Finite Group Databasebeta

← C22×#63827 #11918715

Subgroups — C22×#63827 #11918715

Order 1152.

3077 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#235C211yesmaximal, characteristic
2C23×C3×Dic6C211yesmaximal, characteristic, complemented
3C22×#3748C211yesmaximal, characteristic, complemented
4C22×#3718C212yesmaximal, complemented
5C2×#63827C2124yesmaximal, direct product factor
6C22×#3715C212yesmaximal, complemented
7C22×#951—31nomaximal
8C22×#942—31nomaximal
9C23×C32×C4C2211yescharacteristic
10C22×C3×#67—21no—
11C23×C3×Dic3—21no—
12C2×C3×#235—26no—
13C2×C3×#235C22112yes—
14C22×C3×D12C2211yescharacteristic
15C22×C3×C4×D3—21no—
16C22×C3×Dic6—26no—
17C22×C3×Dic6—21no—
18C22×C3×Dic6C2212yes—
19C22×C3×Dic6C2211yescharacteristic
20C22×C3×Dic6C22112yescomplemented
21C2×#3748C22112yescomplemented
22C22×C3⋊(C3⋊C8)C2212yes—
23C2×#3718C22124yescomplemented
24#63827C22164yesdirect product factor
25C2×#3715C22124yescomplemented
26C22×C3×#41—31no—
27C23×C3×Q8—31no—
28C22×#235D311yescharacteristic
29C23×Dic6D311yescharacteristic, complemented
30C2×#951—324no—
31C22×#216—32no—
32C22×#216—32no—
33C2×#942—324no—
34C22×#215—32no—
35C22×#214—32no—
36C22×#208—31no—
37C22×#208—31no—
38C22×#208—61no—
39C24×C32D411yescharacteristic
40C23×C3×D3—21no—
41C22×C32×C4D411yescharacteristic
42C22×C32×C4D416yes—
43C22×C32×C4C2311yescharacteristic
44C22×C32×C4C2316yes—
45C2×C3×#67—212no—
46C22×C3×Dic3—22no—
47C22×C3×Dic3—212no—
48C22×C3×Dic3—21no—
49C3×#235—224no—
50C3×#235C23116yes—
51C2×C3×D12—23no—
52C2×C3×D12C2316yes—
53C2×C3×C4×D3—212no—
54C2×C3×Dic6—26no—
55C2×C3×Dic6—224no—
56C2×C3×Dic6—212no—
57C2×C3×Dic6—23no—
58C2×C3×Dic6C23112yes—
59C2×C3×Dic6C2316yes—
60C2×C3×Dic6C23116yescomplemented
61#3748C23116yescomplemented
62C2×C3⋊(C3⋊C8)C23112yes—
63#3718C23132yescomplemented
64#3715C23132yescomplemented
65C22×#137—91noSylow
66C2×C3×#41—312no—
67C2×C3×#41—66no—
68C22×C3×Q8—32no—
69C22×C3×Q8—31no—
70C22×C3×Q8—312no—
71C22×C3×Q8—61no—
72C22×C3×Q8—66no—
73C22×C3×D4—31no—
74C22×C3×D4—61no—
75C23×C3×C4—21no—
76C23×C3×C4—61no—
77C23×C3×C4—61no—
78C23×C3×C4C2×D311yescharacteristic
79C23×C3×C4C2×D311yescharacteristic
80C22×#67—21no—
81C23×Dic3—21no—
82C2×#235—26no—
83C2×#235C2×D3112yes—
84C22×D12C2×D311yescharacteristic
85C22×C4×D3—21no—
86C22×Dic6—26no—
87C22×Dic6—21no—
88C22×Dic6C2×D312yes—
89C22×Dic6C2×D311yescharacteristic
90C22×Dic6C2×D3112yescomplemented
91#951—364no—
92C2×#216—324no—
93C2×#216—324no—
94#942—364no—
95C2×#215—324no—
96C2×#214—324no—
97C2×#208—312no—
98C2×#208—312no—
99C2×#208—612no—
100C22×C3⋊C8—32no—
101C22×C3⋊C8—32no—
102C22×C3⋊C8—62no—
103C23×C32—24no—
104C23×C32C2×D416yes—
105C23×C32C2×D411yescharacteristic
106C22×C3×D3—26no—
107C22×C3×D3—44no—
108C2×C32×C4C2×D413yes—
109C2×C32×C4C2×D416yes—
110C2×C32×C4C2×D4112yes—
111C2×C32×C4C2413yes—
112C2×C32×C4C2414yes—
113C3×#67—216no—
114C2×C3×Dic3—212no—
115C2×C3×Dic3—216no—
116C2×C3×Dic3—26no—
117C3×D12—26no—
118C3×D12C2414yes—
119C3×C4×D3—216no—
120C3×Dic6—212no—
121C3×Dic6—216no—
122C3×Dic6—26no—
123C3×Dic6C2418yes—
124C3×Dic6C2414yes—
125C3⋊(C3⋊C8)C2418yes—
126C22×#41—31no—
127C23×Q8—31no—
