Finite Group Databasebeta

← C22×#63817 #11918473

Subgroups — C22×#63817 #11918473

Order 1152.

3453 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3828C211yesmaximal, characteristic
2C22×C3×#208C211yesmaximal, characteristic, complemented
3C23×C3×C3⋊C8C211yesmaximal, characteristic, complemented
4C22×#3712C212yesmaximal, complemented
5C2×#63817C2124yesmaximal, direct product factor
6C22×#3710C212yesmaximal, complemented
7C22×#907—31nomaximal
8C22×#901—31nomaximal
9C23×C32×C4C2211yescharacteristic
10C22×#638C412yes—
11C2×#3828C4112yes—
12C2×#3828C22112yes—
13C22×C4⋊(C3⋊D3)C411yescharacteristic
14C22×C4×C3⋊D3C2212yes—
15C22×#634C411yescharacteristic
16C2×C3×#208C22112yescomplemented
17C22×C3×C3⋊C8C2211yescharacteristic
18C22×C3×C3⋊C8C2211yescharacteristic
19C22×C3×C3⋊C8C2212yes—
20C22×C3×C3⋊C8C22112yescomplemented
21C2×#3712C22124yescomplemented
22#63817C22164yesdirect product factor
23C2×#3710C22124yescomplemented
24C22×#235—31no—
25C22×#235—31no—
26C22×#235—61no—
27C22×C3×M16—31no—
28C23×C3×C8—31no—
29C22×#208D311yescharacteristic, complemented
30C23×C3⋊C8D311yescharacteristic, complemented
31C2×#907—324no—
32C2×#901—324no—
33C22×ΓL(1,𝔽25)—32no—
34C22×ΓL(1,𝔽25)—32no—
35C22×D3×C8—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×#638C2×C4124yes—
44C22×C3⋊Dic3C2×C412yes—
45#3828C2×C4148yes—
46#3828C23116yes—
47C2×C4⋊(C3⋊D3)C2×C4112yes—
48C2×C4×C3⋊D3C2×C4112yes—
49C2×C4×C3⋊D3C23112yes—
50C2×#634C2×C4112yes—
51C3×#208C23116yescomplemented
52C2×C3×C3⋊C8C2316yes—
53C2×C3×C3⋊C8C2316yes—
54C2×C3×C3⋊C8C23112yes—
55C2×C3×C3⋊C8C23116yescomplemented
56#3712C23132yescomplemented
57#3710C23132yescomplemented
58C22×#131—91noSylow
59C23×C3×C4—21no—
60C23×C3×C4C2×D311yescharacteristic
61C23×C3×C4C2×D311yescharacteristic
62C22×#67—32no—
63C22×#67—32no—
64C22×#67—62no—
65C2×#235—312no—
66C2×#235—312no—
67C2×#235—312no—
68C2×#235—312no—
69C2×#235—612no—
70C2×#235—612no—
71C22×D12—31no—
72C22×D12—31no—
73C22×D12—61no—
74C22×C4×D3—32no—
75C22×C4×D3—32no—
76C22×C4×D3—62no—
77C22×Dic6—31no—
78C22×Dic6—31no—
79C22×Dic6—61no—
80C2×C3×M16—312no—
81C22×C3×C8—31no—
82C22×C3×C8—31no—
83C22×C3×C8—312no—
84C22×C3×C8—32no—
85C2×#208C2×D3112yescomplemented
86C22×C3⋊C8C2×D311yescharacteristic
87C22×C3⋊C8C2×D312yes—
88C22×C3⋊C8C2×D311yescharacteristic
89C22×C3⋊C8C2×D3112yescomplemented
90#907—364no—
91#901—364no—
92C2×ΓL(1,𝔽25)—324no—
93C2×ΓL(1,𝔽25)—324no—
94C2×D3×C8—324no—
95C2×D3×C8—324no—
96C23×C32—24no—
97C23×C32C22×C416yes—
98C23×C32C22×C411yescharacteristic
99C22×C3⋊D3—28no—
100C22×C3⋊D3C22×C4112yes—
101C2×C32×C4C22×C416yes—
102C2×C32×C4C22×C413yes—
103C2×C32×C4C22×C4112yes—
104C2×C32×C4C2413yes—
105C2×C32×C4C2414yes—
106#638C22×C4132yes—
107C2×C3⋊Dic3C22×C4112yes—
108C4⋊(C3⋊D3)C22×C4116yes—
109C4×C3⋊D3C22×C4124yes—
110C4×C3⋊D3C2418yes—
111#634C22×C4116yes—
112C3×C3⋊C8C2414yes—
113C3×C3⋊C8C2418yes—
114C3×C3⋊C8C2414yes—
115C22×#41—91no—
116C2×#131—924no—
117C22×M16—31no—
118C22×M16—92no—
119C23×C8—31no—
120C23×C8—92no—
121C24×C3—21no—
