Finite Group Databasebeta

← C2×#20242928 #26198926

Subgroups — C2×#20242928 #26198926

Order 12032.

702 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689470C211yesmaximal, characteristic
2C2×#15689418C211yesmaximal, characteristic, complemented
3C2×#15689323C211yesmaximal, characteristic
4#20242928C218yesmaximal, direct product factor
5C2×#15688876C211yesmaximal, characteristic, complemented
6C2×#15688871C211yesmaximal, characteristic, complemented
7C2×#15688866C211yesmaximal, characteristic
8C2×C47×#446C211yesmaximal, characteristic, complemented
9C22×#10625336—21no—
10C2×#13094024C2211yescharacteristic
11C2×#13094021C2211yescharacteristic
12#15689470—22no—
13#15689470C2214yes—
14#15689418—22no—
15#15689418C2214yescomplemented
16#15689323—22no—
17#15689323C2214yes—
18C2×#13093957—21no—
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×C4⋊(C47⋊C8)C2211yescharacteristic
21C2×#13093948—21no—
22C2×#13093946—21no—
23C2×#13093946—21no—
24C2×#13093944—21no—
25#15688876C2214yescomplemented
26#15688871C2214yescomplemented
27#15688866C2214yes—
28C2×C4×Q8×C47C2211yescharacteristic
29C47×#446—22no—
30C47×#446C2214yescomplemented
31C2×C4⋊C8×C47—21no—
32C2×C4⋊C8×C47C2211yescharacteristic
33C2×#97×C47—21no—
34C2×C4×C8×C47C2211yescharacteristic
35C22×Dic94—21no—
36C22×D188—21no—
37C22×D188—21no—
38C22×D188—21no—
39C2×#10625336—24no—
40C2×#10625336—44no—
41C22×C47⋊C8—21no—
42C22×C47⋊C8—21no—
43#13094024—21no—
44#13094024—21no—
45#13094024—21no—
46#13094024C2312yes—
47#13094021—21no—
48#13094021—21no—
49#13094021—21no—
50#13094021C2312yes—
51C2×C4⋊Dic47—21no—
52C2×C4⋊Dic47—21no—
53#13093957—24no—
54C4⋊(C47⋊C8)—21no—
55C4⋊(C47⋊C8)—21no—
56C4⋊(C47⋊C8)C2312yes—
57C4⋊(C47⋊C8)C2312yes—
58#13093948—24no—
59#13093946—24no—
60#13093946—24no—
61#13093944—24no—
62C22×Q8×C47D411yescharacteristic
63C22×C8×C47—21no—
64C22×C8×C47D411yescharacteristic
65C22×C8×C47D411yescharacteristic
66C4×Q8×C47—21no—
67C4×Q8×C47—21no—
68C4×Q8×C47D412yes—
69C4×Q8×C47C2312yes—
70C2×C4⋊C4×C47—21no—
71C2×C4⋊C4×C47D411yescharacteristic
72C2×C42×C47—21no—
73C2×C42×C47C2311yescharacteristic
74C4⋊C8×C47—24no—
75C4⋊C8×C47—21no—
76C4⋊C8×C47C2312yes—
77#97×C47—24no—
78C4×C8×C47D412yes—
79C4×C8×C47C2312yes—
80C23×D47—41no—
81C2×Q8×C47—21no—
82C2×Q8×C47—22no—
83C2×Q8×C47—22no—
84C2×Q8×C47C2×D412yes—
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47#4111yescharacteristic
90C22×Dic47—41no—
91C2×D188—22no—
92C2×D188—22no—
93C2×D188—22no—
94C2×D188—42no—
95C2×D188—42no—
96C2×D188—42no—
97C2×D188—42no—
98C2×D188—41no—
99C2×D188—41no—
100C2×D188—41no—
101C2×D188—82no—
102C2×Dic94—22no—
103C2×Dic94—41no—
104C2×Dic94—82no—
105C2×C8×C47—22no—
106C2×C8×C47—22no—
107C2×C8×C47—22no—
108C2×C8×C47—42no—
109C2×C8×C47C2×D412yes—
110C2×C8×C47C2×D412yes—
111C4⋊C4×C47—22no—
112C4⋊C4×C47—21no—
113C4⋊C4×C47—22no—
114C4⋊C4×C47C2×D412yes—
115C42×C47—22no—
116C42×C47—22no—
117C42×C47C2×D411yescharacteristic
118C42×C47C2×D411yescharacteristic
119C42×C47#4111yescharacteristic
120C42×C47C2411yescharacteristic
121#10625336—48no—
122C4⋊Dic47—22no—
123C4⋊Dic47—22no—
124C4⋊Dic47—41no—