128C2×#137—924no—
129C22×Q16—92no—
130C22×SD16—92no—
131C22×M16—91no—
132C24×C3—21no—
133C24×C3—61no—
134C24×C3#6711yescharacteristic
135C24×C3#6711yescharacteristic
136C23×D3—21no—
137C3×#41—316no—
138C3×#41—624no—
139C2×C3×Q8—312no—
140C2×C3×Q8—316no—
141C2×C3×Q8—36no—
142C2×C3×Q8—66no—
143C2×C3×Q8—612no—
144C2×C3×Q8—624no—
145C2×C3×Q8—63no—
146C2×C3×D4—36no—
147C2×C3×D4—612no—
148C2×C3×D4—63no—
149C22×C3×C4—26no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—26no—
153C22×C3×C4—61no—
154C22×C3×C4—612no—
155C22×C3×C4—62no—
156C22×C3×C4—612no—
157C22×C3×C4#6711yescharacteristic
158C22×C3×C4#6716yes—
159C22×C3×C4#6711yescharacteristic
160C22×C3×C4#6716yes—
161C22×C3×C4C22×D311yescharacteristic
162C22×C3×C4C22×D311yescharacteristic
163C22×C3×C4C22×D316yes—
164C22×C3×C4C22×D316yes—
165C2×#67—212no—
166C22×Dic3—22no—
167C22×Dic3—212no—
168C22×Dic3—21no—
169#235—224no—
170#235C22×D3116yes—
171C2×D12—23no—
172C2×D12C22×D316yes—
173C2×C4×D3—212no—
174C2×Dic6—26no—
175C2×Dic6—224no—
176C2×Dic6—212no—
177C2×Dic6—23no—
178C2×Dic6C22×D3112yes—
179C2×Dic6C22×D316yes—
180C2×Dic6C22×D3116yescomplemented
181#216—332no—
182#216—332no—
183#215—332no—
184#214—332no—
185#208—316no—
186#208—316no—
187#208—616no—
188C2×C3⋊C8—312no—
189C2×C3⋊C8—312no—
190C2×C3⋊C8—612no—
191C22×C32—212no—
192C22×C32#13714yescomplemented
193C22×C32C22×D414yes—
194C22×C32C22×D413yes—
195C2×C3×D3—24no—
196C2×C3×D3—412no—
197C32×C4C22×D413yes—
198C32×C4C22×D414yes—
199C32×C4C2511yescharacteristic
200C3×Dic3—28no—
201C3×Dic3—24no—
202C2×#41—312no—
203C2×#41—66no—
204C22×Q8—32no—
205C22×Q8—31no—
206C22×Q8—312no—
207C22×Q8—66no—
208C22×Q8—61no—
209C22×D4—31no—
210C22×D4—61no—
211C23×C4—61no—
212C23×C4—61no—
213C23×C4D3211yescharacteristic
214#137—964no—
215C2×Q16—924no—
216C2×SD16—924no—
217C2×M16—912no—
218C22×C8—92no—
219C23×C3—24no—
220C23×C3—24no—
221C23×C3—26no—
222C23×C3—21no—
223C23×C3—44no—
224C23×C3—66no—
225C23×C3—124no—
226C23×C3C2×#6716yes—
227C23×C3C2×#6716yes—
228C23×C3C2×#6711yescharacteristic
229C23×C3C2×#6711yescharacteristic
230C22×D3—26no—
231C22×D3—44no—
232C3×Q8—38no—
233C3×Q8—34no—
234C3×Q8—612no—
235C3×Q8—616no—
236C3×Q8—66no—
237C3×D4—34no—
238C3×D4—616no—
239C3×D4—66no—
240C2×C3×C4—23no—
241C2×C3×C4—24no—
242C2×C3×C4—26no—
243C2×C3×C4—23no—
244C2×C3×C4—212no—
245C2×C3×C4—66no—
246C2×C3×C4—616no—
247C2×C3×C4—612no—
248C2×C3×C4—616no—
249C2×C3×C4C2×#6716yes—
250C2×C3×C4C2×#6713yes—
251C2×C3×C4C2×#67112yes—
252C2×C3×C4C2×#6713yes—
253C2×C3×C4C2×#6716yes—
254C2×C3×C4C2×#67112yes—
255C2×C3×C4C23×D313yes—
256C2×C3×C4C23×D313yes—
257C2×C3×C4C23×D314yes—
258C2×C3×C4C23×D314yes—
259#67—216no—
260C2×Dic3—212no—
261C2×Dic3—216no—
262C2×Dic3—26no—
263D12—26no—
264D12C23×D314yes—
265C4×D3—216no—
266Dic6—212no—
267Dic6—216no—
268Dic6—26no—
269Dic6C23×D318yes—
270Dic6C23×D314yes—
271C3⋊C8—38no—
272C3⋊C8—38no—
273C3⋊C8—68no—
274C2×C32—24no—
275C2×C32C2×#13716yescomplemented
276C2×C32C23×D411yescharacteristic
277C3×D3—44no—
278C24—61no—
279C24#62511yescharacteristic
280#41—316no—
281#41—624no—
282C2×Q8—312no—
283C2×Q8—36no—
284C2×Q8—316no—
285C2×Q8—612no—
286C2×Q8—66no—
