Finite Group Databasebeta

← C4×#5402 #11712856

Subgroups — C4×#5402 #11712856

Order 768.

633 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 633 conjugacy classes of subgroups — too many to draw. Only its 73 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
2C4×C4⋊S4C212yesmaximal, complemented
3C4×C8×A4C211yesmaximal, characteristic, complemented
4#73113C212yesmaximal, complemented
5#73111C211yesmaximal, characteristic, complemented
6C4×#466—31nomaximal, Sylow
7C2×C4⋊S4C2212yes—
8C2×C4×S4—22no—
9C2×C8×A4C2211yescharacteristic
10C2×C8×A4C2211yescharacteristic
11C42×A4C2211yescharacteristic
12(C2×S4)⋊C4—22no—
13C4⋊SL(2,ℤ/4)C2212yes—
14#5402C414yesdirect product factor
15C4×D24—41nomaximal
16C2×#466—31no—
17C2×C4×D8—31no—
18C22×C4×C8D311yescharacteristic, complemented
19C4×#121—32no—
20#1970—31no—
21#1942—31no—
22#1934—32no—
23#1893—32no—
24#1808—31no—
25C4×#107—31no—
26C4×#102—32no—
27C22×S4—22no—
28C2×C4×A4D411yescharacteristic
29C2×C4×A4D411yescharacteristic
30C2×C4×A4C2311yescharacteristic
31C2×SL(2,ℤ/4)—22no—
32C4⋊S4—22no—
33C4⋊S4C2×C414yescomplemented
34C4×S4—44no—
35C2×D24—41no—
36C4×D12—42no—
37C8×A4—21no—
38C8×A4C2×C412yescomplemented
39C3×C4×C8—41no—
40#821—42no—
41#818—41no—
42C22×D8—31no—
43C23×C8C2×D311yescharacteristic
44C23×C8C2×D311yescharacteristic
45C2×#121—32no—
46C2×C4×D4—32no—
47C2×C4×D4—62no—
48C22×C42C2×D311yescharacteristic
49#466—34no—
50#466—34no—
51C4×D8—61no—
52C4×D8—61no—
53C4×D8—61no—
54C4×D8—61no—
55C2×#107—31no—
56C2×#107—31no—
57C2×#107—31no—
58C2×#102—32no—
59C2×#102—32no—
60C2×#102—32no—
61C2×C4×C8—31no—
62C2×C4×C8—31no—
63C2×C4×C8—31no—
64C2×C4×C8—31no—
65#390—32no—
66#389—32no—
67#388—62no—
68#386—62no—
69#380—32no—
70C4×C4⋊C4—32no—
71C4×C22⋊C4—62no—
72#340—32no—
73#340—32no—
74#340—32no—
75C22×A4C2×D411yescharacteristic
76C2×S4—24no—
77C2×S4—42no—
78C2×S4—42no—
79C2×D12—42no—
80C2×C4×D3—82no—
81C4×A4—21no—
82C4×A4D812yescomplemented
83C4×A4C22×C411yescharacteristic
84C4×A4#4111yescharacteristic
85SL(2,ℤ/4)—42no—
86C2×C3×C8—41no—
87C2×C3×C8—41no—
88C3×C42—41no—
89(C2×D3)⋊C4—82no—
90C4⋊Dic3—42no—
91D24—44no—
92C22×D4—32no—
93C22×D4—62no—
94C23×C4—62no—
95C23×C4D1211yescharacteristic
96C23×C4D1211yescharacteristic
97C23×C4C22×D311yescharacteristic
98C2×D8—34no—
99C2×D8—62no—
100C2×D8—61no—
101C2×D8—61no—
102C22×C8—21no—
103C22×C8—31no—
104C22×C8—31no—
105C22×C8—31no—
106C22×C8—32no—
107C22×C8—31no—
108C22×C8—61no—
109C22×C8C4×D312yescomplemented
110#121—34no—
111#121—34no—
112#121—62no—
113#121—62no—
114C4×D4—62no—
115C4×D4—62no—
116C4×D4—62no—
117C4×D4—62no—
118C4×D4—124no—
119C4×D4—124no—
120C2×C4⋊C4—32no—
121C2×C4⋊C4—32no—
122C2×C4⋊C4—32no—
123C2×C4⋊C4—62no—
124C2×C22⋊C4—62no—
125C2×C22⋊C4—62no—
126C2×C22⋊C4—62no—
127C2×C22⋊C4—62no—
128C2×C22⋊C4—62no—
129C2×C42—31no—
130C2×C42—31no—
131C2×C42—31no—
132C2×C42—31no—
133C2×C42—62no—
134#107—34no—
135#107—62no—
136#107—61no—
137#107—61no—
138#102—38no—
139#102—64no—
140#102—62no—
141#102—62no—
142C4×C8—31no—
143C4×C8—31no—
144C4×C8—31no—
145C4×C8—61no—
146C4×C8S411yescharacteristic, complemented
147#95—62no—
148#95—62no—
149#95—62no—
150C22×D3—82no—
151C2×A4C4×D411yescharacteristic
152C2×A4C2×D811yescharacteristic
153C2×A4#13511yescharacteristic
154S4—44no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×Dic3—82no—
159D12—44no—
160D12—82no—
161C4×D3—164no—
162C3×C8—42no—
163C3×C8—81no—
164C24—62no—
165C24C2×D1211yescharacteristic
166C2×D4—34no—
167C2×D4—62no—
168C2×D4—62no—
169C2×D4—62no—
170C2×D4—62no—
