Finite Group Databasebeta

← C2×#23750312 #34058200

Subgroups — C2×#23750312 #34058200

Order 19072.

1124 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750312C2116yesmaximal, direct product factor
2C2×#18011904C211yesmaximal, characteristic, complemented
3C2×#18011904C211yesmaximal, characteristic, complemented
4C2×#18011902C211yesmaximal, characteristic, complemented
5C2×#18011901C211yesmaximal, characteristic, complemented
6C2×#18011901C211yesmaximal, characteristic, complemented
7C2×#18011892C211yesmaximal, characteristic
8C2×#18011888C211yesmaximal, characteristic, complemented
9C2×#18011888C211yesmaximal, characteristic, complemented
10C2×#18011876C211yesmaximal, characteristic, complemented
11C2×D4×Dic149C211yesmaximal, characteristic
12C2×#18011870C211yesmaximal, characteristic, complemented
13C2×#18011867C211yesmaximal, characteristic, complemented
14C2×#18011867C211yesmaximal, characteristic, complemented
15C22×#14558715C211yesmaximal, characteristic, complemented
16C2×#123×C149C211yesmaximal, characteristic
17C22×Dic298C2211yescharacteristic
18C23×Dic149C2211yescharacteristic
19C23×Dic149C2211yescharacteristic
20C23×Dic149C2211yescharacteristic
21#18011904C2218yescomplemented
22#18011904C2218yescomplemented
23#18011902C2218yescomplemented
24#18011901C2218yescomplemented
25#18011901C2218yescomplemented
26#18011892C2218yes—
27#18011888C2218yescomplemented
28#18011888C2218yescomplemented
29#18011876C2218yescomplemented
30D4×Dic149C2218yes—
31#18011870C2218yescomplemented
32#18011867C2218yescomplemented
33#18011867C2218yescomplemented
34C2×C4⋊Dic149C2211yescharacteristic
35C2×C4⋊Dic149C2211yescharacteristic
36C2×C4⋊Dic149C2211yescharacteristic
37C2×C4⋊Dic149C2211yescharacteristic
38C2×#14558715C2211yescharacteristic
39C2×#14558715C2211yescharacteristic
40C2×#14558715C2211yescharacteristic
41C2×#14558715C2211yescharacteristic
42C2×#14558715C2211yescharacteristic
43C2×#14558715C2211yescharacteristic
44C2×#14558715C2211yescharacteristic
45C2×#14558715C2211yescharacteristic
46C2×#14558715C2211yescharacteristic
47C2×#14558715C2211yescharacteristic
48C2×#14558715C2218yescomplemented
49C2×C22⋊Dic149C2211yescharacteristic
50C2×C22⋊Dic149C2211yescharacteristic
51C2×C22⋊Dic149C2211yescharacteristic
52C2×C22⋊Dic149C2211yescharacteristic
53C2×C22⋊Dic149C2211yescharacteristic
54C2×C4×Dic149C2211yescharacteristic
55C2×C4×Dic149C2211yescharacteristic
56C2×C4×Dic149C2211yescharacteristic
57C2×C4×Dic149C2211yescharacteristic
58C2×C4×Dic149C2211yescharacteristic
59C22×D4×C149C2211yescharacteristic
60C23×C4×C149C2211yescharacteristic
61#123×C149C2218yes—
62C2×C4⋊C4×C149C2211yescharacteristic
63C2×C4⋊C4×C149C2211yescharacteristic
64C2×C22⋊C4×C149C2211yescharacteristic
