Finite Group Databasebeta

← C2×#85566 #11715660

Subgroups — C2×#85566 #11715660

Order 768.

911 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 911 conjugacy classes of subgroups — too many to draw. Only its 36 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×#5950C211yesmaximal, characteristic
2#85566C212yesmaximal, direct product factor
3C2×#1791—31nomaximal, Sylow
4#5950C411yescharacteristic
5#5950C2211yescharacteristic
6C2×#997C411yescharacteristic
7C2×A4⋊C8—43nomaximal
8C2×#585D311yescharacteristic
9C2×#413—33no—
10C2×#408—33no—
11#1791—36no—
12#1791—32no—
13#997C2×C411yescharacteristic
14C2×#41⋊C3—81no—
15C2×C4×A4—43no—
16A4⋊C8—46no—
17#585—33no—
18#585—33no—
19#585—61no—
20#585—61no—
21#585Dic311yescharacteristic
22#585C2×D311yescharacteristic
23C2×2−1+4—61no—
24C2×H(5,𝔽2)—33no—
25C2×H(5,𝔽2)Dic311yescharacteristic
26C22×#41—33no—
27C22×#41—61no—
28C22×M16—33no—
29C23×C8—31no—
30#413—36no—
31#413—36no—
32#413—63no—
33#413—63no—
34C2×#101—63no—
35C2×#100—63no—
36#408—36no—
37#408—36no—
38#408—63no—
39#408—63no—
40C2×C22⋊C8—63no—
41C2×C22⋊C8—63no—
42C22×A4—43no—
43#41⋊C3—81no—
44#41⋊C3—81no—
45C2×SL(2,𝔽3)—81no—
46C4×A4—43no—
47C4×A4—43no—
48C2×C3⋊C8—161no—
492−1+4—61no—
502−1+4—61no—
512−1+4—66no—
522−1+4—63no—
532−1+4—63no—
542−1+4—62no—
55H(5,𝔽2)—33no—
56H(5,𝔽2)—33no—
57H(5,𝔽2)—33no—
58H(5,𝔽2)—36no—
59H(5,𝔽2)—61no—
60H(5,𝔽2)—63no—
61H(5,𝔽2)—66no—
62H(5,𝔽2)—66no—
63H(5,𝔽2)—63no—
64H(5,𝔽2)C2×Dic311yescharacteristic
65C2×#41—21no—
66C2×#41—33no—
67C2×#41—33no—
68C2×#41—33no—
69C2×#41—33no—
70C2×#41—33no—
71C2×#41—33no—
72C2×#41—33no—
73C2×#41—33no—
74C2×#41—61no—
75C2×#41—61no—
76C2×#41—61no—
77C2×#41—63no—
78C2×#41—66no—
79C2×#41—66no—
80C2×#41—66no—
81C2×#41—63no—
82C2×#41—63no—
83C2×#41—63no—
84C2×#41—63no—
85C2×#41—63no—
86C2×#41—61no—
87C2×#41—61no—
88C22×Q8—33no—
89C22×Q8—61no—
90C22×D4—33no—
91C22×D4—33no—
92C22×D4—33no—
93C22×D4—63no—
94C23×C4—31no—
95C23×C4—33no—
96C23×C4S413yes—
97C2×M16—36no—
98C2×M16—63no—
99C2×M16—66no—
100C2×M16—66no—
101C22×C8—32no—
102C22×C8—63no—
103C22×C8—63no—
104C22×C8—63no—
105#101—66no—
106#101—66no—
107#100—66no—
108#100—66no—
109C22⋊C8—66no—
110C22⋊C8—66no—
111C22⋊C8—66no—
112C22⋊C8—66no—
113C2×A4—46no—
114C2×A4—43no—
115C2×C3×C4—161no—
116SL(2,𝔽3)—81no—
117C3⋊C8—162no—
118C24—31no—
119C24—33no—
120C24SL(2,ℤ/4)13yes—
121#41—21no—
122#41—21no—
123#41—61no—
124#41—63no—
125#41—66no—
126#41—63no—
127#41—63no—
128#41—66no—
129#41—63no—
130#41—66no—
131#41—66no—
132#41—66no—
133#41—66no—
134#41—63no—
135#41—66no—
136#41—66no—
137#41—66no—
138#41—66no—
139#41—63no—
140#41—63no—
141#41—66no—
142#41—63no—
143#41—61no—
144#41—63no—
145#41—63no—
