Finite Group Databasebeta

← C4×#5398 #11712857

Subgroups — C4×#5398 #11712857

Order 768.

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

normal subgroups only

This group has 525 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×#5398C211yesmaximal, characteristic
2C4×A4⋊Q8C212yesmaximal
3C4×C8×A4C211yesmaximal, characteristic
4#73111C211yesmaximal, characteristic
5#73109C212yesmaximal
6C4×#467—31nomaximal, Sylow
7C2×A4⋊Q8C2212yes—
8C2×C8×A4C2211yescharacteristic
9C2×C8×A4C2211yescharacteristic
10C42×A4C2211yescharacteristic
11C4⋊SL(2,ℤ/4)C2212yes—
12SL(2,ℤ/4)⋊C4—22no—
13C4×SL(2,ℤ/4)—22no—
14#5398C414yesdirect product factor
15C4×Dic12—41nomaximal
16C2×#467—31no—
17C2×C4×Q16—31no—
18C22×C4×C8D311yescharacteristic
19C4×C22⋊Q8—32no—
20#1969—31no—
21#1942—31no—
22#1933—32no—
23#1892—32no—
24#1808—31no—
25C4×#107—31no—
26C4×#103—32no—
27C2×C4×A4D411yescharacteristic
28C2×C4×A4D411yescharacteristic
29C2×C4×A4C2311yescharacteristic
30C2×SL(2,ℤ/4)—22no—
31C2×SL(2,ℤ/4)—22no—
32A4⋊Q8—22no—
33A4⋊Q8C2×C414yes—
34C2×Dic12—41no—
35C4×Dic6—42no—
36C8×A4—21no—
37C8×A4C2×C412yes—
38C3×C4×C8—41no—
39#818—41no—
40#816—42no—
41C22×Q16—31no—
42C23×C8C2×D311yescharacteristic
43C23×C8C2×D311yescharacteristic
44C2×C22⋊Q8—32no—
45C2×C4×Q8—32no—
46C22×C42C2×D311yescharacteristic
47#467—34no—
48#467—34no—
49C4×Q16—61no—
50C4×Q16—61no—
51C4×Q16—61no—
52C4×Q16—61no—
53C2×#107—31no—
54C2×#107—31no—
55C2×#107—31no—
56C2×#103—32no—
57C2×#103—32no—
58C2×#103—32no—
59C2×C4×C8—31no—
60C2×C4×C8—31no—
61C2×C4×C8—31no—
62C2×C4×C8—31no—
63#391—32no—
64#389—32no—
65#387—62no—
66#385—62no—
67#380—32no—
68C4×C4⋊C4—32no—
69C4×C4⋊C4—62no—
70C4×C22⋊C4—62no—
71#340—32no—
72#340—32no—
73#340—32no—
74C22×A4C2×D411yescharacteristic
75C2×Dic6—42no—
76C4×A4—21no—
77C4×A4Q1612yescomplemented
78C4×A4C22×C411yescharacteristic
79C4×A4#4111yescharacteristic
80SL(2,ℤ/4)—24no—
81SL(2,ℤ/4)—42no—
82C2×C3×C8—41no—
83C2×C3×C8—41no—
84C3×C42—41no—
85C4⋊Dic3—42no—
86Dic3⋊C4—82no—
87C4×Dic3—82no—
88Dic12—44no—
89C22×Q8—32no—
90C23×C4D1211yescharacteristic
91C23×C4D1211yescharacteristic
92C23×C4C22×D311yescharacteristic
93C2×Q16—34no—
94C2×Q16—62no—
95C2×Q16—61no—
96C2×Q16—61no—
97C22×C8—21no—
98C22×C8—31no—
99C22×C8—31no—
100C22×C8—31no—
101C22×C8—32no—
102C22×C8—31no—
103C22×C8—61no—
104C22×C8C4×D312yes—
105C22⋊Q8—34no—
106C22⋊Q8—34no—
107C22⋊Q8—62no—
108C22⋊Q8—62no—
109C4×Q8—62no—
110C4×Q8—62no—
111C4×Q8—62no—
112C4×Q8—62no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—62no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4—62no—
119C2×C22⋊C4—62no—
120C2×C22⋊C4—62no—
121C2×C42—31no—
122C2×C42—31no—
123C2×C42—31no—
124C2×C42—31no—
125C2×C42—62no—
126C2×C42—62no—
127#107—34no—
128#107—62no—
129#107—61no—
130#107—61no—
131#103—38no—
132#103—64no—
133#103—62no—
134#103—62no—
135C4×C8—31no—
136C4×C8—31no—
137C4×C8—31no—
138C4×C8—61no—
139C4×C8S411yescharacteristic
140#95—62no—
141#95—62no—
142#95—62no—
143#95—62no—
144#95—62no—
145C2×A4C4×D411yescharacteristic
146C2×A4C2×Q1611yescharacteristic
147C2×A4#13511yescharacteristic
148C2×C3×C4—41no—
149C2×C3×C4—41no—
150C2×C3×C4—41no—
151C2×Dic3—82no—
