Finite Group Databasebeta

← C2×#23750329 #34058219

Subgroups — C2×#23750329 #34058219

Order 19072.

1200 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750329C2116yesmaximal, direct product factor
2C2×#18011909C211yesmaximal, characteristic, complemented
3C2×#18011901C211yesmaximal, characteristic, complemented
4C2×#18011901C211yesmaximal, characteristic, complemented
5C2×#18011889C211yesmaximal, characteristic, complemented
6C2×#18011889C211yesmaximal, characteristic, complemented
7C2×#18011888C211yesmaximal, characteristic, complemented
8C2×#18011888C211yesmaximal, characteristic, complemented
9C2×#18011887C211yesmaximal, characteristic, complemented
10C2×#18011876C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊D149C211yesmaximal, characteristic
12C2×#18011871C211yesmaximal, characteristic
13C2×#18011871C211yesmaximal, characteristic
14C22×#14558715C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D149C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C149C211yesmaximal, characteristic, complemented
17C22×Dic298C2211yescharacteristic
18C22×Dic298C2211yescharacteristic
19C22×C22⋊D149C2211yescharacteristic
20C23×Dic149C2211yescharacteristic
21C23×Dic149C2211yescharacteristic
22C22×C4×D149C2211yescharacteristic
23C22×C4×D149C2211yescharacteristic
24C22×C4×D149C2211yescharacteristic
25#18011909C2218yescomplemented
26#18011901C2218yescomplemented
27#18011901C2218yescomplemented
28#18011889C2218yescomplemented
29#18011889C2218yescomplemented
30#18011888C2218yescomplemented
31#18011888C2218yescomplemented
32#18011887C2218yescomplemented
33#18011876C2218yescomplemented
34C4×C22⋊D149C2218yes—
35#18011871C2218yes—
36#18011871C2218yes—
37C2×C4⋊Dic149C2211yescharacteristic
38C2×C4⋊Dic149C2211yescharacteristic
39C2×#14558715C2211yescharacteristic
40C2×#14558715C2211yescharacteristic
41C2×#14558715C2211yescharacteristic
42C2×#14558715C2211yescharacteristic
43C2×#14558715C2211yescharacteristic
44C2×#14558715C2211yescharacteristic
45C2×#14558715C2211yescharacteristic
46C2×#14558715C2211yescharacteristic
47C2×#14558715C2211yescharacteristic
48C2×#14558715C2211yescharacteristic
49C2×#14558715C2211yescharacteristic
50C2×#14558715C2218yescomplemented
51C4⋊C4×D149C2218yescomplemented
52C2×C22⋊Dic149C2211yescharacteristic
53C2×#14558713C2211yescharacteristic
54C2×#14558713C2211yescharacteristic
55C2×#14558713C2211yescharacteristic
56C2×C4×Dic149C2211yescharacteristic
57C2×C4×Dic149C2211yescharacteristic
58C2×C4×Dic149C2211yescharacteristic
59C22×Q8×C149C2211yescharacteristic
60C23×C4×C149C2211yescharacteristic
61C22⋊Q8×C149C2218yescomplemented
62C2×C4⋊C4×C149C2211yescharacteristic
63C2×C4⋊C4×C149C2211yescharacteristic
64C2×C4⋊C4×C149C2211yescharacteristic
65C2×C22⋊C4×C149C2211yescharacteristic
66C2×C22⋊C4×C149C2211yescharacteristic
67C23×D149C2311yescharacteristic
68C2×Dic298—24no—