146#41—66no—
147#41—61no—
148#41—61no—
149C2×Q8—21no—
150C2×Q8—33no—
151C2×Q8—33no—
152C2×Q8—33no—
153C2×Q8—33no—
154C2×Q8—63no—
155C2×Q8—66no—
156C2×Q8—61no—
157C2×Q8—61no—
158C2×Q8—63no—
159C2×Q8—63no—
160C2×D4—33no—
161C2×D4—33no—
162C2×D4—33no—
163C2×D4—33no—
164C2×D4—33no—
165C2×D4—33no—
166C2×D4—33no—
167C2×D4—33no—
168C2×D4—33no—
169C2×D4—33no—
170C2×D4—33no—
171C2×D4—33no—
172C2×D4—66no—
173C2×D4—66no—
174C2×D4—63no—
175C2×D4—66no—
176C2×D4—63no—
177C2×D4—63no—
178C2×D4—66no—
179C2×D4—66no—
180C2×D4—66no—
181C2×D4—61no—
182C2×D4—63no—
183C2×D4—63no—
184C22×C4—31no—
185C22×C4—33no—
186C22×C4—33no—
187C22×C4—33no—
188C22×C4—31no—
189C22×C4—33no—
190C22×C4—33no—
191C22×C4—33no—
192C22×C4—33no—
193C22×C4—33no—
194C22×C4—33no—
195C22×C4—33no—
196C22×C4—33no—
197C22×C4—33no—
198C22×C4—63no—
199C22×C4—63no—
200C22×C4—63no—
201C22×C4—61no—
202C22×C4SL(2,ℤ/4)13yes—
203C22×C4C2×S413yes—
204M16—612no—
205M16—126no—
206C2×C8—66no—
207C2×C8—66no—
208C2×C8—123no—
209C2×C8—123no—
210C2×C8—123no—
211C2×C8—121no—
212C22×C3—161no—
213A4—43no—
214C3×C4—161no—
215C3×C4—161no—
216C23—31no—
217C23—33no—
218C23—33no—
219C23—33no—
220C23—33no—
221C23—33no—
222C23—46no—
223C23—63no—
224C23—61no—
225C23—122no—
226C23—126no—
227C23C2×SL(2,ℤ/4)13yes—
228Q8—21no—
229Q8—63no—
230Q8—66no—
231Q8—63no—
232Q8—61no—
233D4—66no—
234D4—66no—
235D4—66no—
236D4—66no—
237D4—63no—
238D4—63no—
239D4—66no—
240D4—61no—
241D4—61no—
242D4—62no—
243C2×C4—33no—
244C2×C4—36no—
245C2×C4—33no—
246C2×C4—33no—
247C2×C4—33no—
248C2×C4—33no—
249C2×C4—33no—
250C2×C4—33no—
251C2×C4—33no—
252C2×C4—33no—
253C2×C4—33no—
254C2×C4—33no—
255C2×C4—63no—
256C2×C4—63no—
257C2×C4—63no—
258C2×C4—63no—
259C2×C4—63no—
260C2×C4—63no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4#102011yescharacteristic
267C8—126no—
268C8—122no—
269C2×C3—162no—
270C2×C3—161no—
271C22—33no—
272C22—33no—
273C22—43no—
274C22—66no—
275C22—61no—
276C22—61no—
277C22—123no—
278C22—121no—
279C22—123no—
280C22—123no—
281C22—123no—
282C22—122no—
283C22—243no—
284C22#593611yescharacteristic
285C4—33no—
286C4—33no—
287C4—61no—
288C4—61no—
289C4#593611yescharacteristic
290C4C2×#102011yescharacteristic
291C3—161noSylow
292C2—63no—
293C2—63no—
294C2—121no—
295C2—121no—
296C2#8556612yesdirect product factor
297C2C2×#593611yescharacteristic
subgroupG/H
center, hypercenterC2×C4#1020
commutator, hypocenter#997C2×C4
FrattiniC4C2×#1020
FittingC2×#585D3
socleC22#5936
prime radicalC2×#5950C2
cosocle#5950C22
LCSG ▷ #997
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #85566C2direct (N × H)
296. C2#85566direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#59502
#855662
C2×#17913
C2×A4⋊C84