Finite Group Databasebeta

← C2×#10591397 #11808197

Subgroups — C2×#10591397 #11808197

Order 1152.

2025 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#627C211yesmaximal, characteristic
2C22×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C22×Dic3⋊Dic3C212yesmaximal
4C2×C3×C3⋊#95C212yesmaximal
5C2×#63579C211yesmaximal, characteristic
6#10591397C218yesmaximal, direct product factor
7C2×#4656—32nomaximal
8C23×C32×C4C2211yescharacteristic
9C23×C3⋊Dic3C2211yescharacteristic
10C23×C3⋊Dic3C2211yescharacteristic
11C23×C3×Dic3C2212yes—
12C23×C3×Dic3C2212yes—
13C22×#627—24no—
14C22×#627—28no—
15C22×#627C2214yes—
16C2×C4⋊(C3⋊Dic3)—24no—
17C2×C4⋊(C3⋊Dic3)C2214yes—
18C2×Dic3⋊Dic3—28no—
19C2×Dic3⋊Dic3C2218yes—
20C3×C3⋊#95C2218yes—
21#63579C2214yes—
22C23×Dic6—32no—
23C22×C4⋊Dic3—32no—
24C22×C4⋊Dic3—32no—
25C22×C4⋊Dic3—61no—
26C22×Dic3⋊C4—32no—
27C2×C3×#95—32no—
28C2×C3⋊#95—32no—
29C2×C3⋊#95D312yes—
30#4656—316no—
31C24×C32C2311yescharacteristic
32C22×C32×C4—22no—
33C22×C32×C4—22no—
34C22×C32×C4Q814yescomplemented
35C22×C32×C4C2312yes—
36C22×C3⋊Dic3—24no—
37C22×C3⋊Dic3—22no—
38C22×C3⋊Dic3—24no—
39C22×C3⋊Dic3D414yes—
40C22×C3⋊Dic3C2312yes—
41C22×C3⋊Dic3C2312yes—
42C22×C3×Dic3—24no—
43C22×C3×Dic3—24no—
44C22×C3×Dic3—24no—
45C22×C3×Dic3—24no—
46C22×C3×Dic3D418yes—
47C22×C3×Dic3Q818yescomplemented
48C22×C3×Dic3C2314yes—
49C22×C3×Dic3C2314yes—
50C2×#627—232no—
51C2×#627—28no—
52C2×#627—216no—
53C4⋊(C3⋊Dic3)—28no—
54Dic3⋊Dic3—216no—
55C2×#395—91noSylow
56C23×C3×C4—21no—
57C23×C3×C4—32no—
58C23×C3×C4—32no—
59C23×C3×C4C2×D312yes—
60C23×Dic3—32no—
61C23×Dic3—32no—
62C23×Dic3—61no—
63C23×Dic3—61no—
64C23×Dic3C2×D312yes—
65C23×Dic3C2×D312yes—
66C22×Dic6—38no—
67C22×Dic6—68no—
68C22×Dic6—68no—
69C22×Dic6—68no—
70C2×C4⋊Dic3—38no—
71C2×C4⋊Dic3—38no—
72C2×C4⋊Dic3—64no—
73C2×C4⋊Dic3—68no—
74C2×C4⋊Dic3—68no—
75C2×C4⋊Dic3—68no—
76C2×Dic3⋊C4—38no—
77C2×Dic3⋊C4—68no—
78C3×#95—38no—
79C3⋊#95—38no—
80C3⋊#95C2×D318yes—
81C23×C32C2×D414yes—
82C23×C32C2×D412yes—
83C23×C32C2×Q812yes—
84C23×C32C2×Q814yes—
85C23×C32#4112yes—
86C23×C32C2411yescharacteristic
87C2×C32×C4—28no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4C2×Q814yescomplemented
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3—28no—
93C2×C3⋊Dic3—24no—
94C2×C3⋊Dic3—24no—
95C2×C3⋊Dic3—44no—
96C2×C3⋊Dic3C2×D414yes—
97C2×C3×Dic3—216no—
98C2×C3×Dic3—216no—
99C2×C3×Dic3—24no—
100C2×C3×Dic3—24no—
101C2×C3×Dic3—24no—
102C2×C3×Dic3—24no—
103C2×C3×Dic3C2×D418yes—
104C2×C3×Dic3C2×Q818yescomplemented
105#627—232no—
106C23×Q8—91no—
107C22×C4⋊C4—92no—
108C22×C4⋊C4—91no—
109#395—98no—
110C2×#95—32no—
111C2×#95—91no—
112C24×C3—21no—
113C24×C3C22×D312yes—
114C22×C3×C4—24no—
115C22×C3×C4—24no—
116C22×C3×C4—24no—
117C22×C3×C4—24no—
118C22×C3×C4—24no—
119C22×C3×C4—22no—
120C22×C3×C4—38no—
121C22×C3×C4—38no—
122C22×C3×C4—34no—
123C22×C3×C4—34no—
