Finite Group Databasebeta

← C22×#63871 #11900341

Subgroups — C22×#63871 #11900341

Order 1152.

4361 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×C3⋊D3C211yesmaximal, characteristic
2C23×C3×Dic6C211yesmaximal, characteristic, complemented
3C22×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
4C22×Dic3⋊Dic3C212yesmaximal, complemented
5C22×#3722C212yesmaximal
6C2×#63871C2124yesmaximal, direct product factor
7C22×#946—31nomaximal
8C22×#897—31nomaximal
9C24×C3⋊D3C2211yescharacteristic
10C23×C32×C4C2211yescharacteristic
11C23×C3⋊Dic3C2211yescharacteristic
12C22×C4×C3⋊D3—28no—
13C22×C4×C3⋊D3C22112yes—
14C23×C3×Dic3C2212yes—
15C23×C3×Dic3C2212yes—
16C22×C3×Dic6—28no—
17C22×C3×Dic6C22112yescomplemented
18C2×C3×C4⋊Dic3C22112yescomplemented
19C2×Dic3⋊Dic3C22124yescomplemented
20C2×#3722C22124yes—
21#63871C22164yesdirect product factor
22C23×C3×Q8—31no—
23C23×C4×D3—31no—
24C23×C4×D3—31no—
25C23×C4×D3—61no—
26C23×Dic6D311yescharacteristic, complemented
27C22×C3×C4⋊C4—31no—
28C2×#946—324no—
29C22×(C2×D3)⋊C4—32no—
30C22×(C2×D3)⋊C4—32no—
31C22×C4⋊Dic3—32no—
32C22×C4⋊Dic3D311yescharacteristic, complemented
33C22×Dic3⋊C4—32no—
34C2×#897—324no—
35C24×C32C2311yescharacteristic
36C23×C3⋊D3—24no—
37C23×C3⋊D3—24no—
38C23×C3⋊D3Q818yescomplemented
39C23×C3⋊D3C2316yes—
40C22×C32×C4D418yes—
41C22×C32×C4C2316yes—
42C22×C3⋊Dic3—24no—
43C22×C3⋊Dic3C2316yes—
44C2×C4×C3⋊D3—248no—
45C2×C4×C3⋊D3C23116yes—
46C22×C3×Dic3—28no—
47C22×C3×Dic3—28no—
48C22×C3×Dic3C23112yes—
49C22×C3×Dic3C23112yes—
50C2×C3×Dic6—248no—
51C2×C3×Dic6C23116yescomplemented
52C3×C4⋊Dic3C23116yescomplemented
53Dic3⋊Dic3C23132yescomplemented
54#3722C23132yes—
55C22×C22⋊Q8—91noSylow
56C24×D3—31no—
57C24×D3—31no—
58C24×D3—61no—
59C22×C3×Q8—312no—
60C22×C3×Q8—68no—
61C23×C3×C4—21no—
62C23×C3×C4—32no—
63C23×C3×C4—32no—
64C23×C3×C4C2×D311yescharacteristic
65C23×C3×C4C2×D311yescharacteristic
66C23×Dic3—31no—
67C23×Dic3—31no—
68C23×Dic3—61no—
69C23×Dic3C2×D312yes—
70C23×Dic3C2×D312yes—
71C22×C4×D3—312no—
72C22×C4×D3—312no—
73C22×C4×D3—612no—
74C22×C4×D3—616no—
75C22×C4×D3—68no—
76C22×C4×D3—68no—
77C22×Dic6—28no—
78C22×Dic6C2×D3112yescomplemented
79C2×C3×C4⋊C4—312no—
80#946—364no—
81C2×(C2×D3)⋊C4—324no—
82C2×(C2×D3)⋊C4—324no—
83C2×C4⋊Dic3—324no—
84C2×C4⋊Dic3C2×D3112yescomplemented
85C2×Dic3⋊C4—324no—
86#897—364no—
87C23×C32C2×D414yes—
88C23×C32C2×Q814yes—
89C23×C32#4114yes—
90C23×C32C2413yes—
91C22×C3⋊D3—232no—
92C22×C3⋊D3—212no—
93C22×C3⋊D3—212no—
94C22×C3⋊D3C2×Q8124yescomplemented
95C22×C3⋊D3C2414yes—
96C2×C32×C4C2×D4124yes—
97C2×C32×C4C2414yes—
98C2×C3⋊Dic3—212no—
99C2×C3⋊Dic3C2414yes—
100C4×C3⋊D3—232no—
101C2×C3×Dic3—224no—
102C2×C3×Dic3—224no—
103C2×C3×Dic3C2418yes—
104C2×C3×Dic3C2418yes—
105C3×Dic6—232no—
106C23×Q8—31no—
107C24×C4—91no—
108C2×C22⋊Q8—924no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—92no—
111C22×C22⋊C4—92no—
112C24×C3—21no—
113C24×C3C22×D311yescharacteristic
114C24×C3C22×D311yescharacteristic
115C23×D3—38no—
116C23×D3—38no—
117C23×D3—36no—
118C23×D3—36no—
119C23×D3—68no—
120C23×D3—66no—
