Finite Group Databasebeta

← #34056202

Subgroups — #34056202

Order 19072.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011942C211yesmaximal, characteristic, complemented
2C22×#14558723C211yesmaximal, characteristic, complemented
3#23750271C211yesmaximal, characteristic
4#23750017C211yesmaximal, characteristic, complemented
5#23749873C211yesmaximal, characteristic, complemented
6#23749864C211yesmaximal, characteristic, complemented
7#23749847C211yesmaximal, characteristic, complemented
8#23749827C211yesmaximal, characteristic, complemented
9#23749785C211yesmaximal, characteristic, complemented
10#23749781C211yesmaximal, characteristic, complemented
11#23749775C211yesmaximal, characteristic, complemented
12#23749764C211yesmaximal, characteristic, complemented
13#23749723C211yesmaximal, characteristic, complemented
14#23749697C211yesmaximal, characteristic, complemented
15D4×Q8×C149C211yesmaximal, characteristic, complemented
16C2×#14558736C2211yescharacteristic
17C22×D596C2211yescharacteristic
18#18011942—24no—
19C2×#14558726C2211yescharacteristic, complemented
20C2×#14558726C2211yescharacteristic, complemented
21C2×#14558726C2211yescharacteristic, complemented
22C2×#14558723—24no—
23C2×#14558723C2211yescharacteristic
24C2×#14558723C2211yescharacteristic
25C2×#14558723C2211yescharacteristic
26C2×#14558723C2211yescharacteristic
27C2×#14558723C2211yescharacteristic
28C2×#14558723C2211yescharacteristic
29C2×#14558720C2211yescharacteristic, complemented
30C22×#11924746C2211yescharacteristic
31#18011899C2211yescharacteristic
32#18011896—21no—
33#18011890C2211yescharacteristic
34#18011885C2211yescharacteristic
35#18011884—21no—
36#18011879—21no—
37C4×D596C2211yescharacteristic
38C22⋊C4×D149—21no—
39#18011844C2211yescharacteristic, complemented
40C4⋊#11924746C2211yescharacteristic, complemented
41#18011835C2211yescharacteristic, complemented
42#18011835C2211yescharacteristic, complemented
43#18011835C2211yescharacteristic, complemented
44#18011834C2211yescharacteristic, complemented
45#18011831C2211yescharacteristic
46#18011831C2211yescharacteristic
47C22⋊#11924746C2211yescharacteristic
48C22⋊#11924746C2211yescharacteristic, complemented
49#18011822C2211yescharacteristic, complemented
50C22×Q8×C149C2211yescharacteristic
51C22×Q8×C149C2211yescharacteristic, complemented
52C4⋊Q8×C149—21no—
53C4⋊Q8×C149C2211yescharacteristic, complemented
54C22⋊Q8×C149—21no—
55C22⋊Q8×C149—21no—
56C22⋊Q8×C149C2211yescharacteristic
57C22⋊Q8×C149C2211yescharacteristic, complemented
58C4×Q8×C149C2211yescharacteristic, complemented
59C4×D4×C149—21no—
60C4×D4×C149C2211yescharacteristic
61C23×D149—21no—
62#14558736—22no—
63#14558736—41no—
64C2×Dic298C2311yescharacteristic
65C2×D596—22no—
66C2×D596—21no—
67C2×D596—41no—
68C2×D596C2311yescharacteristic
69C2×D596C2311yescharacteristic
70C2×C22⋊D149—21no—
71C2×C22⋊D149—21no—
72C2×C4×D149—21no—
73C2×C4×D149—21no—
74#14558726—22no—
75#14558726—22no—
76#14558726—41no—
77#14558723—22no—
78#14558723—22no—
79#14558723—22no—
80#14558723—22no—
81#14558723—41no—
82#14558723D414yescomplemented
83#14558720—22no—
84C2×#11924746—22no—
85C2×#11924746C2311yescharacteristic
86C2×#11924746C2311yescharacteristic
87C2×#11924746C2311yescharacteristic
88C2×#11924746C2311yescharacteristic
89C4⋊Dic149C2311yescharacteristic
90#14558715—21no—
91C22⋊Dic149—21no—
92#14558713—21no—
93#14558713—21no—
94#14558713—21no—
95#14558713—21no—
96C2×Q8×C149—21no—
97C2×Q8×C149—21no—
98C2×Q8×C149—22no—
99C2×Q8×C149—22no—
100C2×Q8×C149—41no—
101C2×Q8×C149—41no—
102C2×Q8×C149C2311yescharacteristic
103C2×Q8×C149C2311yescharacteristic
104C2×Q8×C149C2311yescharacteristic
105C2×D4×C149D411yescharacteristic
106C22×C4×C149—21no—