287C2×Q8—624no—
288C2×Q8—63no—
289C2×D4—36no—
290C2×D4—612no—
291C2×D4—63no—
292C22×C4—612no—
293C22×C4—61no—
294C22×C4—62no—
295C22×C4—612no—
296C22×C4#62516yes—
297C22×C4#62511yescharacteristic
298C22×C4C2×D3216yes—
299C22×C4C2×D3211yescharacteristic
300Q16—932no—
301SD16—932no—
302M16—916no—
303C2×C8—912no—
304C22×C3—212no—
305C22×C3—212no—
306C22×C3—23no—
307C22×C3—24no—
308C22×C3—24no—
309C22×C3—412no—
310C22×C3—64no—
311C22×C3—1212no—
312C22×C3#94214yescomplemented
313C22×C3#95114yescomplemented
314C22×C3C22×#6714yes—
315C22×C3C22×#6714yes—
316C22×C3C22×#6713yes—
317C22×C3C22×#6713yes—
318C2×D3—24no—
319C2×D3—412no—
320C3×C4—24no—
321C3×C4—23no—
322C3×C4—21no—
323C3×C4—64no—
324C3×C4—68no—
325C3×C4C22×#6714yes—
326C3×C4C22×#6713yes—
327C3×C4C22×#6713yes—
328C3×C4C22×#6714yes—
329C3×C4C24×D311yescharacteristic
330C3×C4C24×D311yescharacteristic
331Dic3—28no—
332Dic3—24no—
333C32C22×#13711yescharacteristic, complemented, Sylow
334C23—24no—
335C23—66no—
336C23—124no—
337C23C2×#62516yes—
338C23C2×#62511yescharacteristic
339Q8—38no—
340Q8—34no—
341Q8—616no—
342Q8—612no—
343Q8—66no—
344D4—34no—
345D4—616no—
346D4—66no—
347C2×C4—66no—
348C2×C4—616no—
349C2×C4—612no—
350C2×C4—616no—
351C2×C4C2×#62516yes—
352C2×C4C2×#62513yes—
353C2×C4C2×#625112yes—
354C2×C4C22×D3213yes—
355C2×C4C22×D3214yes—
356C8—98no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—21no—
360C2×C3—26no—
361C2×C3—44no—
362C2×C3—124no—
363C2×C3C2×#94216yescomplemented
364C2×C3C2×#95116yescomplemented
365C2×C3C23×#6711yescharacteristic
366C2×C3C23×#6711yescharacteristic
367D3—44no—
368C22—212no—
369C22—64no—
370C22—1212no—
371C22#6382714yesdirect product factor
372C22C22×#62514yes—
373C22C22×#62513yes—
374C4—64no—
375C4—68no—
376C4C22×#62514yes—
377C4C22×#62513yes—
378C4C23×D3211yescharacteristic
379C3—21no—
380C3C22×#94211yescharacteristic, complemented
381C3C22×#95111yescharacteristic, complemented
382C2—24no—
383C2—124no—
384C2C2×#6382716yesdirect product factor
385C2C23×#62511yescharacteristic
subgroupG/H
centerC23C2×#625
commutator, cosocleC32×C4C25
FrattiniC4C23×D32
FittingC23×C32×C4C22
socleC23×C32C2×D4
prime radicalC22×C3×#235C2
hypercenterC23×C4D32
hypocenterC32C22×#137
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C23×C3×Dic6C2semidirect
3. C22×#3748C2semidirect
4. C22×#3718C2semidirect
5. C2×#63827C2direct (N × H)
6. C22×#3715C2semidirect
20. C22×C3×Dic6C22semidirect
21. C2×#3748C22semidirect
23. C2×#3718C22semidirect
24. #63827C22direct (N × H)
25. C2×#3715C22semidirect
29. C23×Dic6D3semidirect
60. C2×C3×Dic6C23semidirect
61. #3748C23semidirect
63. #3718C23semidirect
64. #3715C23semidirect
90. C22×Dic6C2×D3semidirect
180. C2×Dic6C22×D3semidirect
192. C22×C32#137semidirect
275. C2×C32C2×#137semidirect
312. C22×C3#942semidirect
313. C22×C3#951semidirect
333. C32C22×#137semidirect
363. C2×C3C2×#942semidirect
364. C2×C3C2×#951semidirect
371. C22#63827direct (N × H)
380. C3C22×#942semidirect
381. C3C22×#951semidirect
384. C2C2×#63827direct (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
C23×C3×Dic62
C22×#37482
C22×#37182
C2×#638272
C22×#37152
C22×#9513
C22×#9423