic, complemented
317C22C2×GL(2,ℤ/4)11yescharacteristic
318C4—32no—
319C4—32no—
320C4—32no—
321C4—42no—
322C4—122no—
323C4—242no—
324C4#541512yes—
325C4#541712yes—
326C4#591912yes—
327C3—41noSylow
328C2—31no—
329C2—31no—
330C2—31no—
331C2—31no—
332C2—482no—
333C2C2×#541511yescharacteristic
334C2C2×#541711yescharacteristic
335C2#8550811yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4⋊Q8D3
socleC24C2×#67
prime radical#85289C2
hypercenterC4⋊Q8S4
hypocenterA4#486
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
3. #73053C2semidirect
4. C4×A4⋊C8C2semidirect
15. C22×C4⋊Q8D3semidirect
107. C4⋊Q8S4semidirect
233. A4#486semidirect
316. C22#5088semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊Q82
#852892
#730532
C4×A4⋊C82
#259033
#50884