Finite Group Databasebeta

← C22×#5933 #11715822

Subgroups — C22×#5933 #11715822

Order 768.

1606 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 52 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
1C22×#997C211yesmaximal, characteristic
2C2×#5933C216yesmaximal, direct product factor
3C22×#455—31nomaximal, Sylow
4C2×#997C2213yes—
5#5933C2214yesdirect product factor
6C22×SL(2,ℤ/4)—41nomaximal
7C22×#226—42nomaximal
8C22×H(5,𝔽2)D311yescharacteristic
9C22×#137—32no—
10C22×#124—31no—
11C2×#455—36no—
12C2×#455—312no—
13C2×#455—36no—
14C22×#104—32no—
15C22×#100—31no—
16C23×A4—41no—
17C23×A4—81no—
18#997C2311yescharacteristic
19C22×SL(2,𝔽3)—42no—
20C2×SL(2,ℤ/4)—46no—
21C2×#226—412no—
22C2×H(5,𝔽2)—33no—
23C2×H(5,𝔽2)—36no—
24C2×H(5,𝔽2)—63no—
25C2×H(5,𝔽2)C2×D313yes—
26C22×#41—32no—
27C23×Q8—31no—
28C23×D4—31no—
29C23×D4—61no—
30C2×#137—312no—
31C2×#137—312no—
32C2×#137—66no—
33C2×#137—66no—
34C22×Q16—62no—
35C22×SD16—62no—
36C22×M16—32no—
37C2×#124—36no—
38C2×#124—36no—
39C2×#124—63no—
40C2×#124—63no—
41C22×C22⋊C4—61no—
42C22×C22⋊C4—61no—
43C22×C42—31no—
44#455—312no—
45#455—324no—
46#455—324no—
47#455—34no—
48C2×#104—312no—
49C2×#104—312no—
50C2×#100—36no—
51C2×#100—36no—
52C22×A4—44no—
53C22×A4—43no—
54C22×A4—83no—
55C22×A4—84no—
56C22×Dic3—161no—
57C2×SL(2,𝔽3)—46no—
58SL(2,ℤ/4)—44no—
59#226—48no—
60C25—21no—
61C25—31no—
62C25S411yescharacteristic
63H(5,𝔽2)—33no—
64H(5,𝔽2)—36no—
65H(5,𝔽2)—36no—
66H(5,𝔽2)—61no—
67H(5,𝔽2)—612no—
68H(5,𝔽2)—63no—
69H(5,𝔽2)—63no—
70H(5,𝔽2)—62no—
71H(5,𝔽2)—66no—
72H(5,𝔽2)—66no—
73H(5,𝔽2)—62no—
74H(5,𝔽2)C22×D311yescharacteristic
75C2×#41—36no—
76C2×#41—36no—
77C2×#41—36no—
78C2×#41—36no—
79C2×#41—66no—
80C2×#41—66no—
81C22×Q8—36no—
82C22×Q8—63no—
83C22×Q8—62no—
84C22×Q8—66no—
85C22×Q8S412yes—
86C22×D4—32no—
87C22×D4—33no—
88C22×D4—33no—
89C22×D4—33no—
90C22×D4—33no—
91C22×D4—62no—
92C22×D4—66no—
93C22×D4—63no—
94C22×D4—63no—
95C22×D4—66no—
96C22×D4—66no—
97C22×D4—63no—
98C22×D4—63no—
99C23×C4—31no—
100C23×C4—61no—
101C23×C4—61no—
102C23×C4—61no—
103#137—324no—
104#137—38no—
105#137—624no—
106#137—612no—
107#137—612no—
108C2×Q16—612no—
109C2×Q16—612no—
110C2×SD16—612no—
111C2×SD16—612no—
112C2×M16—312no—
113C2×M16—66no—
114C22×C8—62no—
115#124—312no—
116#124—34no—
117#124—612no—
118#124—66no—
119#124—66no—
120C2×C22⋊C4—66no—
121C2×C22⋊C4—66no—
122C2×C22⋊C4—66no—
123C2×C22⋊C4—66no—
124C2×C42—36no—
125C2×C42—63no—
126#104—324no—
127#104—38no—
128#100—312no—
129#100—34no—
130C23×C3—161no—
131C2×A4—46no—
132C2×A4—41no—
133C2×A4—81no—
134C2×A4—86no—
135C2×Dic3—166no—
136SL(2,𝔽3)—42no—
137C24—23no—
138C24—31no—
139C24—33no—
140C24—33no—
141C24—33no—
142C24—44no—
143C24—63no—
144C24—61no—
145C24—63no—
146C24—84no—
147C24—124no—
148C24C2×S413yes—
149#41—36no—
150#41—312no—
151#41—36no—
152#41—32no—
153#41—36no—
