Finite Group Databasebeta

← C2×#20252326 #27080648

Subgroups — C2×#20252326 #27080648

Order 12032.

1894 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
1C2×#15689676C211yesmaximal, characteristic
2#20252326C2116yesmaximal, direct product factor
3C2×#15689253C214yesmaximal, complemented
4C2×#15689251C214yesmaximal, complemented
5C2×#15689151C211yesmaximal, characteristic
6C2×#15689150C211yesmaximal, characteristic
7C2×#15689146C211yesmaximal, characteristic
8C2×#15689078C211yesmaximal, characteristic
9C22×#13093934C211yesmaximal, characteristic
10C2×C47×#405C211yesmaximal, characteristic
11C22×#10625329C2211yescharacteristic
12C23×C47⋊C8C2211yescharacteristic
13#15689676C418yes—
14#15689676C2218yes—
15C2×#13094023C412yes—
16C2×#13094015C412yes—
17C2×#13094001C414yes—
18C2×C4×Dic94C2214yes—
19C2×#13093983C2211yescharacteristic
20C22×C4×Dic47C2211yescharacteristic
21#15689253C22132yescomplemented
22#15689251C22132yescomplemented
23#15689151C2218yes—
24#15689150C2218yes—
25#15689146C2218yes—
26C2×C4⋊(C47⋊C8)C2214yes—
27C2×Dic47⋊C8C2214yes—
28C2×Dic47⋊C8C2214yes—
29#15689078C2218yes—
30C2×#13093934C2212yes—
31C2×#13093934C2212yes—
32C2×#13093934C2218yes—
33C2×#13093934C2212yes—
34C2×C8×Dic47C2212yes—
35C22×M16×C47C2211yescharacteristic
36C23×C8×C47C2211yescharacteristic
37C2×#117×C47C2211yescharacteristic
38C47×#405C2218yes—
39C2×#97×C47C2212yes—
40C2×C4×C8×C47C2212yes—
41C22×Dic94C2×C414yes—
42C23×Dic47C2×C412yes—
43C2×#10625329—24no—
44C2×#10625329C2314yes—
45C22×C47⋊C8—24no—
46C22×C47⋊C8C2312yes—
47C22×C47⋊C8C2312yes—
48C22×C47⋊C8C2314yes—
49#13094023C2×C4116yes—
50#13094015C2×C4116yes—
51#13094001C2×C4132yes—
52C4×Dic94C2×C4116yes—
53C4×Dic94C23116yes—
54#13093983C2×C414yes—
55#13093983C2314yes—
56C2×C4⋊Dic47C2×C412yes—
57C2×C4⋊Dic47C2×C414yes—
58C2×Dic47⋊C4C2×C414yes—
59C2×Dic47⋊C4C2×C414yes—
60C2×C22⋊Dic47C2×C412yes—
61C2×C4×Dic47C2×C411yescharacteristic
62C2×C4×Dic47C2×C411yescharacteristic
63C2×C4×Dic47C2×C414yes—
64C2×C4×Dic47C2312yes—
65C2×C4×Dic47C2314yes—
66C2×C4×Dic47C2312yes—
67C4⋊(C47⋊C8)C23116yes—
68Dic47⋊C8C23116yes—
69Dic47⋊C8C23116yes—
70#13093934C2318yes—
71#13093934C2318yes—
72#13093934C2318yes—
73C8×Dic47C2318yes—
74C23×C4×C47C2311yescharacteristic
75C2×M16×C47—24no—
76C2×M16×C47C2314yes—
77C22×C8×C47Q818yescomplemented
78C22×C8×C47C2312yes—
79C22×C8×C47C2311yescharacteristic
80C22×C8×C47C2311yescharacteristic
81C22×C8×C47C2314yes—
82#117×C47C2×C414yes—
83#117×C47C2314yes—
84C2×C4⋊C4×C47C2×C412yes—
85C2×C22⋊C4×C47C2×C412yes—
86C2×C42×C47C2312yes—
87#97×C47C2318yes—
88C4×C8×C47C2318yes—
89C24×C47C22×C411yescharacteristic
90C22×C4×C47C22×C414yes—
91C22×C4×C47C22×C411yescharacteristic
92C22×C4×C47C22×C412yes—
93C22×C4×C47C2×Q814yes—
94C22×C4×C47#4114yes—
95C22×C4×C47C2412yes—
96C22×C4×C47C2411yescharacteristic
97C22×Dic47—44no—
98C22×Dic47C22×C412yes—
99C22×Dic47C22×C412yes—
100C22×Dic47C22×C414yes—
101C22×Dic47C22×C418yes—
102C2×Dic94—48no—
103C2×Dic94C22×C4116yes—
104M16×C47—28no—
105C2×C8×C47—24no—
106C2×C8×C47C2×Q814yes—
107C2×C8×C47C2×Q814yes—
108C2×C8×C47C2×Q8116yescomplemented