122C24×C3C4×D311yescharacteristic
123C24×C3C4×D311yescharacteristic
124C23×D3—32no—
125C23×D3—32no—
126C23×D3—62no—
127C22×C3×C4—21no—
128C22×C3×C4—26no—
129C22×C3×C4—26no—
130C22×C3×C4—21no—
131C22×C3×C4C4×D311yescharacteristic
132C22×C3×C4C4×D311yescharacteristic
133C22×C3×C4C4×D316yes—
134C22×C3×C4C4×D316yes—
135C22×C3×C4C22×D316yes—
136C22×C3×C4C22×D316yes—
137C22×C3×C4C22×D311yescharacteristic
138C22×C3×C4C22×D311yescharacteristic
139C2×#67—324no—
140C2×#67—324no—
141C2×#67—624no—
142C22×Dic3—32no—
143C22×Dic3—32no—
144C22×Dic3—62no—
145#235—348no—
146#235—348no—
147#235—316no—
148#235—316no—
149#235—648no—
150#235—616no—
151C2×D12—312no—
152C2×D12—312no—
153C2×D12—612no—
154C2×C4×D3—312no—
155C2×C4×D3—312no—
156C2×C4×D3—312no—
157C2×C4×D3—312no—
158C2×C4×D3—612no—
159C2×C4×D3—612no—
160C2×Dic6—312no—
161C2×Dic6—312no—
162C2×Dic6—612no—
163C3×M16—316no—
164C2×C3×C8—36no—
165C2×C3×C8—312no—
166C2×C3×C8—36no—
167C2×C3×C8—316no—
168#208C22×D3116yescomplemented
169C2×C3⋊C8C22×D316yes—
170C2×C3⋊C8C22×D3112yes—
171C2×C3⋊C8C22×D316yes—
172C2×C3⋊C8C22×D3116yescomplemented
173ΓL(1,𝔽25)—332no—
174ΓL(1,𝔽25)—332no—
175D3×C8—332no—
176D3×C8—332no—
177C22×C32—212no—
178C22×C32#13114yescomplemented
179C22×C32C23×C414yes—
180C22×C32C23×C413yes—
181C2×C3⋊D3—224no—
182C2×C3⋊D3C23×C418yes—
183C32×C4C23×C414yes—
184C32×C4C23×C413yes—
185C32×C4C2511yescharacteristic
186C3⋊Dic3C23×C418yes—
187C2×#41—912no—
188C2×#41—912no—
189C22×Q8—91no—
190C22×D4—92no—
191C22×D4—91no—
192C23×C4—92no—
193C23×C4D3211yescharacteristic
194#131—964no—
195C2×M16—312no—
196C2×M16—924no—
197C22×C8—312no—
198C22×C8—31no—
199C22×C8—31no—
200C22×C8—32no—
201C22×C8—924no—
202C23×C3—24no—
203C23×C3—24no—
204C23×C3—21no—
205C23×C3—26no—
206C23×C3—44no—
207C23×C3C2×C4×D316yes—
208C23×C3C2×C4×D316yes—
209C23×C3C2×C4×D311yescharacteristic
210C23×C3C2×C4×D311yescharacteristic
211C22×D3—312no—
212C22×D3—312no—
213C22×D3—612no—
214C22×D3—68no—
215C22×D3—68no—
216C22×D3—128no—
217C2×C3×C4—23no—
218C2×C3×C4—24no—
219C2×C3×C4—26no—
220C2×C3×C4—23no—
221C2×C3×C4—212no—
222C2×C3×C4C2×C4×D316yes—
223C2×C3×C4C2×C4×D316yes—
224C2×C3×C4C2×C4×D313yes—
225C2×C3×C4C2×C4×D313yes—
226C2×C3×C4C2×C4×D3112yes—
227C2×C3×C4C2×C4×D3112yes—
228C2×C3×C4C23×D313yes—
229C2×C3×C4C23×D313yes—
230C2×C3×C4C23×D314yes—
231C2×C3×C4C23×D314yes—
232#67—332no—
233#67—332no—
234#67—632no—
235C2×Dic3—312no—
236C2×Dic3—312no—
237C2×Dic3—612no—
238D12—316no—
239D12—316no—
240D12—616no—
241C4×D3—324no—
242C4×D3—324no—
243C4×D3—38no—
244C4×D3—38no—
245C4×D3—624no—
246C4×D3—68no—
247Dic6—316no—
248Dic6—316no—
249Dic6—616no—
250C3×C8—34no—
251C3×C8—38no—
252C3×C8—34no—
253C3⋊C8C23×D314yes—
254C3⋊C8C23×D318yes—
255C3⋊C8C23×D314yes—
256C2×C32—24no—
257C2×C32C2×#13116yescomplemented
258C2×C32C24×C411yescharacteristic
259C3⋊D3—28no—
260C24—92no—
261C24#62411yescharacteristic
262#41—916no—
263#41—948no—
264C2×Q8—912no—
265C2×D4—924no—
266C2×D4—912no—
267C22×C4—92no—
268C22×C4—912no—
269C22×C4—912no—
270C22×C4#62411yescharacteristic