171C2×D4—64no—
172C2×D4—64no—
173C2×D4—62no—
174C2×D4—122no—
175C2×D4—122no—
176C2×D4—122no—
177C2×D4—122no—
178C22×C4—21no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—32no—
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—61no—
189C22×C4—62no—
190C22×C4—62no—
191C22×C4—62no—
192C22×C4—122no—
193C22×C4—124no—
194C22×C4—124no—
195C22×C4—122no—
196C22×C4D2412yescomplemented
197C22×C4C2×C4×D311yescharacteristic
198C22×C4#23511yescharacteristic
199D8—64no—
200D8—64no—
201C2×C8—31no—
202C2×C8—32no—
203C2×C8—32no—
204C2×C8—32no—
205C2×C8—32no—
206C2×C8—31no—
207C2×C8—61no—
208C2×C8—61no—
209C2×C8—61no—
210C2×C8—61no—
211C2×C8C2×S411yescharacteristic
212C2×C8C2×S411yescharacteristic
213C4⋊C4—34no—
214C4⋊C4—62no—
215C4⋊C4—62no—
216C4⋊C4—62no—
217C4⋊C4—62no—
218C4⋊C4—62no—
219C4⋊C4—124no—
220C22⋊C4—64no—
221C22⋊C4—64no—
222C22⋊C4—124no—
223C22⋊C4—122no—
224C22⋊C4—124no—
225C22⋊C4—122no—
226C22⋊C4—122no—
227C22⋊C4—122no—
228C42—31no—
229C42—31no—
230C42—31no—
231C42—61no—
232C42—124no—
233C42C2×S411yescharacteristic
234C22×C3—41no—
235C2×D3—84no—
236C2×D3—162no—
237C2×D3—162no—
238A4C4×D811yescharacteristic, complemented
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—42no—
242C3×C4—81no—
243Dic3—162no—
244C23—31no—
245C23—31no—
246C23—31no—
247C23—31no—
248C23—64no—
249C23—122no—
250C23—122no—
251C23—122no—
252C23—122no—
253C23—122no—
254C23—122no—
255C23C4×D1211yescharacteristic
256C23C2×D2411yescharacteristic
257C23#90411yescharacteristic
258D4—64no—
259D4—64no—
260D4—122no—
261D4—124no—
262D4—124no—
263D4—124no—
264D4—124no—
265D4—122no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—32no—
273C2×C4—32no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—31no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—64no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—244no—
294C2×C4—244no—
295C2×C4C4⋊S411yescharacteristic
296C2×C4C4⋊S411yescharacteristic
297C2×C4C22×S411yescharacteristic
298C8—21no—
299C8—32no—
300C8—61no—
301C8C4×S412yescomplemented
302C2×C3—41no—
303C2×C3—41no—
304C2×C3—41no—
305D3—164no—
306C22—31no—
307C22—31no—
308C22—31no—
309C22—31no—
310C22—31no—
311C22—31no—
312C22—31no—
313C22—31no—
314C22—31no—
315C22—61no—
316C22—124no—
317C22—124no—
318C22—124no—
319C22—242no—
320C22—242no—
321C22—242no—
322C22—242no—
323C22C4×D2411yescharacteristic, complemented
324C22C2×C4⋊S411yescharacteristic
325C4—21no—
326C4—31no—
327C4—31no—
328C4—32no—
329C4—61no—
330C4—124no—
331C4—242no—
332C4—242no—
333C4#540212yesdirect product factor
334C4C2×C4×S411yescharacteristic
335C4#591211yescharacteristic
336C3—41noSylow
337C2—31no—
338C2—31no—
339C2—31no—
340C2—31no—
341C2—244no—
342C2C4×C4⋊S411yescharacteristic
343C2C2×#540211yescharacteristic
344C2#8544011yescharacteristic
subgroupG/H
centerC2×C4C4⋊S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×C4×C8D3
socleC24C2×D12
prime radicalC2×#5402C2
cosocleC2×C4×A4C23
hypercenterC4×C8S4
hypocenterA4C4×D8
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊S4C2semidirect
3. C4×C8×A4C2semidirect
4. #73113C2semidirect
5. #73111C2semidirect
14. #5402C4direct (N × H)
18. C22×C4×C8D3semidirect
33. C4⋊S4C2×C4semidirect
38. C8×A4C2×C4semidirect
82. C4×A4D8semidirect
109. C22×C8C4×D3semidirect
146. C4×C8S4semidirect
196. C22×C4D24semidirect
238. A4C4×D8semidirect
301. C8C4×S4semidirect
323. C22C4×D24semidirect
333. C4#5402direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#54022
C4×C4⋊S42
C4×C8×A42
#731132
#731112
C4×#4663
C4×D244