65C2×C22⋊C4×C149C2211yescharacteristic
66C2×C22⋊C4×C149C2211yescharacteristic
67C2×Dic298—24no—
68C2×Dic298C2314yes—
69C22×Dic149—24no—
70C22×Dic149—24no—
71C22×Dic149—24no—
72C22×Dic149C2311yescharacteristic
73C22×Dic149C2311yescharacteristic
74C22×Dic149C2311yescharacteristic
75C22×Dic149C2311yescharacteristic
76C22×Dic149C2314yes—
77C22×Dic149C2314yes—
78C22×Dic149C2314yes—
79C22×Dic149C2311yescharacteristic
80C22×Dic149C2311yescharacteristic
81C22×Dic149C2311yescharacteristic
82C22×Dic149C2311yescharacteristic
83C4⋊Dic149C2314yes—
84C4⋊Dic149C2314yes—
85C4⋊Dic149C2314yes—
86C4⋊Dic149C2314yes—
87#14558715C2314yes—
88#14558715C2314yes—
89#14558715C2314yes—
90#14558715C2314yes—
91#14558715C2314yes—
92#14558715C2314yes—
93#14558715C2314yes—
94#14558715C2314yes—
95#14558715C2314yes—
96#14558715C2314yes—
97C22⋊Dic149C2314yes—
98C22⋊Dic149C2314yes—
99C22⋊Dic149C2314yes—
100C22⋊Dic149C2314yes—
101C22⋊Dic149C2314yes—
102C4×Dic149C2314yes—
103C4×Dic149C2314yes—
104C4×Dic149C2314yes—
105C4×Dic149C2314yes—
106C4×Dic149C2314yes—
107C24×C149C2311yescharacteristic
108C24×C149C2311yescharacteristic
109C2×D4×C149—24no—
110C2×D4×C149C2314yes—
111C22×C4×C149—24no—
112C22×C4×C149C2311yescharacteristic
113C22×C4×C149C2311yescharacteristic
114C22×C4×C149C2311yescharacteristic
115C22×C4×C149C2311yescharacteristic
116C22×C4×C149C2311yescharacteristic
117C22×C4×C149C2314yes—
118C4⋊C4×C149C2314yes—
119C4⋊C4×C149C2314yes—
120C22⋊C4×C149C2314yes—
121C22⋊C4×C149C2314yes—
122C22⋊C4×C149C2314yes—
123Dic298—28no—
124C2×Dic149—22no—
125C2×Dic149—22no—
126C2×Dic149—28no—
127C2×Dic149—22no—
128C2×Dic149—22no—
129C2×Dic149—22no—
130C2×Dic149—22no—
131C2×Dic149—22no—
132C2×Dic149—28no—
133C2×Dic149—28no—
134C2×Dic149#4114yes—
135C2×Dic149C2412yes—
136C2×Dic149C2412yes—
137C2×Dic149C2412yes—
138C2×Dic149C2412yes—
139C2×Dic149C2412yes—
140C2×Dic149C2412yes—
141C2×Dic149C2412yes—
142C2×Dic149C2412yes—
143C23×C149—22no—
144C23×C149—22no—
145C23×C149—22no—
146C23×C149—22no—
147C23×C149—22no—
148C23×C149#4114yes—
149C23×C149C2412yes—
150C23×C149C2412yes—
151C23×C149C2411yescharacteristic
152D4×C149—28no—
153C2×C4×C149—22no—
154C2×C4×C149—28no—
155C2×C4×C149—22no—
156C2×C4×C149—22no—
157C2×C4×C149—22no—
158C2×C4×C149—22no—
159C2×C4×C149C2412yes—
160C2×C4×C149C2412yes—
161C2×C4×C149C2412yes—
162C2×C4×C149C2412yes—
163C2×C4×C149C2412yes—
164C22×C149—28no—
165C22×C149—22no—
166C22×C149—22no—
167C22×C149—22no—
168C22×C149—22no—
169C22×C149—22no—
170C22×C149—44no—
171C22×C149C2×#4112yes—
172C22×C149C2×#4114yes—
173C22×C149C2×#4112yes—
174C22×C1492−1+412yes—
175C22×C149C2511yescharacteristic
176C4×C149—22no—
177C4×C149—22no—
178C4×C149—22no—
179C4×C149—22no—
180C4×C149—22no—
181Dic149—22no—
182Dic149—22no—
