Finite Group Databasebeta

← C22×#63814 #11918474

Subgroups — C22×#63814 #11918474

Order 1152.

3157 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#235C211yesmaximal, characteristic
2C22×#3748C211yesmaximal, characteristic, complemented
3C23×C3×C3⋊C8C211yesmaximal, characteristic, complemented
4C22×#3711C212yesmaximal, complemented
5C2×#63814C2124yesmaximal, direct product factor
6C22×D3×C3⋊C8C212yesmaximal, complemented
7C22×#948—31nomaximal
8C22×#901—31nomaximal
9C23×C32×C4C2211yescharacteristic
10C22×C3×#67C412yes—
11C2×C3×#235C4112yes—
12C2×C3×#235C22112yes—
13C22×C3×D12C411yescharacteristic
14C22×C3×C4×D3C2212yes—
15C22×C3×Dic6C411yescharacteristic
16C2×#3748C22112yescomplemented
17C22×C3⋊(C3⋊C8)C2212yes—
18C22×C3×C3⋊C8C2211yescharacteristic
19C22×C3×C3⋊C8C2211yescharacteristic
20C22×C3×C3⋊C8C22112yescomplemented
21C2×#3711C22124yescomplemented
22#63814C22164yesdirect product factor
23C2×D3×C3⋊C8C22124yescomplemented
24C22×C3×#41—31no—
25C22×#235D311yescharacteristic
26C23×C3×C8—31no—
27C2×#948—324no—
28C22×#208—31no—
29C22×#208—31no—
30C22×#208—32no—
31C22×#208—61no—
32C23×C3⋊C8—32no—
33C23×C3⋊C8D311yescharacteristic, complemented
34C2×#901—324no—
35C22×ΓL(1,𝔽25)—32no—
36C22×D3×C8—32no—
37C24×C32C2×C411yescharacteristic
38C23×C3×D3C2×C412yes—
39C22×C32×C4C2×C411yescharacteristic
40C22×C32×C4C2×C416yes—
41C22×C32×C4C2311yescharacteristic
42C22×C32×C4C2316yes—
43C2×C3×#67C2×C4124yes—
44C22×C3×Dic3C2×C412yes—
45C3×#235C2×C4148yes—
46C3×#235C23116yes—
47C2×C3×D12C2×C4112yes—
48C2×C3×C4×D3C2×C4112yes—
49C2×C3×C4×D3C23112yes—
50C2×C3×Dic6C2×C4112yes—
51#3748C23116yescomplemented
52C2×C3⋊(C3⋊C8)C23112yes—
53C2×C3×C3⋊C8C2316yes—
54C2×C3×C3⋊C8C2316yes—
55C2×C3×C3⋊C8C23116yescomplemented
56#3711C23132yescomplemented
57D3×C3⋊C8C23132yescomplemented
58C22×#131—91noSylow
59C2×C3×#41—312no—
60C2×C3×#41—312no—
61C22×C3×Q8—31no—
62C22×C3×D4—32no—
63C22×C3×D4—31no—
64C23×C3×C4—21no—
65C23×C3×C4—32no—
66C23×C3×C4C2×D311yescharacteristic
67C23×C3×C4C2×D311yescharacteristic
68C22×#67Dic312yes—
69C2×#235Dic3112yes—
70C2×#235C2×D3112yes—
71C22×D12Dic311yescharacteristic
72C22×C4×D3C2×D312yes—
73C22×Dic6Dic311yescharacteristic
74C22×C3×C8—31no—
75C22×C3×C8—31no—
76C22×C3×C8—312no—
77#948—364no—
78C2×#208—312no—
79C2×#208—312no—
80C2×#208—324no—
81C2×#208—612no—
82C22×C3⋊C8—32no—
83C22×C3⋊C8—32no—
84C22×C3⋊C8—324no—
85C22×C3⋊C8—62no—
86C22×C3⋊C8C2×D311yescharacteristic
87C22×C3⋊C8C2×D311yescharacteristic
88C22×C3⋊C8C2×D3112yescomplemented
89#901—364no—
90C2×ΓL(1,𝔽25)—324no—
91C2×D3×C8—324no—
92C23×C32—24no—
93C23×C32C22×C416yes—
94C23×C32C22×C411yescharacteristic
95C22×C3×D3—28no—
96C22×C3×D3C22×C4112yes—
97C2×C32×C4C22×C416yes—
98C2×C32×C4C22×C413yes—
99C2×C32×C4C22×C4112yes—
100C2×C32×C4C2413yes—
101C2×C32×C4C2414yes—
102C3×#67C22×C4132yes—
103C2×C3×Dic3C22×C4112yes—
104C3×D12C22×C4116yes—
105C3×C4×D3C22×C4124yes—
106C3×C4×D3C2418yes—
107C3×Dic6C22×C4116yes—
108C3⋊(C3⋊C8)C2418yes—
109C3×C3⋊C8C2414yes—
110C3×C3⋊C8C2414yes—
