Finite Group Databasebeta

← #34079862

Subgroups — #34079862

Order 19328.

468 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
1C22⋊Q8×D151C211yesmaximal, characteristic, complemented
2#121×D151C211yesmaximal, characteristic
3#23752544C211yesmaximal, characteristic, complemented
4#23752541C211yesmaximal, characteristic, complemented
5#23752532C211yesmaximal, characteristic, complemented
6#23752529C211yesmaximal, characteristic, complemented
7#23752443C211yesmaximal, characteristic, complemented
8#23752441C211yesmaximal, characteristic, complemented
9#23752439C211yesmaximal, characteristic, complemented
10#23752364C211yesmaximal, characteristic, complemented
11#23752361C211yesmaximal, characteristic, complemented
12#23752352C211yesmaximal, characteristic, complemented
13#23752349C211yesmaximal, characteristic, complemented
14C2×(C2×D151)⋊C8C211yesmaximal, characteristic, complemented
15#465×C151C211yesmaximal, characteristic, complemented
16C2×Q8×D151C2211yescharacteristic, complemented
17C2×D4×D151—21no—
18C2×D4×D151C2211yescharacteristic
19C22×C4×D151C2211yescharacteristic
20C2×#14558993C2211yescharacteristic, complemented
21C2×#14558992C2211yescharacteristic, complemented
22C2×#14558991C2211yescharacteristic, complemented
23C22×C151⋊C8C2211yescharacteristic, complemented
24#18012719C2211yescharacteristic
25#18012718C2211yescharacteristic, complemented
26#18012717—21no—
27#18012716—21no—
28#18012715C2211yescharacteristic
29#18012714C2211yescharacteristic
30#18012713C2211yescharacteristic
31#18012712—21no—
32#18012711—21no—
33#18012710C2211yescharacteristic
34C4⋊C4×D151—21no—
35C4⋊C4×D151C2211yescharacteristic
36C4⋊C4×D151C2211yescharacteristic, complemented
37C22⋊C4×D151—21no—
38C22⋊C4×D151—21no—
39#18012671C2211yescharacteristic, complemented
40#18012670C2211yescharacteristic, complemented
41#18012670C2211yescharacteristic, complemented
42#18012664C2211yescharacteristic, complemented
43#18012663C2211yescharacteristic, complemented
44#18012662C2211yescharacteristic, complemented
45#18012661C2211yescharacteristic, complemented
46#18012660C2211yescharacteristic, complemented
47#18012659C2211yescharacteristic, complemented
48(C2×D151)⋊C8C2211yescharacteristic
49(C2×D151)⋊C8C2211yescharacteristic
50(C2×D151)⋊C8C2211yescharacteristic, complemented
51(C2×D151)⋊C8C2211yescharacteristic, complemented
52C2×SD16×C151C2211yescharacteristic, complemented
53C22×C8×C151C2211yescharacteristic, complemented
54C22⋊Q8×C151C2211yescharacteristic, complemented
55#121×C151C2211yescharacteristic
56#106×C151C2211yescharacteristic, complemented
57#103×C151C2211yescharacteristic, complemented
58#102×C151C2211yescharacteristic, complemented
59C23×D151—21no—
60C23×D151D411yescharacteristic
61C2×Dic302—21no—
62C2×Dic302C2311yescharacteristic
63Q8×D151—41no—
64Q8×D151—41no—
