Finite Group Databasebeta

← #11713132

Subgroups — #11713132

Order 768.

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

normal subgroups only

This group has 621 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
1C2×#5402C211yesmaximal, characteristic
2#85308C212yesmaximal, complemented
3A4×#107C211yesmaximal, characteristic, complemented
4C8×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5#73053C212yesmaximal, complemented
6#23025—31nomaximal, Sylow
7C2×C4⋊S4C2212yes—
8C2×C4×S4—22no—
9C2×C8×A4C2211yescharacteristic
10A4×C4⋊C4C2212yes—
11(C2×S4)⋊C4—22no—
12C4×SL(2,ℤ/4)C2211yescharacteristic
13C2×A4⋊C8C2211yescharacteristic
14#5402C414yescomplemented
15#4872—41nomaximal
16C2×#466—31no—
17C2×C4×D8—31no—
18C22×#107D311yescharacteristic, complemented
19#2361—32no—
20#1970—31no—
21#1942—31no—
22#1925—32no—
23#1874—32no—
24#1862—32no—
25C4×#107—31no—
26C8×C22⋊C4—31no—
27C22×S4—22no—
28C2×C4×A4—22no—
29C2×C4×A4C2311yescharacteristic
30C2×SL(2,ℤ/4)D411yescharacteristic
31C2×SL(2,ℤ/4)D411yescharacteristic
32C4⋊S4—22no—
33C4⋊S4C2×C414yescomplemented
34C4×S4—44no—
35C2×D24—41no—
36#893—42no—
37C8×A4C2×C412yescomplemented
38A4⋊C8—21no—
39C3×#107—41no—
40C8×Dic3—41no—
41#809—42no—
42C22×D8—31no—
43C23×C8C2×D311yescharacteristic
44C2×#121—32no—
45C2×C4×D4—32no—
46C2×C4×D4—62no—
47C22×C4⋊C4C2×D312yes—
48#466—34no—
49#466—34no—
50C4×D8—62no—
51C4×D8—61no—
52C4×D8—61no—
53C2×#107—31no—
54C2×#107—31no—
55C2×#107—32no—
56C2×#107—31no—
57C2×#102—32no—
58C2×#102—32no—
59C2×#102—32no—
60C2×C22⋊C8—31no—
61C2×C4×C8—31no—
62C2×C4×C8—31no—
63#390—32no—
64#389—32no—
65#388—62no—
66#386—62no—
67#385—62no—
68C4×C4⋊C4—32no—
69C4×C22⋊C4—31no—
70#340—32no—
71#340—32no—
72#336—31no—
73#336—31no—
74C22×A4C2×D411yescharacteristic
75C2×S4—24no—
76C2×S4—42no—
77C2×S4—42no—
78C2×D12—42no—
79C2×C4×D3—82no—
80C4×A4—42no—
81C4×A4C22×C411yescharacteristic
82C4×A4#4111yescharacteristic
83SL(2,ℤ/4)—21no—
84SL(2,ℤ/4)D812yescomplemented
85C2×C3×C8—41no—
86C3×C4⋊C4—42no—
87(C2×D3)⋊C4—82no—
88C4×Dic3—41no—
89C2×C3⋊C8—41no—
90D24—44no—
91C22×D4—32no—
92C22×D4—62no—
93C23×C4—22no—
94C23×C4—62no—
95C23×C4C22×D311yescharacteristic
96C2×D8—34no—
97C2×D8—62no—
98C2×D8—61no—
99C2×D8—61no—
100C22×C8—31no—
101C22×C8—31no—
102C22×C8—32no—
103C22×C8—31no—
104C22×C8—31no—
105C22×C8C4×D312yescomplemented
106#121—34no—
107#121—34no—
108#121—62no—
109#121—62no—
110C4×D4—62no—
111C4×D4—62no—
112C4×D4—122no—
113C4×D4—124no—
114C4×D4—124no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—32no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—62no—
120C2×C22⋊C4—31no—
121C2×C22⋊C4—31no—
122C2×C22⋊C4—62no—
123C2×C22⋊C4—62no—
124C2×C22⋊C4—62no—
125C2×C22⋊C4—62no—
126C2×C42—31no—
127C2×C42—31no—
128C2×C42—62no—
129#107—34no—
130#107—31no—
131#107—32no—
132#107—61no—
133#107S411yescharacteristic, complemented
134#102—38no—
135#102—62no—
136#102—62no—
137#102—122no—
138C22⋊C8—61no—
139C22⋊C8—61no—
140C4×C8—62no—
141C4×C8—61no—
142C4×C8—61no—
143#95—31no—
144#95—31no—
145#95—62no—
146#95—62no—
147C22×D3—82no—
148C2×A4C4×D411yescharacteristic
149C2×A4C2×D811yescharacteristic
150C2×A4#13511yescharacteristic
151S4—44no—
152C2×C3×C4—41no—
153C2×C3×C4—82no—
154C2×Dic3—41no—
155C2×Dic3—41no—
156D12—44no—
157D12—82no—
158C4×D3—164no—
159C3×C8—42no—
160C3⋊C8—81no—
161C24—62no—
162C24D3×D411yescharacteristic
163C2×D4—34no—
164C2×D4—62no—