111C22×#41—31no—
112C2×#131—924no—
113C22×M16—92no—
114C22×M16—91no—
115C23×C8—31no—
116C23×C8—92no—
117C24×C3—21no—
118C24×C3—32no—
119C24×C3C4×D311yescharacteristic
120C24×C3C2×Dic311yescharacteristic
121C23×D3C2×Dic312yes—
122C3×#41—348no—
123C3×#41—316no—
124C2×C3×Q8—312no—
125C2×C3×D4—324no—
126C2×C3×D4—312no—
127C22×C3×C4—21no—
128C22×C3×C4—26no—
129C22×C3×C4—26no—
130C22×C3×C4—21no—
131C22×C3×C4—32no—
132C22×C3×C4—312no—
133C22×C3×C4—312no—
134C22×C3×C4C4×D311yescharacteristic
135C22×C3×C4C4×D316yes—
136C22×C3×C4C2×Dic311yescharacteristic
137C22×C3×C4C2×Dic316yes—
138C22×C3×C4C22×D316yes—
139C22×C3×C4C22×D316yes—
140C22×C3×C4C22×D311yescharacteristic
141C22×C3×C4C22×D311yescharacteristic
142C2×#67C2×Dic3124yes—
143C22×Dic3C2×Dic312yes—
144#235C2×Dic3148yes—
145#235C22×D3116yes—
146C2×D12C2×Dic3112yes—
147C2×C4×D3C2×Dic3112yes—
148C2×C4×D3C22×D3112yes—
149C2×Dic6C2×Dic3112yes—
150C2×C3×C8—36no—
151C2×C3×C8—36no—
152C2×C3×C8—316no—
153#208—316no—
154#208—316no—
155#208—332no—
156#208—616no—
157C2×C3⋊C8—312no—
158C2×C3⋊C8—312no—
159C2×C3⋊C8—332no—
160C2×C3⋊C8—612no—
161C2×C3⋊C8C22×D316yes—
162C2×C3⋊C8C22×D316yes—
163C2×C3⋊C8C22×D3116yescomplemented
164ΓL(1,𝔽25)—332no—
165D3×C8—332no—
166C22×C32—212no—
167C22×C32#13114yescomplemented
168C22×C32C23×C414yes—
169C22×C32C23×C413yes—
170C2×C3×D3—224no—
171C2×C3×D3C23×C418yes—
172C32×C4C23×C414yes—
173C32×C4C23×C413yes—
174C32×C4C2511yescharacteristic
175C3×Dic3C23×C418yes—
176C2×#41—312no—
177C2×#41—312no—
178C22×Q8—31no—
179C22×D4—32no—
180C22×D4—31no—
181C23×C4—32no—
182C23×C4D3211yescharacteristic
183#131—964no—
184C2×M16—924no—
185C2×M16—912no—
186C22×C8—312no—
187C22×C8—31no—
188C22×C8—31no—
189C22×C8—924no—
190C22×C8—92no—
191C23×C3—24no—
192C23×C3—24no—
193C23×C3—21no—
194C23×C3—26no—
195C23×C3—312no—
196C23×C3—44no—
197C23×C3—68no—
198C23×C3C2×C4×D316yes—
199C23×C3C2×C4×D311yescharacteristic
200C23×C3C22×Dic316yes—
201C23×C3C22×Dic311yescharacteristic
202C22×D3—28no—
203C22×D3C22×Dic3112yes—
204C3×Q8—316no—
205C3×D4—332no—
206C3×D4—316no—
207C2×C3×C4—23no—
208C2×C3×C4—24no—
209C2×C3×C4—23no—
210C2×C3×C4—26no—
211C2×C3×C4—212no—
212C2×C3×C4—312no—
213C2×C3×C4—324no—
214C2×C3×C4—38no—
215C2×C3×C4C2×C4×D313yes—
216C2×C3×C4C2×C4×D316yes—
217C2×C3×C4C2×C4×D3112yes—
218C2×C3×C4C22×Dic316yes—
219C2×C3×C4C22×Dic313yes—
220C2×C3×C4C22×Dic3112yes—
221C2×C3×C4C23×D313yes—
222C2×C3×C4C23×D314yes—
223C2×C3×C4C23×D313yes—
224C2×C3×C4C23×D314yes—
225#67C22×Dic3132yes—
226C2×Dic3C22×Dic3112yes—
227D12C22×Dic3116yes—
228C4×D3C22×Dic3124yes—
229C4×D3C23×D318yes—
230Dic6C22×Dic3116yes—
231C3×C8—34no—
232C3×C8—34no—
233C3⋊C8—38no—
234C3⋊C8—38no—
235C3⋊C8—68no—
236C3⋊C8C23×D314yes—
237C3⋊C8C23×D314yes—
238C2×C32—24no—
239C2×C32C2×#13116yescomplemented
240C2×C32C24×C411yescharacteristic
241C3×D3—28no—
242C24—32no—
243C24D3×Dic311yescharacteristic
244#41—316no—
245#41—348no—
246C2×Q8—312no—
247C2×D4—324no—
248C2×D4—312no—
249C22×C4—32no—
250C22×C4—312no—
251C22×C4—312no—
252C22×C4D3×Dic311yescharacteristic