124C22×C3×C4—64no—
125C22×C3×C4—64no—
126C22×C3×C4—64no—
127C22×C3×C4—64no—
128C22×C3×C4Dic618yescomplemented
129C22×C3×C4C22×D314yes—
130C22×Dic3—24no—
131C22×Dic3—24no—
132C22×Dic3—24no—
133C22×Dic3—24no—
134C22×Dic3—38no—
135C22×Dic3—34no—
136C22×Dic3—34no—
137C22×Dic3—64no—
138C22×Dic3—62no—
139C22×Dic3—62no—
140C22×Dic3—64no—
141C22×Dic3—64no—
142C22×Dic3—64no—
143C22×Dic3—64no—
144C22×Dic3—64no—
145C22×Dic3—64no—
146C22×Dic3—124no—
147C22×Dic3—124no—
148C22×Dic3Dic618yescomplemented
149C22×Dic3D1218yes—
150C22×Dic3C22×D314yes—
151C22×Dic3C22×D314yes—
152C2×Dic6—616no—
153C2×Dic6—616no—
154C2×Dic6—664no—
155C2×Dic6—616no—
156C4⋊Dic3—616no—
157C4⋊Dic3—616no—
158C4⋊Dic3—616no—
159Dic3⋊C4—616no—
160C22×C32#12014yes—
161C22×C32C22⋊Q814yes—
162C22×C32C22⋊Q818yes—
163C22×C32C4⋊Q818yes—
164C22×C32C4⋊Q814yes—
165C22×C32C22×D412yes—
166C22×C32C22×D411yescharacteristic
167C22×C32C22×Q811yescharacteristic
168C22×C32C22×Q812yes—
169C22×C32C2×#4111yescharacteristic
170C32×C4—24no—
171C3⋊Dic3—24no—
172C3⋊Dic3—42no—
173C3×Dic3—28no—
174C3×Dic3—28no—
175C22×Q8—94no—
176C22×Q8—188no—
177C22×Q8—184no—
178C23×C4—32no—
179C23×C4—32no—
180C23×C4—91no—
181C23×C4—91no—
182C23×C4D3211yescharacteristic
183C2×C4⋊C4—98no—
184C2×C4⋊C4—94no—
185C2×C4⋊C4—184no—
186C2×C4⋊C4—188no—
187#95—38no—
188#95—94no—
189C23×C3—21no—
190C23×C3—24no—
191C23×C3—24no—
192C23×C3—22no—
193C23×C3—22no—
194C23×C3—22no—
195C23×C3C2×Dic614yes—
196C23×C3C2×Dic614yes—
197C23×C3C2×D1214yes—
198C23×C3D3×D414yes—
199C23×C3D3×D414yes—
200C23×C3#23714yes—
201C23×C3D3×Q814yes—
202C23×C3C23×D312yes—
203C2×C3×C4—24no—
204C2×C3×C4—24no—
205C2×C3×C4—216no—
206C2×C3×C4—216no—
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—24no—
210C2×C3×C4—38no—
211C2×C3×C4—38no—
212C2×C3×C4—616no—
213C2×C3×C4—616no—
214C2×C3×C4—64no—
215C2×C3×C4—64no—
216C2×C3×C4—64no—
217C2×C3×C4—64no—
218C2×C3×C4C2×Dic618yescomplemented
219C2×Dic3—216no—
220C2×Dic3—216no—
221C2×Dic3—24no—
222C2×Dic3—24no—
223C2×Dic3—24no—
224C2×Dic3—24no—
225C2×Dic3—38no—
226C2×Dic3—64no—
227C2×Dic3—616no—
228C2×Dic3—64no—
229C2×Dic3—64no—
230C2×Dic3—64no—
231C2×Dic3—64no—
232C2×Dic3—64no—
233C2×Dic3—64no—
234C2×Dic3—124no—
235C2×Dic3—128no—
236C2×Dic3—124no—
237C2×Dic3—128no—
238C2×Dic3—128no—
239C2×Dic3C2×Dic618yescomplemented
240C2×Dic3C2×D1218yes—
241Dic6—664no—
242C2×C32#39518yescomplemented
243C2×C32C2×#12011yescharacteristic
244C2×C32C2×C22⋊Q811yescharacteristic
245C2×C32C2×C22⋊Q812yes—
246C2×C32C2×C4⋊Q812yes—
247C2×C32C2×C4⋊Q811yescharacteristic
248C24C2×D3211yescharacteristic
249C2×Q8—1816no—
250C2×Q8—188no—
251C2×Q8—1832no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—34no—
255C22×C4—34no—
256C22×C4—38no—
257C22×C4—38no—
258C22×C4—64no—
259C22×C4—64no—
260C22×C4—64no—
261C22×C4—64no—
262C22×C4—92no—
263C22×C4—94no—
264C22×C4—92no—
265C22×C4—184no—