183Dic149—22no—
184Dic149—22no—
185Dic149—22no—
186Dic149—22no—
187Dic149—22no—
188Dic149C2×#4114yes—
189C2×C149—24no—
190C2×C149—42no—
191C2×C149#55214yescomplemented
192C2×C149C22×#4111yescharacteristic
193C2×C149C22×#4111yescharacteristic
194C2×C149C2×2−1+411yescharacteristic
195C149C2×#55211yescharacteristic, complemented, Sylow
196C2×#552—1491nomaximal, Sylow
197#552—14916no—
198C2×#126—1491no—
199C2×#126—1491no—
200C2×#125—1491no—
201C2×#125—1491no—
202C2×#125—1491no—
203C2×#123—1491no—
204C2×#123D14911yescharacteristic
205C2×C22⋊Q8—1491no—
206C2×C22⋊Q8—1491no—
207C2×C4×Q8—1491no—
208C2×C4×D4—1491no—
209C2×#117—1491no—
210C2×#117—1491no—
211C2×#117—1491no—
212C22×C4⋊C4—1491no—
213C22×Q8—1491no—
214C22×D4C2×D14911yescharacteristic
215C23×C4—1491no—
216C23×C4—1491no—
217C23×C4—1491no—
218C23×C4C2×D14911yescharacteristic
219#126—1498no—
220#126—1498no—
221#125—1498no—
222#125—1498no—
223#125—1498no—
224#123—1498no—
225#123C2×D14918yes—
226C22⋊Q8—1498no—
227C22⋊Q8—1498no—
228C4×Q8—1498no—
229C4×D4—1498no—
230#117—1498no—
231#117—1498no—
232#117—1498no—
233C2×C4⋊C4—1491no—
234C2×C4⋊C4—1491no—
235C2×C4⋊C4—1491no—
236C2×C4⋊C4—1491no—
237C2×C4⋊C4—1498no—
238C2×C4⋊C4—1491no—
239C2×C4⋊C4—1491no—
240C2×C4⋊C4—1491no—
241C2×C4⋊C4—1491no—
242C2×C4⋊C4—1491no—
243C2×C4⋊C4—1491no—
244C2×C4⋊C4—1491no—
245C2×C4⋊C4—1491no—
246C2×C4⋊C4—1491no—
247C2×C4⋊C4—1491no—
248C2×C4⋊C4C2×D14911yescharacteristic
249C2×C4⋊C4C2×D14911yescharacteristic
250C2×C22⋊C4—1491no—
251C2×C22⋊C4—1491no—
252C2×C22⋊C4—1491no—
253C2×C22⋊C4—1491no—
254C2×C22⋊C4—1491no—
255C2×C22⋊C4C2×D14911yescharacteristic
256C2×C22⋊C4C2×D14911yescharacteristic
257C2×C22⋊C4C2×D14911yescharacteristic
258C2×C42—1491no—
259C2×C42—1491no—
260C2×C42—1491no—
261C2×C42—1491no—
262C2×C42—1491no—
263C24C22×D14911yescharacteristic
264C24C22×D14911yescharacteristic
265C2×Q8—2984no—
266C2×Q8—1494no—
267C2×D4—24no—
268C2×D4C22×D14914yes—
269C22×C4—24no—
270C22×C4—2984no—
271C22×C4—2984no—
272C22×C4—2984no—
273C22×C4—1494no—
274C22×C4—1494no—
275C22×C4—1494no—
276C22×C4—1491no—
277C22×C4—1491no—
278C22×C4—1491no—
279C22×C4—1491no—
280C22×C4—1491no—
281C22×C4—1491no—
282C22×C4—1491no—
283C22×C4—1491no—
284C22×C4C22×D14914yes—
285C22×C4C22×D14911yescharacteristic
286C22×C4C22×D14911yescharacteristic
287C22×C4C22×D14911yescharacteristic
288C22×C4C22×D14911yescharacteristic
289C22×C4C22×D14911yescharacteristic
290C4⋊C4—1494no—
291C4⋊C4—1494no—
292C4⋊C4—1494no—
293C4⋊C4—1494no—
294C4⋊C4—1494no—
295C4⋊C4—1494no—
296C4⋊C4—1494no—
297C4⋊C4—1494no—
298C4⋊C4—1494no—
299C4⋊C4—1494no—
300C4⋊C4—1494no—
301C4⋊C4—1494no—
302C4⋊C4—1494no—
303C4⋊C4—1494no—
304C4⋊C4C22×D14914yes—
305C4⋊C4C22×D14914yes—
306C22⋊C4—1494no—
307C22⋊C4—1494no—