69C2×Dic298—42no—
70C2×Dic298C2314yes—
71C2×Dic298C2314yes—
72C2×C22⋊D149Q818yescomplemented
73C2×C22⋊D149C2314yes—
74C22×Dic149—24no—
75C22×Dic149—42no—
76C22×Dic149C2311yescharacteristic
77C22×Dic149C2311yescharacteristic
78C22×Dic149C2311yescharacteristic
79C22×Dic149C2311yescharacteristic
80C22×Dic149C2314yes—
81C22×Dic149C2311yescharacteristic
82C22×Dic149C2314yes—
83C22×Dic149C2311yescharacteristic
84C22×Dic149C2311yescharacteristic
85C2×C4×D149—24no—
86C2×C4×D149—42no—
87C2×C4×D149—42no—
88C2×C4×D149C2314yes—
89C2×C4×D149C2314yes—
90C2×C4×D149C2314yes—
91C4⋊Dic149C2314yes—
92C4⋊Dic149C2314yes—
93#14558715C2314yes—
94#14558715C2314yes—
95#14558715C2314yes—
96#14558715C2314yes—
97#14558715C2314yes—
98#14558715C2314yes—
99#14558715C2314yes—
100#14558715C2314yes—
101#14558715C2314yes—
102#14558715C2314yes—
103#14558715C2314yes—
104C22⋊Dic149C2314yes—
105#14558713C2314yes—
106#14558713C2314yes—
107#14558713C2314yes—
108C4×Dic149C2314yes—
109C4×Dic149C2314yes—
110C4×Dic149C2314yes—
111C24×C149C2311yescharacteristic
112C2×Q8×C149—42no—
113C2×Q8×C149C2314yes—
114C22×C4×C149—42no—
115C22×C4×C149C2311yescharacteristic
116C22×C4×C149C2311yescharacteristic
117C22×C4×C149C2311yescharacteristic
118C22×C4×C149C2311yescharacteristic
119C22×C4×C149C2311yescharacteristic
120C22×C4×C149C2311yescharacteristic
121C22×C4×C149C2314yes—
122C4⋊C4×C149C2314yes—
123C4⋊C4×C149C2314yes—
124C4⋊C4×C149C2314yes—
125C22⋊C4×C149C2314yes—
126C22⋊C4×C149C2314yes—
127C22×D149—22no—
128C22×D149—22no—
129C22×D149C2×Q814yes—
130C22×D149C2412yes—
131Dic298—28no—
132Dic298—44no—
133C22⋊D149C2×Q8116yescomplemented
134C2×Dic149—22no—
135C2×Dic149—22no—
136C2×Dic149—22no—
137C2×Dic149—22no—
138C2×Dic149—22no—
139C2×Dic149—28no—
140C2×Dic149—44no—
141C2×Dic149C2×Q814yes—
142C2×Dic149#4114yes—
143C2×Dic149C2412yes—
144C2×Dic149C2412yes—
145C2×Dic149C2412yes—
146C2×Dic149C2412yes—
147C2×Dic149C2412yes—
148C2×Dic149C2412yes—
149C2×Dic149C2412yes—
150C4×D149—28no—
151C4×D149—44no—
152C4×D149—44no—
153C23×C149—22no—
154C23×C149—22no—
155C23×C149C2×Q814yes—
156C23×C149C2412yes—
157C23×C149C2411yescharacteristic
158Q8×C149—44no—
159C2×C4×C149—22no—
160C2×C4×C149—22no—
161C2×C4×C149—22no—
162C2×C4×C149—22no—
163C2×C4×C149—22no—
164C2×C4×C149—22no—
165C2×C4×C149—44no—
166C2×C4×C149C2412yes—
167C2×C4×C149C2412yes—
168C2×C4×C149C2412yes—
169C2×C4×C149C2412yes—
170C2×C4×C149C2412yes—
171C2×C4×C149C2412yes—
172C22×C149—28no—
173C22×C149—22no—
174C22×C149—22no—
175C22×C149C22×Q814yes—
176C22×C149C22×Q812yes—
177C22×C149C2×#4112yes—
178C22×C149H(5,𝔽2)12yes—
179C22×C149C2511yescharacteristic
180C2×D149—28no—
181C2×D149—22no—
182C2×D149—22no—
183C2×D149C22×Q814yes—
184C4×C149—22no—
185C4×C149—22no—
186C4×C149—22no—
187C4×C149—22no—
188C4×C149—22no—
189C4×C149—22no—
190Dic149—22no—
191Dic149—22no—
192Dic149—22no—
193Dic149—22no—
