Finite Group Databasebeta

← C2×#10592794 #11864899

Subgroups — C2×#10592794 #11864899

Order 1152.

2124 conjugacy classes of subgroups; only the 313 normal subgroups are shown.

normal subgroups only

This group has 2124 conjugacy classes of subgroups — too many to draw. Only its 313 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×#64364C211yesmaximal, characteristic
2C2×C32×C4⋊Q8C211yesmaximal, characteristic, complemented
3C2×#64151C212yesmaximal, complemented
4C2×#63663C211yesmaximal, characteristic, complemented
5#10592794C218yesmaximal, direct product factor
6C2×#63632C212yesmaximal, complemented
7C2×#5095—34nomaximal
8C22×C32×Q8C2211yescharacteristic
9C22×C32×Q8C2211yescharacteristic
10#64364—22no—
11#64364—24no—
12#64364C2214yes—
13C2×#3832—22no—
14C2×Q8×C3⋊D3—22no—
15C2×#3828—22no—
16C2×#3828C2212yes—
17C32×C4⋊Q8—22no—
18C32×C4⋊Q8C2214yescomplemented
19C2×C32×C4⋊C4—22no—
20C2×C32×C42C2211yescharacteristic
21C2×#3756—22no—
22#64151—24no—
23#64151C2218yescomplemented
24C2×#3755—22no—
25C2×#3748C2212yes—
26#63663C2214yescomplemented
27#63632C2218yescomplemented
28C2×#1007—34no—
29C2×C3×C4⋊Q8D314yescomplemented
30C2×#942—38no—
31C2×#836—34no—
32#5095—332no—
33C2×#805—38no—
34C2×C32×Q8—22no—
35C2×C32×Q8—22no—
36C2×C32×Q8—21no—
37C2×C32×Q8—22no—
38C2×C32×Q8—22no—
39C2×C32×Q8D412yes—
40C2×C32×Q8C2312yes—
41C2×C32×Q8C2312yes—
42C22×C32×C4—21no—
43C22×C32×C4—21no—
44C22×C32×C4D412yes—
45C22×C32×C4C2311yescharacteristic
46C2×#638—22no—
47#3832—28no—
48#3832—28no—
49Q8×C3⋊D3—28no—
50Q8×C3⋊D3—28no—
51#3828—28no—
52#3828—28no—
53#3828—22no—
54#3828—22no—
55#3828D414yes—
56#3828C2314yes—
57C2×C4⋊(C3⋊D3)—22no—
58C2×C4⋊(C3⋊D3)D412yes—
59C2×C4×C3⋊D3—24no—
60C2×C4×C3⋊D3—22no—
61C2×#634—22no—
62C2×#634D412yes—
63C32×C4⋊C4—28no—
64C32×C42—21no—
65C32×C42C2312yes—
66#3756—28no—
67#3755—28no—
68#3748—22no—
69#3748C2314yes—
70C2×C3⋊(C3⋊C8)—22no—
71C2×#456—91noSylow
72C22×C3×Q8C2×D314yes—
73C22×C3×Q8C2×D314yes—
74#1007—316no—
75#1007—616no—
76#1007—68no—
77C2×#239—68no—
78C2×D3×Q8—68no—
79C2×#235—38no—
80C2×#235—68no—
81C3×C4⋊Q8—28no—
82C3×C4⋊Q8C2×D3116yescomplemented
83C2×C3×C4⋊C4—28no—
84C2×C3×C42C2×D314yes—
85C2×#216—68no—
86#942—332no—
87#942—616no—
88C2×#215—68no—
89C2×#208—38no—
90#836—316no—
91#805—332no—
92C23×C32C2×D411yescharacteristic
93C22×C3⋊D3—22no—
94C32×Q8—24no—
95C32×Q8—24no—
96C32×Q8—24no—
97C32×Q8—42no—
98C2×C32×C4—24no—
99C2×C32×C4—22no—
100C2×C32×C4—22no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—41no—
104C2×C32×C4—41no—
105C2×C32×C4C2×D412yes—
106C2×C32×C4C2×D412yes—