154#41—66no—
155#41—66no—
156#41—612no—
157#41—66no—
158#41—66no—
159#41—612no—
160C2×Q8—36no—
161C2×Q8—34no—
162C2×Q8—66no—
163C2×Q8—612no—
164C2×Q8—612no—
165C2×Q8—126no—
166C2×Q8—126no—
167C2×Q8C2×S416yes—
168C2×D4—36no—
169C2×D4—36no—
170C2×D4—33no—
171C2×D4—36no—
172C2×D4—33no—
173C2×D4—31no—
174C2×D4—33no—
175C2×D4—66no—
176C2×D4—66no—
177C2×D4—66no—
178C2×D4—66no—
179C2×D4—66no—
180C2×D4—66no—
181C2×D4—66no—
182C2×D4—66no—
183C2×D4—66no—
184C2×D4—61no—
185C2×D4—63no—
186C2×D4—66no—
187C2×D4—612no—
188C2×D4—66no—
189C2×D4—63no—
190C2×D4—66no—
191C2×D4—63no—
192C2×D4—66no—
193C2×D4—66no—
194C2×D4—612no—
195C22×C4—33no—
196C22×C4—33no—
197C22×C4—36no—
198C22×C4—32no—
199C22×C4—63no—
200C22×C4—66no—
201C22×C4—63no—
202C22×C4—66no—
203C22×C4—66no—
204C22×C4—123no—
205C22×C4—124no—
206C22×C4—121no—
207Q16—624no—
208Q16—68no—
209SD16—68no—
210SD16—624no—
211M16—38no—
212M16—612no—
213C2×C8—612no—
214C22⋊C4—612no—
215C22⋊C4—64no—
216C22⋊C4—612no—
217C22⋊C4—64no—
218C42—34no—
219C42—66no—
220C22×C3—164no—
221C22×C3—163no—
222A4—41no—
223A4—81no—
224Dic3—164no—
225C23—21no—
226C23—33no—
227C23—33no—
228C23—33no—
229C23—31no—
230C23—33no—
231C23—46no—
232C23—61no—
233C23—63no—
234C23—63no—
235C23—64no—
236C23—62no—
237C23—63no—
238C23—63no—
239C23—63no—
240C23—86no—
241C23—126no—
242C23—126no—
243C23—126no—
244C23—124no—
245C23—246no—
246C23—246no—
247C23C22×S411yescharacteristic
248C23#102011yescharacteristic
249Q8—36no—
250Q8—62no—
251Q8—68no—
252Q8—1212no—
253Q8C22×S412yes—
254D4—32no—
255D4—36no—
256D4—62no—
257D4—66no—
258D4—66no—
259D4—66no—
260D4—612no—
261D4—66no—
262D4—66no—
263C2×C4—31no—
264C2×C4—33no—
265C2×C4—36no—
266C2×C4—36no—
267C2×C4—36no—
268C2×C4—36no—
269C2×C4—61no—
270C2×C4—66no—
271C2×C4—63no—
272C2×C4—66no—
273C2×C4—64no—
274C2×C4—64no—
275C2×C4—126no—
276C2×C4—1212no—
277C2×C4—126no—
278C8—68no—
279C2×C3—161no—
280C2×C3—166no—
281C22—31no—
282C22—33no—
283C22—41no—
284C22—66no—
285C22—66no—
286C22—61no—
287C22—63no—
288C22—81no—
289C22—123no—
290C22—121no—
291C22—123no—
292C22—126no—
293C22—126no—
294C22—241no—
295C22—243no—
296C22—243no—
297C22—243no—
298C22—242no—
299C22#593314yesdirect product factor
300C22C2×#102013yes—
301C4—32no—
302C4—36no—
303C4—124no—
304C4—124no—
305C3—161noSylow
306C2—61no—
307C2—63no—
308C2—121no—
309C2—123no—
310C2C2×#593316yesdirect product factor
311C2C22×#102011yescharacteristic
subgroupG/H
center, socle, hypercenterC23#1020
commutator, cosocle, hypocenter#997C23
FrattiniC2C22×#1020
FittingC22×H(5,𝔽2)D3
prime radicalC22×#997C2
LCSG ▷ #997
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#5933C2direct (N × H)
5. #5933C22direct (N × H)
299. C22#5933direct (N × H)
310. C2C2×#5933direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9972
C2×#59332
C22×#4553
C22×SL(2,ℤ/4)4
C22×#2264