194Dic149—22no—
195Dic149C22×Q814yes—
196Dic149C2×#4114yes—
197C2×C149—24no—
198C2×C149#55414yescomplemented
199C2×C149C23×Q811yescharacteristic
200C2×C149C22×#4111yescharacteristic
201C2×C149C2×H(5,𝔽2)11yescharacteristic
202D149—24no—
203C149C2×#55411yescharacteristic, complemented, Sylow
204C2×#554—1491nomaximal, Sylow
205#554—14916no—
206C2×C4⋊Q8—1491no—
207C2×#125—1491no—
208C2×#125—1491no—
209C2×C22⋊Q8—1491no—
210C2×C22⋊Q8—1491no—
211C2×C22⋊Q8—1491no—
212C2×C22⋊Q8—1491no—
213C2×C22⋊Q8—1491no—
214C2×C22⋊Q8D14911yescharacteristic, complemented
215C2×C4×Q8—1491no—
216C2×C4×D4—1491no—
217C2×C4×D4—1491no—
218C2×C4×D4—1491no—
219C22×C4⋊C4—1491no—
220C22×C4⋊C4—1491no—
221C22×Q8—1491no—
222C22×Q8—1491no—
223C22×Q8C2×D14911yescharacteristic
224C22×D4—1491no—
225C23×C4—1491no—
226C23×C4—1491no—
227C23×C4—1491no—
228C23×C4—1491no—
229C23×C4—1491no—
230C23×C4C2×D14911yescharacteristic
231C4⋊Q8—1498no—
232#125—1498no—
233#125—1498no—
234C22⋊Q8—1498no—
235C22⋊Q8—1498no—
236C22⋊Q8—1498no—
237C22⋊Q8—1498no—
238C22⋊Q8—1498no—
239C22⋊Q8C2×D14918yescomplemented
240C4×Q8—1498no—
241C4×D4—1498no—
242C4×D4—1498no—
243C4×D4—1498no—
244C2×C4⋊C4—1491no—
245C2×C4⋊C4—1491no—
246C2×C4⋊C4—1491no—
247C2×C4⋊C4—1491no—
248C2×C4⋊C4—1491no—
249C2×C4⋊C4—1491no—
250C2×C4⋊C4—1498no—
251C2×C4⋊C4—1491no—
252C2×C4⋊C4—1491no—
253C2×C4⋊C4—1491no—
254C2×C4⋊C4—1491no—
255C2×C4⋊C4—1491no—
256C2×C4⋊C4—1491no—
257C2×C4⋊C4—1491no—
258C2×C4⋊C4—1498no—
259C2×C4⋊C4C2×D14911yescharacteristic
260C2×C4⋊C4C2×D14911yescharacteristic
261C2×C4⋊C4C2×D14911yescharacteristic
262C2×C22⋊C4—1491no—
263C2×C22⋊C4—1491no—
264C2×C22⋊C4—1491no—
265C2×C22⋊C4—1491no—
266C2×C22⋊C4C2×D14911yescharacteristic
267C2×C22⋊C4C2×D14911yescharacteristic
268C2×C42—1491no—
269C2×C42—1491no—
270C2×C42—1491no—
271C24—1491no—
272C24C22×D14911yescharacteristic
273C2×Q8—42no—
274C2×Q8—2984no—
275C2×Q8—5962no—
276C2×Q8—1494no—
277C2×Q8—1494no—
278C2×Q8C22×D14914yes—
279C2×D4—1498no—
280C2×D4—1494no—
281C22×C4—42no—
282C22×C4—2984no—
283C22×C4—2984no—
284C22×C4—5962no—
285C22×C4—5962no—
286C22×C4—5962no—
287C22×C4—1494no—
288C22×C4—1491no—
289C22×C4—1491no—
290C22×C4—1491no—
291C22×C4—1491no—
292C22×C4—1491no—
293C22×C4—1494no—
294C22×C4—1494no—
295C22×C4—1494no—
296C22×C4—1494no—
297C22×C4—1491no—
298C22×C4—1491no—
299C22×C4C22×D14911yescharacteristic
300C22×C4C22×D14911yescharacteristic
301C22×C4C22×D14911yescharacteristic
302C22×C4C22×D14911yescharacteristic
303C22×C4C22×D14911yescharacteristic
304C22×C4C22×D14911yescharacteristic
305C22×C4C22×D14914yes—
306C4⋊C4—1494no—
307C4⋊C4—1494no—
308C4⋊C4—1494no—
309C4⋊C4—1494no—
310C4⋊C4—1494no—
311C4⋊C4—1494no—
312C4⋊C4—1494no—
313C4⋊C4—1494no—
314C4⋊C4—1494no—
315C4⋊C4—1494no—
316C4⋊C4—1494no—
317C4⋊C4—1494no—
318C4⋊C4—1494no—
319C4⋊C4C22×D14914yes—
320C4⋊C4C22×D14914yes—
321C4⋊C4C22×D14914yes—
322C22⋊C4—1494no—
323C22⋊C4—1494no—