308C22⋊C4—1494no—
309C22⋊C4—1494no—
310C22⋊C4—1494no—
311C22⋊C4C22×D14914yes—
312C22⋊C4C22×D14914yes—
313C22⋊C4C22×D14914yes—
314C42—1494no—
315C42—1494no—
316C42—1494no—
317C42—1494no—
318C42—1494no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23#1455873514yes—
325C23C23×D14912yes—
326C23C23×D14912yes—
327C23C23×D14911yescharacteristic
328Q8—2988no—
329D4—28no—
330C2×C4—22no—
331C2×C4—28no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—2982no—
337C2×C4—2982no—
338C2×C4—2982no—
339C2×C4—2988no—
340C2×C4—2982no—
341C2×C4—2982no—
342C2×C4—2982no—
343C2×C4—2982no—
344C2×C4—2988no—
345C2×C4—2988no—
346C2×C4—1492no—
347C2×C4—1492no—
348C2×C4—1492no—
349C2×C4—1492no—
350C2×C4—1492no—
351C2×C4—1492no—
352C2×C4—1492no—
353C2×C4—1494no—
354C2×C4—1492no—
355C2×C4C23×D14912yes—
356C2×C4C23×D14912yes—
357C2×C4C23×D14912yes—
358C2×C4C23×D14912yes—
359C2×C4C23×D14912yes—
360C22—22no—
361C22—22no—
362C22—28no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—44no—
367C22#41×D14912yes—
368C22C2×#1455873514yes—
369C22C2×#1455873512yes—
370C22#1801195812yes—
371C22C24×D14911yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—2982no—
378C4—2982no—
379C4—2982no—
380C4—2982no—
381C4—2982no—
382C4—2982no—
383C4—2982no—
384C4—1494no—
385C2—24no—
386C2—42no—
387C2#2375031214yesdirect product factor
388C2C2×#41×D14911yescharacteristic
389C2C22×#1455873511yescharacteristic
390C2C2×#1801195811yescharacteristic
subgroupG/H
centerC23C23×D149
commutator, cosocleC22×C149C25
FrattiniC22C24×D149
FittingC2×#123×C149C2
socleC23×C149C24
prime radicalC22×D4×C149C22
hypercenterC2×#123D149
hypocenterC149C2×#552
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #23750312C2direct (N × H)
2. C2×#18011904C2semidirect
3. C2×#18011904C2semidirect
4. C2×#18011902C2semidirect
5. C2×#18011901C2semidirect
6. C2×#18011901C2semidirect
8. C2×#18011888C2semidirect
9. C2×#18011888C2semidirect
10. C2×#18011876C2semidirect
12. C2×#18011870C2semidirect
13. C2×#18011867C2semidirect
14. C2×#18011867C2semidirect
15. C22×#14558715C2semidirect
21. #18011904C22semidirect
22. #18011904C22semidirect
23. #18011902C22semidirect
24. #18011901C22semidirect
25. #18011901C22semidirect
27. #18011888C22semidirect
28. #18011888C22semidirect
29. #18011876C22semidirect
31. #18011870C22semidirect
32. #18011867C22semidirect
33. #18011867C22semidirect
48. C2×#14558715C22semidirect
191. C2×C149#552semidirect
195. C149C2×#552semidirect
387. C2#23750312direct (N × H)

Maximal subgroups

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

subgroupindex
#237503122
C2×#180119042
C2×#180119022
C2×#180119012
C2×#180118922
C2×#180118882
C2×#180118762
C2×D4×Dic1492
C2×#180118702
C2×#180118672
C22×#145587152
C2×#123×C1492
C2×#552149