253C22×C4D3×Dic316yes—
254C22×C4C2×D3216yes—
255C22×C4C2×D3211yescharacteristic
256M16—932no—
257M16—916no—
258C2×C8—36no—
259C2×C8—36no—
260C2×C8—316no—
261C2×C8—932no—
262C2×C8—912no—
263C22×C3—212no—
264C22×C3—212no—
265C22×C3—23no—
266C22×C3—24no—
267C22×C3—24no—
268C22×C3—38no—
269C22×C3—412no—
270C22×C3—624no—
271C22×C3#90114yescomplemented
272C22×C3#94814yescomplemented
273C22×C3C22×C4×D314yes—
274C22×C3C22×C4×D313yes—
275C22×C3C23×Dic314yes—
276C22×C3C23×Dic313yes—
277C2×D3—224no—
278C2×D3C23×Dic318yes—
279C3×C4—23no—
280C3×C4—24no—
281C3×C4—21no—
282C3×C4—38no—
283C3×C4C22×C4×D313yes—
284C3×C4C22×C4×D314yes—
285C3×C4C23×Dic314yes—
286C3×C4C23×Dic313yes—
287C3×C4C24×D311yescharacteristic
288C3×C4C24×D311yescharacteristic
289Dic3C23×Dic318yes—
290C32C22×#13111yescharacteristic, complemented, Sylow
291C23—24no—
292C23—312no—
293C23—68no—
294C23C2×D3×Dic316yes—
295C23C2×D3×Dic311yescharacteristic
296Q8—316no—
297D4—332no—
298D4—316no—
299C2×C4—312no—
300C2×C4—324no—
301C2×C4—38no—
302C2×C4C2×D3×Dic313yes—
303C2×C4C2×D3×Dic316yes—
304C2×C4C2×D3×Dic3112yes—
305C2×C4C22×D3213yes—
306C2×C4C22×D3214yes—
307C8—34no—
308C8—34no—
309C8—98no—
310C2×C3—24no—
311C2×C3—24no—
312C2×C3—21no—
313C2×C3—26no—
314C2×C3—44no—
315C2×C3—68no—
316C2×C3C2×#90116yescomplemented
317C2×C3C2×#94816yescomplemented
318C2×C3C23×C4×D311yescharacteristic
319C2×C3C24×Dic311yescharacteristic
320D3—28no—
321C22—212no—
322C22—38no—
323C22—624no—
324C22#6381414yesdirect product factor
325C22C22×D3×Dic314yes—
326C22C22×D3×Dic313yes—
327C4—38no—
328C4C22×D3×Dic313yes—
329C4C22×D3×Dic314yes—
330C4C23×D3211yescharacteristic
331C3—21no—
332C3C22×#90111yescharacteristic, complemented
333C3C22×#94811yescharacteristic, complemented
334C2—24no—
335C2—68no—
336C2C2×#6381416yesdirect product factor
337C2C23×D3×Dic311yescharacteristic
subgroupG/H
centerC22×C4C2×D32
commutatorC2×C32C24×C4
FrattiniC4C23×D32
FittingC23×C32×C4C22
socleC23×C32C22×C4
prime radicalC22×C3×#235C2
cosocleC32×C4C25
hypercenterC23×C4D32
hypocenterC32C22×#131
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C22×#3748C2semidirect
3. C23×C3×C3⋊C8C2semidirect
4. C22×#3711C2semidirect
5. C2×#63814C2direct (N × H)
6. C22×D3×C3⋊C8C2semidirect
16. C2×#3748C22semidirect
20. C22×C3×C3⋊C8C22semidirect
21. C2×#3711C22semidirect
22. #63814C22direct (N × H)
23. C2×D3×C3⋊C8C22semidirect
33. C23×C3⋊C8D3semidirect
51. #3748C23semidirect
55. C2×C3×C3⋊C8C23semidirect
56. #3711C23semidirect
57. D3×C3⋊C8C23semidirect
88. C22×C3⋊C8C2×D3semidirect
163. C2×C3⋊C8C22×D3semidirect
167. C22×C32#131semidirect
239. C2×C32C2×#131semidirect
271. C22×C3#901semidirect
272. C22×C3#948semidirect
290. C32C22×#131semidirect
316. C2×C3C2×#901semidirect
317. C2×C3C2×#948semidirect
324. C22#63814direct (N × H)
332. C3C22×#901semidirect
333. C3C22×#948semidirect
336. C2C2×#63814direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2352
C22×#37482
C23×C3×C3⋊C82
C22×#37112
C2×#638142
C22×D3×C3⋊C82
C22×#9483
C22×#9013