324C22⋊C4—1494no—
325C22⋊C4—1494no—
326C22⋊C4C22×D14914yes—
327C22⋊C4C22×D14914yes—
328C42—1494no—
329C42—1494no—
330C42—1494no—
331C23—22no—
332C23—22no—
333C23—2982no—
334C23—2982no—
335C23—1494no—
336C23—1492no—
337C23Q8×D14914yes—
338C23C23×D14912yes—
339C23C23×D14911yescharacteristic
340Q8—44no—
341Q8—2988no—
342Q8—5964no—
343D4—14916no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—44no—
351C2×C4—2988no—
352C2×C4—2982no—
353C2×C4—2982no—
354C2×C4—2982no—
355C2×C4—2982no—
356C2×C4—2982no—
357C2×C4—2988no—
358C2×C4—5964no—
359C2×C4—5964no—
360C2×C4—5964no—
361C2×C4—1492no—
362C2×C4—1492no—
363C2×C4—1492no—
364C2×C4—1492no—
365C2×C4—1492no—
366C2×C4—1494no—
367C2×C4—1494no—
368C2×C4—1492no—
369C2×C4—1492no—
370C2×C4C23×D14912yes—
371C2×C4C23×D14912yes—
372C2×C4C23×D14912yes—
373C2×C4C23×D14912yes—
374C2×C4C23×D14912yes—
375C2×C4C23×D14912yes—
376C22—22no—
377C22—22no—
378C22—28no—
379C22—2982no—
380C22—2982no—
381C22—2988no—
382C22—1494no—
383C22C2×Q8×D14914yes—
384C22C2×Q8×D14912yes—
385C22#41×D14912yes—
386C22#1801195612yes—
387C22C24×D14911yescharacteristic
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—2982no—
395C4—2982no—
396C4—2982no—
397C4—2982no—
398C4—2982no—
399C4—1494no—
400C4—1494no—
401C2—24no—
402C2—2984no—
403C2#2375032914yesdirect product factor
404C2C22×Q8×D14911yescharacteristic
405C2C2×#41×D14911yescharacteristic
406C2C2×#1801195611yescharacteristic
subgroupG/H
centerC23C23×D149
commutator, cosocleC22×C149C25
FrattiniC22C24×D149
FittingC2×C22⋊Q8×C149C2
socleC23×C149C24
prime radicalC22×C22⋊D149C22
hypercenterC2×C22⋊Q8D149
hypocenterC149C2×#554
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #23750329C2direct (N × H)
2. C2×#18011909C2semidirect
3. C2×#18011901C2semidirect
4. C2×#18011901C2semidirect
5. C2×#18011889C2semidirect
6. C2×#18011889C2semidirect
7. C2×#18011888C2semidirect
8. C2×#18011888C2semidirect
9. C2×#18011887C2semidirect
10. C2×#18011876C2semidirect
14. C22×#14558715C2semidirect
15. C2×C4⋊C4×D149C2semidirect
16. C2×C22⋊Q8×C149C2semidirect
25. #18011909C22semidirect
26. #18011901C22semidirect
27. #18011901C22semidirect
28. #18011889C22semidirect
29. #18011889C22semidirect
30. #18011888C22semidirect
31. #18011888C22semidirect
32. #18011887C22semidirect
33. #18011876C22semidirect
50. C2×#14558715C22semidirect
51. C4⋊C4×D149C22semidirect
61. C22⋊Q8×C149C22semidirect
72. C2×C22⋊D149Q8semidirect
133. C22⋊D149C2×Q8semidirect
198. C2×C149#554semidirect
203. C149C2×#554semidirect
214. C2×C22⋊Q8D149semidirect
239. C22⋊Q8C2×D149semidirect
403. C2#23750329direct (N × H)

Maximal subgroups

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

subgroupindex
#237503292
C2×#180119092
C2×#180119012
C2×#180118892
C2×#180118882
C2×#180118872
C2×#180118762
C2×C4×C22⋊D1492
C2×#180118712
C22×#145587152
C2×C4⋊C4×D1492
C2×C22⋊Q8×C1492
C2×#554149