125C4⋊Dic47—41no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8—22no—
128C2×C47⋊C8—42no—
129C2×C47⋊C8—42no—
130C23×C47#12111yescharacteristic
131C22×D47—42no—
132C22×D47—82no—
133C22×D47—82no—
134C22×D47—82no—
135Q8×C47—22no—
136Q8×C47—22no—
137Q8×C47—42no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—22no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47#12111yescharacteristic
144C2×C4×C47#12111yescharacteristic
145C2×C4×C47#12111yescharacteristic
146C2×C4×C47#13512yes—
147C2×C4×C47AGL(1,ℤ/8)12yes—
148C2×C4×C47#13712yes—
149C2×C4×C47#13712yes—
150C2×C4×C47C22×D411yescharacteristic
151C2×C4×C47C22×D411yescharacteristic
152C2×C4×C47C2×#4111yescharacteristic
153C2×Dic47—42no—
154C2×Dic47—82no—
155D188—42no—
156D188—42no—
157D188—42no—
158D188—42no—
159D188—82no—
160D188—82no—
161D188—82no—
162D188—82no—
163Dic94—82no—
164Dic94—82no—
165C8×C47—22no—
166C8×C47—22no—
167C8×C47—42no—
168C47⋊C8—42no—
169C47⋊C8—42no—
170C2×#1698—471nomaximal, Sylow
171C22×C47#46412yes—
172C22×C47#46812yes—
173C22×C47#47212yes—
174C22×C47C2×#12111yescharacteristic
175C2×D47—82no—
176C2×D47—82no—
177C2×D47—82no—
178C2×D47—164no—
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—42no—
182C4×C47—42no—
183C4×C47#46412yes—
184C4×C47#46812yes—
185C4×C47C2×#13511yescharacteristic
186C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
187C4×C47C2×#13711yescharacteristic
188C4×C47C2×#13711yescharacteristic
189Dic47—82no—
190C2×#486—471no—
191C2×#477—471no—
192C2×#460—471no—
193C2×#446D4711yescharacteristic, complemented
194#1698—478no—
195C2×#335—471no—
196C2×#333—471no—
197C2×#331—471no—
198C2×C47#169814yescomplemented
199C2×C47C2×#46411yescharacteristic
200C2×C47C2×#46811yescharacteristic
201C2×C47C2×#47211yescharacteristic
202D47—162no—
203C22×SD16—941no—
204C2×C4⋊Q8—471no—
205C2×#127—471no—
206C2×C4×Q8C2×D4711yescharacteristic
207#486—474no—
208#486—942no—
209#477—474no—
210#477—942no—
211#460—474no—
212#460—942no—
213#446—22no—
214#446C2×D4714yescomplemented
215C2×#107—941no—
216C2×C4⋊C8—21no—
217C2×C4⋊C8—471no—
218C2×C4⋊C8—471no—
219C2×C4⋊C8C2×D4711yescharacteristic
220C2×#103—941no—
221C2×#102—941no—
222C2×#102—941no—
223C2×#102—941no—
224C2×#97—21no—
225C2×C4×C8C2×D4711yescharacteristic
226#335—474no—
227#333—474no—
228#331—474no—
229C47C2×#169811yescharacteristic, complemented, Sylow
230C22×Q8—941no—
231C22×Q8D18811yescharacteristic
232C22×D4—941no—
233C22×D4—941no—
234C22×D4—941no—
235C2×SD16—944no—
236C2×SD16—1884no—
237C22×C8—21no—
238C22×C8—941no—
239C22×C8—941no—
240C22×C8C22⋊D4711yescharacteristic
241C22×C8C22⋊D4711yescharacteristic
242C4⋊Q8—472no—
243C4⋊Q8—941no—
244C4⋊Q8—941no—
245C4⋊Q8—941no—
246#127—472no—
247#127—941no—
248#127—941no—
249#127—941no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8D18812yes—
253C4×Q8C22×D4712yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—941no—
256C2×C4⋊C4—941no—
257C2×C4⋊C4D18811yescharacteristic
258C2×C42—21no—
259C2×C42C22×D4711yescharacteristic
260#107—944no—
261C4⋊C8—24no—
262C4⋊C8—21no—
263C4⋊C8—472no—
264C4⋊C8—472no—
265C4⋊C8—941no—
266C4⋊C8—941no—
267C4⋊C8C22×D4712yes—
268#103—944no—
269#102—944no—
270#102—944no—
271#102—944no—