107C2×C32×C4C2×D414yes—
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2411yescharacteristic
110#638—24no—
111#638—24no—
112C2×C3⋊Dic3—22no—
113C4⋊(C3⋊D3)—28no—
114C4⋊(C3⋊D3)—22no—
115C4⋊(C3⋊D3)C2×D414yes—
116C4×C3⋊D3—216no—
117C4×C3⋊D3—24no—
118C4×C3⋊D3—24no—
119#634—28no—
120#634—22no—
121#634C2×D414yes—
122C3⋊(C3⋊C8)—24no—
123C2×2−1+4—91no—
124C2×#137—92no—
125C2×C4⋊Q8C3⋊D311yescharacteristic, complemented
126#456—98no—
127C2×#104—92no—
128C2×#101—91no—
129C2×C3×Q8—28no—
130C2×C3×Q8—28no—
131C2×C3×Q8—24no—
132C2×C3×Q8—28no—
133C2×C3×Q8—28no—
134C2×C3×Q8#6718yes—
135C2×C3×Q8C22×D318yes—
136C2×C3×Q8C22×D318yes—
137C22×C3×C4—24no—
138C22×C3×C4—24no—
139C22×C3×C4#6718yes—
140C22×C3×C4C22×D314yes—
141C2×#67—68no—
142#239—632no—
143#239—632no—
144D3×Q8—632no—
145D3×Q8—632no—
146#235—316no—
147#235—316no—
148#235—632no—
149#235—68no—
150#235—68no—
151#235—632no—
152C2×D12—38no—
153C2×D12—68no—
154C2×C4×D3—616no—
155C2×C4×D3—68no—
156C2×Dic6—38no—
157C2×Dic6—68no—
158C3×C4⋊C4—232no—
159C3×C42—24no—
160C3×C42C22×D318yes—
161#216—632no—
162#215—632no—
163#208—316no—
164#208—68no—
165C2×C3⋊C8—68no—
166C22×C32—22no—
167C22×C32#12011yescharacteristic
168C22×C32#12011yescharacteristic
169C22×C32C22×D411yescharacteristic
170C2×C3⋊D3—24no—
171C2×C3⋊D3—44no—
172C32×C4—24no—
173C32×C4—24no—
174C32×C4—42no—
175C32×C4#12012yes—
176C32×C4C22×D412yes—
177C3⋊Dic3—24no—
1782−1+4—94no—
1792−1+4—184no—
1802−1+4—182no—
181C2×#41—92no—
182C2×#41—182no—
183C2×#41—182no—
184C22×Q8—182no—
185C22×Q8C2×C3⋊D311yescharacteristic
186C22×Q8C2×C3⋊D311yescharacteristic
187#137—98no—
188#137—184no—
189C2×Q16—182no—
190C2×SD16—182no—
191C2×M16—92no—
192C4⋊Q8—22no—
193C4⋊Q8C2×C3⋊D314yescomplemented
194C2×C4⋊C4—22no—
195C2×C42C2×C3⋊D311yescharacteristic
196#104—98no—
197#101—94no—
198C23×C3C2×#6714yes—
199C22×D3—68no—
200C3×Q8—216no—
201C3×Q8—216no—
202C3×Q8—216no—
203C3×Q8—48no—
204C2×C3×C4—216no—
205C2×C3×C4—28no—
206C2×C3×C4—28no—
207C2×C3×C4—28no—
208C2×C3×C4—28no—
209C2×C3×C4—44no—
210C2×C3×C4—44no—
211C2×C3×C4D3×D418yes—
212C2×C3×C4D3×D418yes—
213C2×C3×C4C2×#67116yes—
214C2×C3×C4C2×#6714yes—
215C2×C3×C4C23×D314yes—
216#67—616no—
217#67—616no—
218C2×Dic3—68no—
219D12—316no—
220D12—632no—
221D12—68no—
222C4×D3—664no—
223C4×D3—616no—
224C4×D3—616no—
225Dic6—316no—
226Dic6—632no—
227Dic6—68no—
228C3⋊C8—616no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32#45612yescomplemented
232C2×C32C2×#12011yescharacteristic
233C3⋊D3—44no—
234#41—94no—
235#41—94no—
236#41—188no—
237#41—188no—
238#41—188no—
239#41—182no—