109C2×C8×C47C2414yes—
110C2×C8×C47C2412yes—
111C2×C8×C47C2412yes—
112C4⋊C4×C47C22×C418yes—
113C22⋊C4×C47C22×C418yes—
114C42×C47C22×C414yes—
115C42×C47C2414yes—
116C22⋊Dic47C22×C418yes—
117C4⋊Dic47C22×C418yes—
118C4⋊Dic47C22×C4116yes—
119Dic47⋊C4C22×C4116yes—
120Dic47⋊C4C22×C4116yes—
121C4×Dic47C22×C414yes—
122C4×Dic47C22×C414yes—
123C4×Dic47C22×C414yes—
124C4×Dic47C22×C414yes—
125C4×Dic47C2414yes—
126C4×Dic47C2414yes—
127#10625329—28no—
128C2×C47⋊C8—24no—
129C2×C47⋊C8—28no—
130C2×C47⋊C8—24no—
131C2×C47⋊C8C2414yes—
132C2×C47⋊C8C2414yes—
133C23×C47—24no—
134C23×C47C4×Q814yes—
135C23×C47C23×C412yes—
136C23×C47C23×C411yescharacteristic
137C2×C4×C47—28no—
138C2×C4×C47C4×Q818yes—
139C2×C4×C47C4×Q814yes—
140C2×C4×C47C23×C418yes—
141C2×C4×C47C23×C411yescharacteristic
142C2×C4×C47C23×C412yes—
143C2×C4×C47C22×Q812yes—
144C2×C4×C47C22×Q814yes—
145C2×C4×C47C2×#4112yes—
146C2×C4×C47C2×#4114yes—
147C2×C4×C47C2511yescharacteristic
148C2×Dic47—24no—
149C2×Dic47—28no—
150C2×Dic47—24no—
151C2×Dic47—48no—
152C2×Dic47C23×C414yes—
153C2×Dic47C23×C414yes—
154C2×Dic47C23×C418yes—
155Dic94—416no—
156C8×C47—24no—
157C8×C47C22×Q814yes—
158C8×C47C22×Q814yes—
159C47⋊C8—24no—
160C47⋊C8—24no—
161C2×#2961—471nomaximal, Sylow
162C22×C47—28no—
163C22×C47—24no—
164C22×C47C2×C4×Q814yes—
165C22×C47C2×C4×Q812yes—
166C22×C47#56814yes—
167C22×C47C24×C411yescharacteristic
168C4×C47—28no—
169C4×C47C2×C4×Q812yes—
170C4×C47C2×C4×Q814yes—
171C4×C47C23×Q811yescharacteristic
172C4×C47C22×#4111yescharacteristic
173Dic47—24no—
174Dic47—28no—
175Dic47—24no—
176C2×#533—471no—
177#2961—4716no—
178C2×#446—474no—
179C2×#446—474no—
180C2×#424—471no—
181C2×#424—471no—
182C2×#423—471no—
183C2×#405—471no—
184C2×#405D4711yescharacteristic
185C22×#97—471no—
186C2×C47—24no—
187C2×C47#296114yescomplemented
188C2×C47C22×C4×Q811yescharacteristic
189C2×C47C2×#56812yes—
190C22×M16—471no—
191C22×M16C2×D4711yescharacteristic
192C23×C8—471no—
193C23×C8C2×D4711yescharacteristic
194#533—478no—
195#533—478no—
196C2×C4⋊Q8—472no—
197C2×#125—472no—
198C2×C22⋊Q8—474no—
199C2×C4×Q8—474no—
200C2×#117—471no—
201C2×#117C2×D4711yescharacteristic
202C22×C42—471no—
203#446—4732no—
204#446—4732no—
205#424—478no—
206#424—478no—
207#423—478no—
208C2×C4⋊C8—474no—
209C2×C4⋊C8—474no—
210C2×C4⋊C8—474no—
211#405—478no—
212#405C2×D4718yes—
213C2×#97—472no—
214C2×#97—472no—
215C2×#97—478no—
216C2×#97—472no—
217C2×#97C2×D4712yes—
218C2×C4×C8—472no—
219C2×C4×C8C2×D4712yes—
220C47C2×#296111yescharacteristic, complemented, Sylow
221C22×Q8—474no—
222C23×C4—472no—
223C23×C4C22×D4711yescharacteristic
224C2×M16—24no—
225C2×M16—474no—
226C2×M16—944no—
227C2×M16C22×D4714yes—
228C22×C8—474no—
229C22×C8—472no—
230C22×C8—472no—
231C22×C8—944no—
232C22×C8Dic9418yescomplemented
233C22×C8C22×D4712yes—
234C22×C8C22×D4711yescharacteristic
235C22×C8C22×D4711yescharacteristic
236C22×C8C22×D4714yes—
237C4⋊Q8—4716no—
238#125—4716no—
239C22⋊Q8—4732no—
240C4×Q8—4716no—
241C4×Q8—4716no—
242#117—474no—
243#117—474no—
244#117C4×D4714yes—
245#117C22×D4714yes—
246C2×C4⋊C4—472no—
247C2×C4⋊C4—474no—