65C2×D604—21no—
66C2×D604C2311yescharacteristic
67C2×C22⋊D151—21no—
68C2×C22⋊D151—21no—
69C2×C22⋊D151—21no—
70D4×D151—41no—
71D4×D151—41no—
72D4×D151—44no—
73C22×Dic151D411yescharacteristic
74C2×C4×D151—21no—
75C2×C4×D151—21no—
76C2×C4×D151—21no—
77C2×C4×D151—41no—
78C2×C4×D151—41no—
79C2×C4×D151D411yescharacteristic
80C2×C4×D151D411yescharacteristic
81C2×C4×D151C2311yescharacteristic
82C2×C4×D151C2311yescharacteristic
83#14558993—41no—
84#14558992—41no—
85#14558991—41no—
86C2×C151⋊C8—41no—
87C2×C151⋊C8C2311yescharacteristic
88C2×C151⋊C8C2311yescharacteristic
89C4⋊Dic151—21no—
90C4⋊Dic151C2311yescharacteristic
91C4⋊Dic151C2311yescharacteristic
92Dic151⋊C4—21no—
93Dic151⋊C4—21no—
94Dic151⋊C4—21no—
95Dic151⋊C4—21no—
96C22⋊Dic151—21no—
97C22⋊Dic151—21no—
98(C2×D151)⋊C4—21no—
99(C2×D151)⋊C4—21no—
100(C2×D151)⋊C4—21no—
101(C2×D151)⋊C4—21no—
102C2×Q8×C151C2311yescharacteristic
103C2×D4×C151—21no—
104C2×D4×C151C2311yescharacteristic
105C22×C4×C151C2311yescharacteristic
106SD16×C151—41no—
107C2×C8×C151—41no—
108C2×C8×C151C2311yescharacteristic
109C2×C8×C151C2311yescharacteristic
110C4⋊C4×C151—21no—
111C4⋊C4×C151C2311yescharacteristic
112C4⋊C4×C151C2311yescharacteristic
113C22⋊C4×C151—21no—
114C22⋊C4×C151—21no—
115C22×D151—21no—
116C22×D151—41no—
117C22×D151—41no—
118C22×D151—42no—
119C22×D151—42no—
120C22×D151—42no—
121C22×D151SD1614yescomplemented
122C22×D151C2×D411yescharacteristic
123C22×D151C2×D411yescharacteristic
124Dic302—21no—
125Dic302—21no—
126Dic302—42no—
127D604—21no—
128D604—21no—
129D604—42no—
130C22⋊D151—42no—
131C22⋊D151—42no—
132C22⋊D151—42no—
133C2×Dic151—21no—
134C2×Dic151—21no—
135C2×Dic151—21no—
136C2×Dic151—41no—
137C2×Dic151C2×D411yescharacteristic
138C2×Dic151C2×D411yescharacteristic
139C4×D151—21no—
140C4×D151—21no—
141C4×D151—21no—
142C4×D151—21no—
143C4×D151—41no—
144C4×D151—41no—
145C4×D151—42no—
146C4×D151—42no—
147C4×D151—42no—
148C151⋊C8—21no—
149C151⋊C8—21no—
150C23×C151—21no—
151C23×C151C2×D411yescharacteristic
152Q8×C151—21no—
153Q8×C151—21no—
154D4×C151—21no—
155D4×C151—21no—
156D4×C151—42no—
157C2×C4×C151—21no—
158C2×C4×C151—21no—
159C2×C4×C151—21no—
160C2×C4×C151—21no—
161C2×C4×C151—21no—
162C2×C4×C151C2×D411yescharacteristic
163C2×C4×C151C2411yescharacteristic
164C8×C151—21no—
165C8×C151—21no—
166C22×C151—21no—
167C22×C151—21no—
168C22×C151—41no—
169C22×C151—41no—
170C22×C151—41no—
171C22×C151C2×SD1612yes—
172C22×C151C22×D411yescharacteristic
173C2×D151—21no—
174C2×D151—21no—
175C2×D151—21no—
176C2×D151—21no—
177C2×D151—44no—
178C2×D151—41no—
179C2×D151—41no—
180C2×D151—41no—
181C2×D151—84no—
182C2×D151C2×SD1612yes—
183C2×D151C2×SD1612yes—
184C4×C151—21no—
185C4×C151—41no—
186C4×C151—41no—
187C4×C151—41no—
188C4×C151H(5,𝔽2)11yescharacteristic
189C4×C151H(5,𝔽2)11yescharacteristic
190Dic151—21no—
191Dic151—21no—
192Dic151—41no—
193Dic151—41no—