240#41—182no—
241#41—188no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—92no—
248C2×Q8—182no—
249C2×Q8—188no—
250C2×Q8—188no—
251C2×Q8#63812yes—
252C2×Q8C22×C3⋊D312yes—
253C2×Q8C22×C3⋊D312yes—
254C2×D4—92no—
255C2×D4—182no—
256C2×D4—182no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—184no—
260C22×C4—182no—
261C22×C4#63812yes—
262C22×C4C22×C3⋊D311yescharacteristic
263Q16—188no—
264SD16—188no—
265M16—94no—
266M16—182no—
267C2×C8—182no—
268C4⋊C4—28no—
269C42—21no—
270C42C22×C3⋊D312yes—
271C22×C3—28no—
272C22×C3#93714yes—
273C22×C3#93714yes—
274C22×C3C22×#6714yes—
275C2×D3—616no—
276C2×D3—1216no—
277C3×C4—216no—
278C3×C4—216no—
279C3×C4—48no—
280C3×C4#93718yes—
281C3×C4C2×D3×D418yes—
282Dic3—616no—
283C32C2×#45611yescharacteristic, complemented, Sylow
284C23—182no—
285C23C2×#63811yescharacteristic
286Q8—24no—
287Q8—24no—
288Q8—24no—
289Q8—42no—
290Q8—94no—
291Q8—188no—
292Q8—182no—
293D4—94no—
294D4—188no—
295D4—184no—
296D4—184no—
297D4—182no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—1816no—
306C2×C4—182no—
307C2×C4—184no—
308C2×C4—184no—
309C2×C4D4×C3⋊D312yes—
310C2×C4D4×C3⋊D312yes—
311C2×C4C2×#63814yes—
312C2×C4C2×#63811yescharacteristic
313C2×C4C23×C3⋊D311yescharacteristic
314C8—184no—
315C2×C3—24no—
316C2×C3—24no—
317C2×C3#509518yescomplemented
318C2×C3C2×#93714yes—
319D3—1216no—
320C22—22no—
321C22—184no—
322C22—364no—
323C22#6414611yescharacteristic
324C22#6414611yescharacteristic
325C22C22×#63811yescharacteristic
326C4—24no—
327C4—24no—
328C4—42no—
329C4—184no—
330C4#6414612yes—
331C4C2×D4×C3⋊D312yes—
332C3C2×#509514yescomplemented
333C2—21no—
334C2—21no—
335C2—364no—
336C2#1059279412yesdirect product factor
337C2C2×#6414611yescharacteristic
subgroupG/H
centerC22#64146
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC2×C32×C4⋊Q8C2
socleC22×C32#120
prime radicalC2×#64364C2
hypercenterC2×C4⋊Q8C3⋊D3
hypocenterC32C2×#456
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C32×C4⋊Q8C2semidirect
3. C2×#64151C2semidirect
4. C2×#63663C2semidirect
5. #10592794C2direct (N × H)
6. C2×#63632C2semidirect
18. C32×C4⋊Q8C22semidirect
23. #64151C22semidirect
26. #63663C22semidirect
27. #63632C22semidirect
29. C2×C3×C4⋊Q8D3semidirect
82. C3×C4⋊Q8C2×D3semidirect
125. C2×C4⋊Q8C3⋊D3semidirect
193. C4⋊Q8C2×C3⋊D3semidirect
231. C2×C32#456semidirect
283. C32C2×#456semidirect
317. C2×C3#5095semidirect
332. C3C2×#5095semidirect
336. C2#10592794direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#643642
C2×C32×C4⋊Q82
C2×#641512
C2×#636632
#105927942
C2×#636322
C2×#50953