107C22×C4×C149—21no—
108C22×C4×C149C2311yescharacteristic
109C22×C4×C149C2311yescharacteristic
110C4⋊C4×C149—21no—
111C4⋊C4×C149—21no—
112C4⋊C4×C149—21no—
113C4⋊C4×C149—21no—
114C4⋊C4×C149—21no—
115C4⋊C4×C149D411yescharacteristic
116C4⋊C4×C149C2311yescharacteristic
117C22⋊C4×C149—21no—
118C22⋊C4×C149—21no—
119C22⋊C4×C149D411yescharacteristic
120C22⋊C4×C149D411yescharacteristic
121C42×C149—21no—
122C42×C149C2311yescharacteristic
123C22×D149—21no—
124C22×D149—21no—
125C22×D149—42no—
126C22×D149—42no—
127C22×D149—81no—
128Dic298—21no—
129Dic298—21no—
130D596—21no—
131D596—21no—
132D596—21no—
133D596—42no—
134D596C2×D412yes—
135C22⋊D149—42no—
136C22⋊D149—81no—
137C2×Dic149—21no—
138C2×Dic149—21no—
139C4×D149—42no—
140C4×D149—81no—
141#11924746—21no—
142#11924746—21no—
143#11924746—21no—
144#11924746C2×D412yes—
145C23×C149C2×D411yescharacteristic
146C23×C149C2×D411yescharacteristic
147Q8×C149—21no—
148Q8×C149—21no—
149Q8×C149—21no—
150Q8×C149—21no—
151Q8×C149—21no—
152Q8×C149—42no—
153Q8×C149—42no—
154Q8×C149C2×D412yes—
155D4×C149—22no—
156C2×C4×C149—21no—
157C2×C4×C149—21no—
158C2×C4×C149—21no—
159C2×C4×C149—21no—
160C2×C4×C149—21no—
161C2×C4×C149—21no—
162C2×C4×C149—21no—
163C2×C4×C149—21no—
164C2×C4×C149—42no—
165C2×C4×C149—42no—
166C2×C4×C149C2×D411yescharacteristic
167C2×C4×C149C2×D411yescharacteristic
168C2×C4×C149C2×D411yescharacteristic
169C2×C4×C149C2×D411yescharacteristic
170C2×C4×C149C2411yescharacteristic
171C22×C149—21no—
172C22×C149—21no—
173C22×C149—21no—
174C22×C149—21no—
175C22×C149—21no—
176C22×C149#13712yescomplemented
177C22×C149C22×D411yescharacteristic
178C2×D149—22no—
179C2×D149—41no—
180C2×D149—41no—
181C2×D149—41no—
182C2×D149—41no—
183C2×D149—41no—
184C2×D149—84no—
185C4×C149—21no—
186C4×C149—21no—
187C4×C149—22no—
188C4×C149—21no—
189C4×C149—21no—
190C4×C149—41no—
191C4×C149—41no—
192C4×C149—41no—
193C4×C149C22×D411yescharacteristic
194C4×C149H(5,𝔽2)11yescharacteristic
195Dic149—41no—
196Dic149—41no—
197C2×C149—22no—
198C2×C149—41no—
199C2×C149D4211yescharacteristic
200C2×C149C2×#13711yescharacteristic
201C2×C149#57711yescharacteristic
202D149—42no—
203D149—81no—
204C149#326611yescharacteristic, complemented, Sylow
205#3266—1491nomaximal, Sylow
206C2×#137—1491no—
207C22×SD16—1491no—
208D4×Q8D14911yescharacteristic, complemented
209#546—1491no—
210#497—1491no—
211#468—1491no—
212#467—1491no—
213#464—1491no—
214#461—1491no—
215#452—1491no—
216#451—1491no—
217#450—1491no—
218#448—1491no—
219#440—1491no—
220#435—1491no—
221C2×#41—1491no—
222C22×Q8C2×D14911yescharacteristic
223C22×Q8C2×D14911yescharacteristic, complemented
224C22×D4—1491no—
225#137—2984no—
226C2×Q16—1491no—
227C2×Q16—1491no—
228C2×Q16—1491no—
229C2×SD16—2984no—
230C2×SD16—1491no—
231C2×SD16—1491no—
232C2×SD16—1491no—
233C2×SD16—1491no—
234C2×SD16—1491no—
235C2×SD16—1491no—
236C2×M16—1491no—
237C22×C8—1491no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D14911yescharacteristic, complemented
240#124—1491no—
241#123—2981no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—1491no—
245C22⋊Q8C2×D14911yescharacteristic
246C22⋊Q8C2×D14911yescharacteristic, complemented
247#121—2981no—
248#121—1491no—
249#120—2981no—
250C4×Q8C2×D14911yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—1491no—
253C4×D4C2×D14911yescharacteristic
254C2×C22⋊C4—2981no—
255#107—1491no—
256C4⋊C8—1491no—
257#103—1491no—
258#103—1491no—
259#103—1491no—
260#103—1491no—
261#102—1491no—
262#102—1491no—
263C22⋊C8—1491no—
264C22⋊C8—1491no—
265#97—1491no—
266C24—2981no—
267#41—2982no—
268#41—5961no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1491no—
276C2×Q8C22×D14911yescharacteristic
277C2×Q8C22×D14911yescharacteristic