194Dic151—41no—
195C2×C151—22no—
196C2×C151—81no—
197C2×C151#53411yescharacteristic
198C2×C151C22×SD1611yescharacteristic
199C2×C151#57511yescharacteristic
200D151—22no—
201D151—22no—
202D151—81no—
203C151#318711yescharacteristic, complemented, Sylow
204#3187—1511nomaximal, Sylow
205C2×C22⋊Q8—1511no—
206C2×#121—1511no—
207#483—1511no—
208#483—1511no—
209#481—1511no—
210#481—1511no—
211#465—1511no—
212#465—1511no—
213#465—1511no—
214#465D15111yescharacteristic, complemented
215#450—1511no—
216#450—1511no—
217#448—1511no—
218#448—1511no—
219C2×C22⋊C8—1511no—
220C22×Q8—1511no—
221C22×D4—3021no—
222C22×D4—1511no—
223C23×C4—1511no—
224C2×SD16—1511no—
225C2×SD16—1511no—
226C2×SD16—1511no—
227C2×SD16C2×D15111yescharacteristic, complemented
228C22×C8—1511no—
229C22×C8C2×D15111yescharacteristic, complemented
230C22⋊Q8—3021no—
231C22⋊Q8—3021no—
232C22⋊Q8—1511no—
233C22⋊Q8—1511no—
234C22⋊Q8—1511no—
235C22⋊Q8C2×D15111yescharacteristic, complemented
236#121—3021no—
237#121—3021no—
238#121—1511no—
239#121—1511no—
240#121—1511no—
241#121C2×D15111yescharacteristic
242C2×C4⋊C4—3021no—
243C2×C4⋊C4—1511no—
244C2×C4⋊C4—1511no—
245C2×C22⋊C4—3021no—
246C2×C22⋊C4—3021no—
247#106—1511no—
248#106—1511no—
249#106—1511no—
250#106C2×D15111yescharacteristic, complemented
251#103—1511no—
252#103—1511no—
253#103—1511no—
254#103C2×D15111yescharacteristic, complemented
255#102—1511no—
256#102—1511no—
257#102—1511no—
258#102C2×D15111yescharacteristic, complemented
259C22⋊C8—1511no—
260C22⋊C8—1511no—
261C22⋊C8—1511no—
262C22⋊C8—1511no—
263C24—3021no—
264C24—1511no—
265C2×Q8—3021no—
266C2×Q8—6041no—
267C2×Q8—6041no—
268C2×Q8—1511no—
269C2×Q8C22×D15111yescharacteristic
270C2×D4—21no—
271C2×D4—3021no—
272C2×D4—3021no—
273C2×D4—3021no—
274C2×D4—3021no—
275C2×D4—6041no—
276C2×D4—6041no—
277C2×D4—6044no—
278C2×D4—1511no—
279C2×D4C22×D15111yescharacteristic
280C22×C4—3021no—
281C22×C4—3021no—
282C22×C4—3021no—
283C22×C4—6041no—
284C22×C4—6041no—
285C22×C4—1511no—
286C22×C4—1511no—
287C22×C4—1511no—
288C22×C4—1511no—
289C22×C4—1511no—
290C22×C4C22×D15111yescharacteristic
291SD16—41no—
292SD16—6041no—
293SD16—6041no—
294SD16—6041no—
295C2×C8—41no—
296C2×C8—6041no—
297C2×C8—1511no—
298C2×C8—1511no—
299C2×C8C22×D15111yescharacteristic
300C2×C8C22×D15111yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—3021no—
303C4⋊C4—3021no—
304C4⋊C4—3021no—
305C4⋊C4—3021no—
306C4⋊C4—3021no—
307C4⋊C4—1511no—
308C4⋊C4—1511no—
309C4⋊C4C22×D15111yescharacteristic
310C4⋊C4C22×D15111yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—3021no—
314C22⋊C4—3021no—
315C22⋊C4—3021no—
316C22⋊C4—3021no—
317C22⋊C4—3021no—
318C22⋊C4—3021no—
319C23—21no—
320C23—3021no—
321C23—6041no—
322C23—6041no—
323C23—6042no—
324C23—6042no—
325C23—6042no—
326C23—1511no—
327C23—1511no—
328C23—1514no—
329C23D4×D15111yescharacteristic
330Q8—21no—