248C2×C4⋊C4—474no—
249C2×C4⋊C4—474no—
250C2×C4⋊C4C4×D4712yes—
251C2×C22⋊C4—472no—
252C2×C22⋊C4C4×D4712yes—
253C2×C42—471no—
254C2×C42—471no—
255C2×C42—474no—
256C2×C42—474no—
257C2×C42—472no—
258C2×C42—472no—
259C2×C42C22×D4712yes—
260C4⋊C8—4716no—
261C4⋊C8—4716no—
262C4⋊C8—4716no—
263#97—478no—
264#97—478no—
265#97—478no—
266#97C22×D4718yes—
267C4×C8—478no—
268C4×C8C22×D4718yes—
269C24C2×C4×D4711yescharacteristic
270C2×Q8—4716no—
271C2×Q8—1888no—
272C22×C4—472no—
273C22×C4—472no—
274C22×C4—474no—
275C22×C4—478no—
276C22×C4—1884no—
277C22×C4C2×Dic9414yes—
278C22×C4C2×C4×D4714yes—
279C22×C4C2×C4×D4711yescharacteristic
280C22×C4C2×C4×D4712yes—
281C22×C4#1062535014yes—
282C22×C4C23×D4712yes—
283C22×C4C23×D4711yescharacteristic
284M16—28no—
285M16—948no—
286C2×C8—24no—
287C2×C8—474no—
288C2×C8—474no—
289C2×C8—944no—
290C2×C8—948no—
291C2×C8—944no—
292C2×C8C2×Dic9414yes—
293C2×C8C2×Dic9414yes—
294C2×C8C2×Dic94116yescomplemented
295C2×C8C23×D4714yes—
296C2×C8C23×D4712yes—
297C2×C8C23×D4712yes—
298C4⋊C4—4716no—
299C4⋊C4—4716no—
300C4⋊C4—478no—
301C4⋊C4—4716no—
302C4⋊C4C2×C4×D4718yes—
303C22⋊C4—478no—
304C22⋊C4C2×C4×D4718yes—
305C42—474no—
306C42—474no—
307C42—474no—
308C42—474no—
309C42—474no—
310C42—474no—
311C42C2×C4×D4714yes—
312C42C23×D4714yes—
313C23—24no—
314C23C4×Dic9414yes—
315C23C22×C4×D4712yes—
316C23C22×C4×D4711yescharacteristic
317Q8—18816no—
318C2×C4—28no—
319C2×C4—474no—
320C2×C4—474no—
321C2×C4—478no—
322C2×C4—944no—
323C2×C4—944no—
324C2×C4—948no—
325C2×C4—1888no—
326C2×C4C4×Dic9418yes—
327C2×C4C4×Dic9414yes—
328C2×C4C22×C4×D4718yes—
329C2×C4C22×C4×D4711yescharacteristic
330C2×C4C22×C4×D4712yes—
331C2×C4C22×Dic9412yes—
332C2×C4C22×Dic9414yes—
333C2×C4C2×#1062535012yes—
334C2×C4C2×#1062535014yes—
335C2×C4C24×D4711yescharacteristic
336C8—24no—
337C8—944no—
338C8—944no—
339C8C22×Dic9414yes—
340C8C22×Dic9414yes—
341C22—24no—
342C22—28no—
343C22C2×C4×Dic9414yes—
344C22C2×C4×Dic9412yes—
345C22#1568988814yes—
346C22C23×C4×D4711yescharacteristic
347C4—28no—
348C4—944no—
349C4—944no—
350C4—948no—
351C4C2×C4×Dic9412yes—
352C4C2×C4×Dic9414yes—
353C4C23×Dic9411yescharacteristic
354C4C22×#1062535011yescharacteristic
355C2—24no—
356C2#2025232614yesdirect product factor
357C2C22×C4×Dic9411yescharacteristic
358C2C2×#1568988812yes—
subgroupG/H
centerC22×C4C23×D47
commutatorC22×C47C24×C4
FrattiniC2×C4C24×D47
FittingC2×C47×#405C2
socleC23×C47C23×C4
prime radicalC24×C47C22×C4
cosocleC2×C4×C47C25
hypercenterC2×#405D47
hypocenterC47C2×#2961
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
2. #20252326C2direct (N × H)
3. C2×#15689253C2semidirect
4. C2×#15689251C2semidirect
21. #15689253C22semidirect
22. #15689251C22semidirect
77. C22×C8×C47Q8semidirect
108. C2×C8×C47C2×Q8semidirect
187. C2×C47#2961semidirect
220. C47C2×#2961semidirect
232. C22×C8Dic94semidirect
294. C2×C8C2×Dic94semidirect
356. C2#20252326direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896762
#202523262
C2×#156892532
C2×#156892512
C2×#156891512
C2×#156891502
C2×#156891462
C2×#156890782
C22×#130939342
C2×C47×#4052
C2×#296147