152C2×Dic3—82no—
153Dic6—44no—
154Dic6—82no—
155C3×C8—42no—
156C3×C8—81no—
157C24C2×D1211yescharacteristic
158C2×Q8—34no—
159C2×Q8—62no—
160C2×Q8—62no—
161C2×Q8—62no—
162C2×Q8—62no—
163C2×Q8—62no—
164C22×C4—21no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—32no—
173C22×C4—31no—
174C22×C4—61no—
175C22×C4—62no—
176C22×C4—62no—
177C22×C4—62no—
178C22×C4—62no—
179C22×C4Dic1212yescomplemented
180C22×C4C2×C4×D311yescharacteristic
181C22×C4#23511yescharacteristic
182Q16—64no—
183Q16—64no—
184C2×C8—31no—
185C2×C8—32no—
186C2×C8—32no—
187C2×C8—32no—
188C2×C8—32no—
189C2×C8—31no—
190C2×C8—61no—
191C2×C8—61no—
192C2×C8—61no—
193C2×C8—61no—
194C2×C8C2×S411yescharacteristic
195C2×C8C2×S411yescharacteristic
196C4⋊C4—34no—
197C4⋊C4—62no—
198C4⋊C4—64no—
199C4⋊C4—64no—
200C4⋊C4—62no—
201C4⋊C4—62no—
202C4⋊C4—62no—
203C4⋊C4—62no—
204C4⋊C4—124no—
205C4⋊C4—122no—
206C4⋊C4—122no—
207C22⋊C4—64no—
208C22⋊C4—64no—
209C22⋊C4—122no—
210C22⋊C4—122no—
211C42—31no—
212C42—31no—
213C42—31no—
214C42—61no—
215C42—124no—
216C42—122no—
217C42—122no—
218C42C2×S411yescharacteristic
219C22×C3—41no—
220A4C4×Q1611yescharacteristic, complemented
221C3×C4—42no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4—81no—
225Dic3—84no—
226Dic3—162no—
227C23—31no—
228C23—31no—
229C23—31no—
230C23—31no—
231C23C4×D1211yescharacteristic
232C23#90411yescharacteristic
233C23C2×Dic1211yescharacteristic
234Q8—64no—
235Q8—64no—
236Q8—122no—
237Q8—122no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—32no—
241C2×C4—32no—
242C2×C4—31no—
243C2×C4—31no—
244C2×C4—31no—
245C2×C4—31no—
246C2×C4—31no—
247C2×C4—31no—
248C2×C4—32no—
249C2×C4—32no—
250C2×C4—31no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—64no—
257C2×C4—64no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4—122no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4C4⋊S411yescharacteristic
269C2×C4C4⋊S411yescharacteristic
270C2×C4C22×S411yescharacteristic
271C8—21no—
272C8—32no—
273C8—61no—
274C8C4×S412yes—
275C2×C3—41no—
276C2×C3—41no—
277C2×C3—41no—
278C22—31no—
279C22—31no—
280C22—31no—
281C22—31no—
282C22—31no—
283C22—31no—
284C22—31no—
285C22—31no—
286C22—31no—
287C22—61no—
288C22C4×Dic1211yescharacteristic, complemented
289C22C2×C4⋊S411yescharacteristic
290C4—21no—
291C4—31no—
292C4—31no—
293C4—32no—
294C4—61no—
295C4—124no—
296C4—124no—
297C4—242no—
298C4—242no—
299C4#539812yesdirect product factor
300C4C2×C4×S411yescharacteristic
301C4#591211yescharacteristic
302C3—41noSylow
303C2—31no—
304C2—31no—
305C2—31no—
306C2—31no—
307C2C4×C4⋊S411yescharacteristic
308C2C2×#539811yescharacteristic
309C2#8544011yescharacteristic
subgroupG/H
centerC2×C4C4⋊S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×C4×C8D3
socleC24C2×D12
prime radicalC22×A4C2×D4
cosocleC2×C4×A4C23
hypercenterC4×C8S4
hypocenterA4C4×Q16
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
14. #5398C4direct (N × H)
77. C4×A4Q16semidirect
179. C22×C4Dic12semidirect
220. A4C4×Q16semidirect
288. C22C4×Dic12semidirect
299. C4#5398direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#53982
C4×A4⋊Q82
C4×C8×A42
#731112
#731092
C4×#4673
C4×Dic124