271C22×C4#62416yes—
272C22×C4C2×D3216yes—
273C22×C4C2×D3211yescharacteristic
274M16—316no—
275M16—932no—
276C2×C8—36no—
277C2×C8—36no—
278C2×C8—316no—
279C2×C8—312no—
280C2×C8—932no—
281C22×C3—212no—
282C22×C3—212no—
283C22×C3—23no—
284C22×C3—24no—
285C22×C3—24no—
286C22×C3—412no—
287C22×C3#90114yescomplemented
288C22×C3#90714yescomplemented
289C22×C3C22×C4×D314yes—
290C22×C3C22×C4×D314yes—
291C22×C3C22×C4×D313yes—
292C22×C3C22×C4×D313yes—
293C2×D3—38no—
294C2×D3—38no—
295C2×D3—68no—
296C2×D3—624no—
297C2×D3—624no—
298C2×D3—1224no—
299C3×C4—24no—
300C3×C4—23no—
301C3×C4—21no—
302C3×C4C22×C4×D314yes—
303C3×C4C22×C4×D314yes—
304C3×C4C22×C4×D313yes—
305C3×C4C22×C4×D313yes—
306C3×C4C24×D311yescharacteristic
307C3×C4C24×D311yescharacteristic
308Dic3—38no—
309Dic3—38no—
310Dic3—68no—
311C32C22×#13111yescharacteristic, complemented, Sylow
312C23—24no—
313C23—912no—
314C23—188no—
315C23C2×#62416yes—
316C23C2×#62411yescharacteristic
317Q8—916no—
318D4—932no—
319D4—916no—
320C2×C4—912no—
321C2×C4—924no—
322C2×C4—98no—
323C2×C4C2×#62413yes—
324C2×C4C2×#62416yes—
325C2×C4C2×#624112yes—
326C2×C4C22×D3213yes—
327C2×C4C22×D3214yes—
328C8—34no—
329C8—34no—
330C8—38no—
331C2×C3—24no—
332C2×C3—24no—
333C2×C3—21no—
334C2×C3—26no—
335C2×C3—44no—
336C2×C3C2×#90116yescomplemented
337C2×C3C2×#90716yescomplemented
338C2×C3C23×C4×D311yescharacteristic
339C2×C3C23×C4×D311yescharacteristic
340D3—68no—
341D3—68no—
342D3—128no—
343C22—212no—
344C22—98no—
345C22—1824no—
346C22#6381714yesdirect product factor
347C22C22×#62414yes—
348C22C22×#62413yes—
349C4—98no—
350C4C22×#62413yes—
351C4C22×#62414yes—
352C4C23×D3211yescharacteristic
353C3—21no—
354C3C22×#90111yescharacteristic, complemented
355C3C22×#90711yescharacteristic, complemented
356C2—24no—
357C2—188no—
358C2C2×#6381716yesdirect product factor
359C2C23×#62411yescharacteristic
subgroupG/H
centerC22×C4C2×D32
commutatorC2×C32C24×C4
FrattiniC4C23×D32
FittingC23×C32×C4C22
socleC23×C32C22×C4
prime radicalC22×#3828C2
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×C3×#208C2semidirect
3. C23×C3×C3⋊C8C2semidirect
4. C22×#3712C2semidirect
5. C2×#63817C2direct (N × H)
6. C22×#3710C2semidirect
16. C2×C3×#208C22semidirect
20. C22×C3×C3⋊C8C22semidirect
21. C2×#3712C22semidirect
22. #63817C22direct (N × H)
23. C2×#3710C22semidirect
29. C22×#208D3semidirect
30. C23×C3⋊C8D3semidirect
51. C3×#208C23semidirect
55. C2×C3×C3⋊C8C23semidirect
56. #3712C23semidirect
57. #3710C23semidirect
85. C2×#208C2×D3semidirect
89. C22×C3⋊C8C2×D3semidirect
168. #208C22×D3semidirect
172. C2×C3⋊C8C22×D3semidirect
178. C22×C32#131semidirect
257. C2×C32C2×#131semidirect
287. C22×C3#901semidirect
288. C22×C3#907semidirect
311. C32C22×#131semidirect
336. C2×C3C2×#901semidirect
337. C2×C3C2×#907semidirect
346. C22#63817direct (N × H)
354. C3C22×#901semidirect
355. C3C22×#907semidirect
358. C2C2×#63817direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#38282
C22×C3×#2082
C23×C3×C3⋊C82
C22×#37122
C2×#638172
C22×#37102
C22×#9073
C22×#9013