272#97—24no—
273C4×C8C22⋊D4712yes—
274C4×C8C22×D4712yes—
275C24—1881no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—942no—
280C2×Q8—3762no—
281C2×Q8—1881no—
282C2×Q8C2×D18812yes—
283C2×D4—942no—
284C2×D4—942no—
285C2×D4—942no—
286C2×D4—3762no—
287C2×D4—1881no—
288C2×D4—1881no—
289C2×D4—1881no—
290C2×D4—1882no—
291C2×D4—1882no—
292C2×D4—1882no—
293C2×D4—1882no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—1881no—
297C22×C4C2×D18811yescharacteristic
298C22×C4#1062535011yescharacteristic
299C22×C4C2×C22⋊D4711yescharacteristic
300SD16—1888no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—42no—
305C2×C8—942no—
306C2×C8—942no—
307C2×C8—1882no—
308C2×C8—1882no—
309C2×C8C2×C22⋊D4712yes—
310C2×C8C2×C22⋊D4712yes—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—942no—
315C4⋊C4—942no—
316C4⋊C4—1881no—
317C4⋊C4—1881no—
318C4⋊C4C2×D18812yes—
319C42—22no—
320C42—22no—
321C42C2×D18811yescharacteristic
322C42#1062535011yescharacteristic
323C42C2×C22⋊D4711yescharacteristic
324C42C23×D4711yescharacteristic
325C23—3762no—
326C23—3762no—
327C23—3762no—
328C23—1882no—
329C23#1309399811yescharacteristic
330Q8—22no—
331Q8—22no—
332Q8—42no—
333Q8—3762no—
334Q8—3762no—
335D4—3762no—
336D4—3762no—
337D4—3762no—
338D4—3762no—
339D4—1882no—
340D4—1882no—
341D4—1882no—
342D4—1882no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—3762no—
349C2×C4—1882no—
350C2×C4#1309399811yescharacteristic
351C2×C4#1309399811yescharacteristic
352C2×C4#1309399811yescharacteristic
353C2×C4#1309404412yes—
354C2×C4#1309405212yes—
355C2×C4#1309405512yes—
356C2×C4#1309405712yes—
357C2×C4C22×C22⋊D4711yescharacteristic
358C2×C4C22×D18811yescharacteristic
359C2×C4C2×#1062535011yescharacteristic
360C8—22no—
361C8—22no—
362C8—42no—
363C8—1882no—
364C8—1882no—
365C22—3762no—
366C22—3762no—
367C22—3762no—
368C22—7524no—
369C22#1568934512yes—
370C22#1568936912yes—
371C22#1568939312yes—
372C22C2×#1309399811yescharacteristic
373C4—21no—
374C4—21no—
375C4—42no—
376C4—42no—
377C4—3762no—
378C4#1568934512yes—
379C4#1568936912yes—
380C4C2×#1309404411yescharacteristic
381C4C2×#1309405211yescharacteristic
382C4C2×#1309405511yescharacteristic
383C4C2×#1309405711yescharacteristic
384C2—7522no—
385C2#2024292814yesdirect product factor
386C2C2×#1568934511yescharacteristic
387C2C2×#1568936911yescharacteristic
388C2C2×#1568939311yescharacteristic
subgroupG/H
centerC23#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#446C2
socleC23×C47#121
prime radicalC2×#13094021C22
hypercenterC2×#446D47
hypocenterC47C2×#1698
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. C2×#15689418C2semidirect
4. #20242928C2direct (N × H)
5. C2×#15688876C2semidirect
6. C2×#15688871C2semidirect
8. C2×C47×#446C2semidirect
15. #15689418C22semidirect
25. #15688876C22semidirect
26. #15688871C22semidirect
30. C47×#446C22semidirect
193. C2×#446D47semidirect
198. C2×C47#1698semidirect
214. #446C2×D47semidirect
229. C47C2×#1698semidirect
385. C2#20242928direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156894702
C2×#156894182
C2×#156893232
#202429282
C2×#156888762
C2×#156888712
C2×#156888662
C2×C47×#4462
C2×#169847