278C2×Q8C22×D14911yescharacteristic
279C2×D4—2982no—
280C2×D4—2981no—
281C2×D4—2981no—
282C2×D4—2981no—
283C2×D4—5961no—
284C2×D4—1491no—
285C2×D4—1491no—
286C2×D4C22⋊D14911yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—2981no—
290C22×C4—2981no—
291C22×C4C22×D14911yescharacteristic
292C22×C4C22×D14911yescharacteristic
293Q16—2982no—
294Q16—2982no—
295Q16—5961no—
296SD16—2982no—
297SD16—2982no—
298SD16—2982no—
299SD16—2982no—
300SD16—5961no—
301SD16—1494no—
302M16—2982no—
303C2×C8—2982no—
304C2×C8—1491no—
305C2×C8—1491no—
306C2×C8—1491no—
307C2×C8—1491no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—2981no—
314C4⋊C4—1491no—
315C4⋊C4C22⋊D14911yescharacteristic
316C4⋊C4C22×D14911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—2981no—
320C22⋊C4—2981no—
321C22⋊C4—2981no—
322C22⋊C4—2981no—
323C22⋊C4—2981no—
324C22⋊C4C22⋊D14911yescharacteristic
325C22⋊C4C22⋊D14911yescharacteristic
326C42—21no—
327C42C22×D14911yescharacteristic
328C23—2981no—
329C23—2981no—
330C23—5962no—
331C23—5962no—
332C23—11921no—
333C23C2×C22⋊D14911yescharacteristic
334C23C2×C22⋊D14911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—2981no—
343Q8—2981no—
344Q8D4×D14912yes—
345D4—22no—
346D4—2981no—
347D4—2981no—
348D4—2981no—
349D4—5962no—
350D4—5962no—
351D4—1492no—
352D4—11921no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—2981no—
364C2×C4—2981no—
365C2×C4—5962no—
366C2×C4—11921no—
367C2×C4C2×C22⋊D14911yescharacteristic
368C2×C4C2×C22⋊D14911yescharacteristic
369C2×C4C2×C22⋊D14911yescharacteristic
370C2×C4C2×C22⋊D14911yescharacteristic
371C2×C4C23×D14911yescharacteristic
372C8—2981no—
373C8—2981no—
374C8—2981no—
375C8—1492no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—2982no—
382C22—5961no—
383C22—5961no—
384C22—5961no—
385C22—5961no—
386C22—5961no—
387C22—11924no—
388C22#1801194212yescomplemented
389C22C22×C22⋊D14911yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—5961no—
399C4—5961no—
400C4C2×D4×D14911yescharacteristic
401C4#1801195611yescharacteristic
402C2—22no—
403C2—41no—
404C2—5962no—
405C2—11921no—
406C2D4×C22⋊D14911yescharacteristic
407C2C2×#1801194211yescharacteristic
408C2#2375043311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingD4×Q8×C149C2
socleC22×C149C22×D4
prime radical#23750271C2
hypercenterD4×Q8D149
hypocenterC149#3266
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#18011942C2semidirect
2. C22×#14558723C2semidirect
4. #23750017C2semidirect
5. #23749873C2semidirect
6. #23749864C2semidirect
7. #23749847C2semidirect
8. #23749827C2semidirect
9. #23749785C2semidirect
10. #23749781C2semidirect
11. #23749775C2semidirect
12. #23749764C2semidirect
13. #23749723C2semidirect
14. #23749697C2semidirect
15. D4×Q8×C149C2semidirect
19. C2×#14558726C22semidirect
20. C2×#14558726C22semidirect
21. C2×#14558726C22semidirect
29. C2×#14558720C22semidirect
39. #18011844C22semidirect
40. C4⋊#11924746C22semidirect
41. #18011835C22semidirect
42. #18011835C22semidirect
43. #18011835C22semidirect
44. #18011834C22semidirect
48. C22⋊#11924746C22semidirect
49. #18011822C22semidirect
51. C22×Q8×C149C22semidirect
53. C4⋊Q8×C149C22semidirect
57. C22⋊Q8×C149C22semidirect
58. C4×Q8×C149C22semidirect
82. #14558723D4semidirect
176. C22×C149#137semidirect
204. C149#3266semidirect
208. D4×Q8D149semidirect
223. C22×Q8C2×D149semidirect
239. C4⋊Q8C2×D149semidirect
246. C22⋊Q8C2×D149semidirect
250. C4×Q8C2×D149semidirect
388. C22#18011942semidirect

Maximal subgroups

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

subgroupindex
C2×#180119422
C22×#145587232
#237502712
#237500172
#237498732
#237498642
#237498472
#237498272
#237497852
#237497812
#237497752
#237497642
#237497232
#237496972
D4×Q8×C1492
#3266149