121C23×D3—68no—
122C23×D3—68no—
123C23×D3—64no—
124C23×D3—64no—
125C23×D3—64no—
126C23×D3—64no—
127C2×C3×Q8—316no—
128C2×C3×Q8—648no—
129C22×C3×C4—28no—
130C22×C3×C4—26no—
131C22×C3×C4—312no—
132C22×C3×C4—312no—
133C22×C3×C4—68no—
134C22×C3×C4—68no—
135C22×C3×C4D1218yes—
136C22×C3×C4#6718yes—
137C22×C3×C4C22×D316yes—
138C22×C3×C4C22×D316yes—
139C22×Dic3—28no—
140C22×Dic3—28no—
141C22×Dic3—36no—
142C22×Dic3—36no—
143C22×Dic3—66no—
144C22×Dic3—68no—
145C22×Dic3—64no—
146C22×Dic3—64no—
147C22×Dic3C22×D3112yes—
148C22×Dic3C22×D3112yes—
149C2×C4×D3—316no—
150C2×C4×D3—316no—
151C2×C4×D3—616no—
152C2×C4×D3—696no—
153C2×C4×D3—648no—
154C2×C4×D3—648no—
155C2×Dic6—248no—
156C2×Dic6C22×D3116yescomplemented
157C3×C4⋊C4—316no—
158(C2×D3)⋊C4—332no—
159(C2×D3)⋊C4—332no—
160C4⋊Dic3—332no—
161C4⋊Dic3C22×D3116yescomplemented
162Dic3⋊C4—332no—
163C22×C32C22⋊Q8116yescomplemented
164C22×C32C22×D416yes—
165C22×C32C22×Q816yes—
166C22×C32C2×#4116yes—
167C22×C32C2511yescharacteristic
168C2×C3⋊D3—248no—
169C2×C3⋊D3—24no—
170C2×C3⋊D3—24no—
171C2×C3⋊D3C22×Q818yescomplemented
172C32×C4C22×D418yes—
173C3⋊Dic3—24no—
174C3×Dic3—28no—
175C3×Dic3—28no—
176C25—91no—
177C22×Q8—312no—
178C22×Q8—68no—
179C23×C4—32no—
180C23×C4—32no—
181C23×C4—91no—
182C23×C4—912no—
183C23×C4—188no—
184C23×C4D3211yescharacteristic
185C22⋊Q8—964no—
186C2×C4⋊C4—312no—
187C2×C4⋊C4—924no—
188C2×C22⋊C4—924no—
189C23×C3—23no—
190C23×C3—24no—
191C23×C3—24no—
192C23×C3—24no—
193C23×C3C2×D1214yes—
194C23×C3#23714yes—
195C23×C3D3×Q814yes—
196C23×C3D3×Q814yes—
197C23×C3#23914yes—
198C23×C3C2×#6714yes—
199C23×C3C23×D313yes—
200C23×C3C23×D313yes—
201C22×D3—324no—
202C22×D3—324no—
203C22×D3—34no—
204C22×D3—34no—
205C22×D3—624no—
206C22×D3—64no—
207C22×D3—624no—
208C22×D3—624no—
209C22×D3—664no—
210C22×D3—632no—
211C22×D3—632no—
212C22×D3—612no—
213C22×D3—612no—
214C22×D3—612no—
215C22×D3—612no—
216C3×Q8—632no—
217C2×C3×C4—24no—
218C2×C3×C4—224no—
219C2×C3×C4—38no—
220C2×C3×C4—38no—
221C2×C3×C4—624no—
222C2×C3×C4—624no—
223C2×C3×C4C2×D12124yes—
224C2×C3×C4C2×#67124yes—
225C2×C3×C4C23×D314yes—
226C2×C3×C4C23×D314yes—
227C2×Dic3—224no—
228C2×Dic3—224no—
229C2×Dic3—34no—
230C2×Dic3—34no—
231C2×Dic3—64no—
232C2×Dic3—624no—
233C2×Dic3—612no—
234C2×Dic3—612no—
235C2×Dic3C23×D318yes—
236C2×Dic3C23×D318yes—
237C4×D3—664no—
238C4×D3—632no—
239C4×D3—632no—
240Dic6—232no—
241C2×C32C2×C22⋊Q8112yescomplemented
242C2×C32C23×D411yescharacteristic
243C2×C32C23×Q811yescharacteristic
244C2×C32C22×#4111yescharacteristic
245C3⋊D3—28no—
246C24—96no—
247C24—98no—
248C24—184no—
249C24—184no—
250C24C2×D3211yescharacteristic
251C2×Q8—316no—
252C2×Q8—648no—
253C22×C4—312no—
254C22×C4—312no—
255C22×C4—68no—
256C22×C4—68no—
257C22×C4—96no—
258C22×C4—916no—
259C22×C4—184no—
260C22×C4—1848no—
261C22×C4#62618yes—
262C22×C4C2×D3216yes—
263C4⋊C4—316no—
264C4⋊C4—932no—
265C22⋊C4—932no—
266C22×C3—26no—
267C22×C3—26no—
268C22×C3—26no—
269C22×C3—216no—
270C22×C3—21no—
271C22×C3#897116yescomplemented
272C22×C3#946116yescomplemented
273C22×C3C22×D1216yes—
274C22×C3C2×#23716yes—