165C2×D4—62no—
166C2×D4—62no—
167C2×D4—62no—
168C2×D4—64no—
169C2×D4—64no—
170C2×D4—62no—
171C2×D4—122no—
172C2×D4—122no—
173C2×D4—122no—
174C2×D4—122no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—31no—
183C22×C4—42no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4—62no—
187C22×C4—122no—
188C22×C4—124no—
189C22×C4—122no—
190C22×C4—122no—
191C22×C4—242no—
192C22×C4C2×C4×D311yescharacteristic
193C22×C4#23911yescharacteristic
194D8—64no—
195D8—64no—
196C2×C8—31no—
197C2×C8—32no—
198C2×C8—32no—
199C2×C8—32no—
200C2×C8—61no—
201C2×C8—61no—
202C2×C8—61no—
203C2×C8—61no—
204C2×C8—61no—
205C2×C8—61no—
206C2×C8—61no—
207C2×C8C2×S411yescharacteristic
208C4⋊C4—34no—
209C4⋊C4—32no—
210C4⋊C4—62no—
211C4⋊C4—62no—
212C4⋊C4—62no—
213C4⋊C4—124no—
214C4⋊C4C2×S412yes—
215C22⋊C4—32no—
216C22⋊C4—32no—
217C22⋊C4—64no—
218C22⋊C4—64no—
219C22⋊C4—61no—
220C22⋊C4—61no—
221C22⋊C4—122no—
222C22⋊C4—124no—
223C22⋊C4—122no—
224C22⋊C4—242no—
225C42—61no—
226C42—61no—
227C42—61no—
228C42—61no—
229C42—124no—
230C22×C3—41no—
231C2×D3—84no—
232C2×D3—162no—
233C2×D3—162no—
234A4C4×D811yescharacteristic, complemented
235C3×C4—41no—
236C3×C4—41no—
237C3×C4—162no—
238Dic3—42no—
239Dic3—81no—
240C23—31no—
241C23—31no—
242C23—31no—
243C23—31no—
244C23—64no—
245C23—122no—
246C23—122no—
247C23—122no—
248C23—122no—
249C23—122no—
250C23—122no—
251C23#89311yescharacteristic
252C23D3×D811yescharacteristic
253C23#91911yescharacteristic
254D4—64no—
255D4—64no—
256D4—122no—
257D4—124no—
258D4—124no—
259D4—124no—
260D4—124no—
261D4—122no—
262C2×C4—22no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—32no—
271C2×C4—32no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—64no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—62no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—244no—
292C2×C4—244no—
293C2×C4C22×S411yescharacteristic
294C8—32no—
295C8—121no—
296C8—121no—
297C8C4×S412yescomplemented
298C2×C3—41no—
299C2×C3—41no—
300C2×C3—41no—
301D3—164no—
302C22—31no—
303C22—31no—
304C22—31no—
305C22—31no—
306C22—31no—
307C22—31no—
308C22—31no—
309C22—31no—
310C22—31no—
311C22—61no—
312C22—124no—
313C22—124no—
314C22—124no—
315C22—242no—
316C22—242no—
317C22—242no—
318C22—242no—
319C22#487211yescharacteristic, complemented
320C22D4×S411yescharacteristic
321C4—31no—
322C4—31no—
323C4—42no—
324C4—62no—
325C4—62no—
326C4—121no—
327C4—124no—
328C4—122no—
329C4—121no—
330C4C2×C4×S411yescharacteristic
331C4#591911yescharacteristic
332C3—41noSylow
333C2—31no—
334C2—31no—
335C2—31no—
336C2—31no—
337C2—244no—
338C2#8530811yescharacteristic
339C2D8×S411yescharacteristic
340C2#8546511yescharacteristic
subgroupG/H
centerC22D4×S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×#107D3
socleC24D3×D4
prime radicalC2×#5402C2
cosocleC2×C4×A4C23
hypercenter#107S4
hypocenterA4C4×D8
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. #85308C2semidirect
3. A4×#107C2semidirect
4. C8×SL(2,ℤ/4)C2semidirect
5. #73053C2semidirect
14. #5402C4semidirect
18. C22×#107D3semidirect
33. C4⋊S4C2×C4semidirect
37. C8×A4C2×C4semidirect
84. SL(2,ℤ/4)D8semidirect
105. C22×C8C4×D3semidirect
133. #107S4semidirect
234. A4C4×D8semidirect
297. C8C4×S4semidirect
319. C22#4872semidirect

Maximal subgroups

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

subgroupindex
C2×#54022
#853082
A4×#1072
C8×SL(2,ℤ/4)2
#730532
#230253
#48724