266C22×C4—184no—
267C22×C4—182no—
268C22×C4#62714yescomplemented
269C22×C4C2×D3212yes—
270C4⋊C4—188no—
271C4⋊C4—1816no—
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—24no—
278C22×C3—28no—
279C22×C3—28no—
280C22×C3—24no—
281C22×C3—24no—
282C22×C3#86918yes—
283C22×C3#87818yes—
284C22×C3#87818yes—
285C22×C3#88218yes—
286C22×C3#88818yes—
287C22×C3#88818yes—
288C22×C3#89718yes—
289C22×C3C22×Dic612yes—
290C22×C3C22×Dic612yes—
291C22×C3C22×D1212yes—
292C22×C3C2×D3×D412yes—
293C22×C3C2×D3×D412yes—
294C22×C3C2×#23712yes—
295C22×C3C2×D3×Q812yes—
296C3×C4—28no—
297C3×C4—28no—
298C3×C4—68no—
299C3×C4—68no—
300Dic3—28no—
301Dic3—28no—
302Dic3—68no—
303Dic3—124no—
304Dic3—124no—
305Dic3—124no—
306C32C2×#39511yescharacteristic, complemented, Sylow
307C23D3×Dic614yes—
308C23D3×D1214yes—
309C23#380512yes—
310C23C2×#62712yes—
311C23#381112yes—
312C23C22×D3211yescharacteristic
313Q8—1832no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—28no—
317C2×C4—38no—
318C2×C4—38no—
319C2×C4—64no—
320C2×C4—64no—
321C2×C4—64no—
322C2×C4—64no—
323C2×C4—616no—
324C2×C4—616no—
325C2×C4—94no—
326C2×C4—184no—
327C2×C4—188no—
328C2×C4—184no—
329C2×C4—182no—
330C2×C4—364no—
331C2×C4C2×#62714yescomplemented
332C2×C3—22no—
333C2×C3—28no—
334C2×C3—21no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3#4656116yescomplemented
339C2×C3C2×#86912yes—
340C2×C3C2×#87812yes—
341C2×C3C2×#87812yes—
342C2×C3C2×#88212yes—
343C2×C3C2×#88812yes—
344C2×C3C2×#88812yes—
345C2×C3C2×#89712yes—
346C22#6384014yes—
347C22#6389318yes—
348C22#6391818yes—
349C22#6392314yes—
350C22#6398214yes—
351C22C2×D3×Dic612yes—
352C22C2×D3×D1212yes—
353C22C2×#380511yescharacteristic
354C22C22×#62711yescharacteristic
355C22C2×#381111yescharacteristic
356C4—24no—
357C4—68no—
358C4—68no—
359C4—184no—
360C4—362no—
361C3—21no—
362C3C2×#465612yescomplemented
363C2#1059139718yesdirect product factor
364C2C2×#6384011yescharacteristic
365C2C2×#6389312yes—
366C2C2×#6391812yes—
367C2C2×#6392311yescharacteristic
368C2C2×#6398211yescharacteristic
subgroupG/H
centerC24C2×D32
commutator, cosocleC23×C32C24
FrattiniC23C22×D32
FittingC23×C32×C4C22
socle, prime radicalC24×C32C23
hypercenterC23×C4D32
hypocenterC32C2×#395
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
6. #10591397C2direct (N × H)
34. C22×C32×C4Q8semidirect
47. C22×C3×Dic3Q8semidirect
90. C2×C32×C4C2×Q8semidirect
104. C2×C3×Dic3C2×Q8semidirect
128. C22×C3×C4Dic6semidirect
148. C22×Dic3Dic6semidirect
218. C2×C3×C4C2×Dic6semidirect
239. C2×Dic3C2×Dic6semidirect
242. C2×C32#395semidirect
268. C22×C4#627semidirect
306. C32C2×#395semidirect
331. C2×C4C2×#627semidirect
338. C2×C3#4656semidirect
362. C3C2×#4656semidirect
363. C2#10591397direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#6272
C22×C4⋊(C3⋊Dic3)2
C22×Dic3⋊Dic32
C2×C3×C3⋊#952
C2×#635792
#105913972
C2×#46563