275C22×C3C2×D3×Q816yes—
276C22×C3C2×D3×Q816yes—
277C22×C3C2×#23916yes—
278C22×C3C22×#6716yes—
279C22×C3C24×D311yescharacteristic
280C22×C3C24×D311yescharacteristic
281C2×D3—38no—
282C2×D3—38no—
283C2×D3—68no—
284C2×D3—68no—
285C2×D3—696no—
286C2×D3—68no—
287C2×D3—64no—
288C2×D3—64no—
289C2×D3—64no—
290C2×D3—64no—
291C2×D3—648no—
292C2×D3—648no—
293C3×C4—28no—
294C3×C4—68no—
295C3×C4—68no—
296C3×C4C22×D1218yes—
297C3×C4C22×#6718yes—
298Dic3—28no—
299Dic3—28no—
300Dic3—68no—
301Dic3—64no—
302Dic3—64no—
303C32C22×C22⋊Q811yescharacteristic, complemented, Sylow
304C23—924no—
305C23—94no—
306C23—1812no—
307C23—1812no—
308C23—1832no—
309C23#379614yes—
310C23#379714yes—
311C23C2×#62614yes—
312C23C22×D3213yes—
313Q8—632no—
314C2×C4—38no—
315C2×C4—38no—
316C2×C4—624no—
317C2×C4—624no—
318C2×C4—94no—
319C2×C4—1812no—
320C2×C4—1832no—
321C2×C4C2×#626124yes—
322C2×C4C22×D3214yes—
323C2×C3—212no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3C2×#897112yescomplemented
328C2×C3C2×#946112yescomplemented
329C2×C3C23×D1211yescharacteristic
330C2×C3C22×#23711yescharacteristic
331C2×C3C22×D3×Q811yescharacteristic
332C2×C3C22×D3×Q811yescharacteristic
333C2×C3C22×#23911yescharacteristic
334C2×C3C23×#6711yescharacteristic
335D3—616no—
336D3—68no—
337D3—68no—
338C22—98no—
339C22—1848no—
340C22—184no—
341C22—184no—
342C22#63871116yesdirect product factor
343C22C2×#379616yes—
344C22C2×#379716yes—
345C22C22×#62616yes—
346C22C23×D3211yescharacteristic
347C4—68no—
348C4—68no—
349C4—184no—
350C4C22×#62618yes—
351C3—21no—
352C3C22×#89711yescharacteristic, complemented
353C3C22×#94611yescharacteristic, complemented
354C2—188no—
355C2C2×#63871112yesdirect product factor
356C2C22×#379611yescharacteristic
357C2C22×#379711yescharacteristic
358C2C23×#62611yescharacteristic
subgroupG/H
centerC24C2×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC23×C32×C4C22
socleC24×C32C23
prime radicalC24×C3⋊D3C22
hypercenterC23×C4D32
hypocenterC32C22×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C23×C3×Dic6C2semidirect
3. C22×C3×C4⋊Dic3C2semidirect
4. C22×Dic3⋊Dic3C2semidirect
6. C2×#63871C2direct (N × H)
17. C22×C3×Dic6C22semidirect
18. C2×C3×C4⋊Dic3C22semidirect
19. C2×Dic3⋊Dic3C22semidirect
21. #63871C22direct (N × H)
26. C23×Dic6D3semidirect
32. C22×C4⋊Dic3D3semidirect
38. C23×C3⋊D3Q8semidirect
51. C2×C3×Dic6C23semidirect
52. C3×C4⋊Dic3C23semidirect
53. Dic3⋊Dic3C23semidirect
78. C22×Dic6C2×D3semidirect
84. C2×C4⋊Dic3C2×D3semidirect
94. C22×C3⋊D3C2×Q8semidirect
156. C2×Dic6C22×D3semidirect
161. C4⋊Dic3C22×D3semidirect
163. C22×C32C22⋊Q8semidirect
171. C2×C3⋊D3C22×Q8semidirect
241. C2×C32C2×C22⋊Q8semidirect
271. C22×C3#897semidirect
272. C22×C3#946semidirect
303. C32C22×C22⋊Q8semidirect
327. C2×C3C2×#897semidirect
328. C2×C3C2×#946semidirect
342. C22#63871direct (N × H)
352. C3C22×#897semidirect
353. C3C22×#946semidirect
355. C2C2×#63871direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×C3⋊D32
C23×C3×Dic62
C22×C3×C4⋊Dic32
C22×Dic3⋊Dic32
C22×#37222
C2×#638712
C22×#9463
C22×#8973