331Q8—21no—
332Q8—3021no—
333Q8—3021no—
334Q8—6042no—
335D4—21no—
336D4—21no—
337D4—42no—
338D4—3021no—
339D4—3021no—
340D4—6042no—
341D4—6042no—
342D4—6042no—
343D4—6042no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—3021no—
350C2×C4—3021no—
351C2×C4—3021no—
352C2×C4—3021no—
353C2×C4—3021no—
354C2×C4—3021no—
355C2×C4—3021no—
356C2×C4—6041no—
357C2×C4—6041no—
358C2×C4—6042no—
359C2×C4—6042no—
360C2×C4—6042no—
361C2×C4—6041no—
362C2×C4—1511no—
363C2×C4—1511no—
364C2×C4D4×D15111yescharacteristic
365C2×C4C23×D15111yescharacteristic
366C8—21no—
367C8—21no—
368C8—3021no—
369C8—3021no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—3021no—
376C22—3021no—
377C22—3021no—
378C22—3021no—
379C22—6044no—
380C22—6041no—
381C22—6041no—
382C22—6041no—
383C22—1512no—
384C22—1512no—
385C22—12084no—
386C22SD16×D15112yes—
387C22C2×D4×D15111yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—3021no—
393C4—3021no—
394C4—6041no—
395C4—6041no—
396C4—6041no—
397C4#1801278411yescharacteristic
398C4#1801278511yescharacteristic
399C2—22no—
400C2—81no—
401C2—3022no—
402C2—3022no—
403C2—12081no—
404C2#2375277011yescharacteristic
405C2C2×SD16×D15111yescharacteristic
406C2#2375300911yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#465×C151C2
socleC22×C151C22×D4
prime radical#121×D151C2
hypercenter#465D151
hypocenterC151#3187
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D151C2semidirect
3. #23752544C2semidirect
4. #23752541C2semidirect
5. #23752532C2semidirect
6. #23752529C2semidirect
7. #23752443C2semidirect
8. #23752441C2semidirect
9. #23752439C2semidirect
10. #23752364C2semidirect
11. #23752361C2semidirect
12. #23752352C2semidirect
13. #23752349C2semidirect
14. C2×(C2×D151)⋊C8C2semidirect
15. #465×C151C2semidirect
16. C2×Q8×D151C22semidirect
20. C2×#14558993C22semidirect
21. C2×#14558992C22semidirect
22. C2×#14558991C22semidirect
23. C22×C151⋊C8C22semidirect
25. #18012718C22semidirect
36. C4⋊C4×D151C22semidirect
39. #18012671C22semidirect
40. #18012670C22semidirect
41. #18012670C22semidirect
42. #18012664C22semidirect
43. #18012663C22semidirect
44. #18012662C22semidirect
45. #18012661C22semidirect
46. #18012660C22semidirect
47. #18012659C22semidirect
50. (C2×D151)⋊C8C22semidirect
51. (C2×D151)⋊C8C22semidirect
52. C2×SD16×C151C22semidirect
53. C22×C8×C151C22semidirect
54. C22⋊Q8×C151C22semidirect
56. #106×C151C22semidirect
57. #103×C151C22semidirect
58. #102×C151C22semidirect
121. C22×D151SD16semidirect
203. C151#3187semidirect
214. #465D151semidirect
227. C2×SD16C2×D151semidirect
229. C22×C8C2×D151semidirect
235. C22⋊Q8C2×D151semidirect
250. #106C2×D151semidirect
254. #103C2×D151semidirect
258. #102C2×D151semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1512
#121×D1512
#237525442
#237525412
#237525322
#237525292
#237524432
#237524412
#237524392
#237523642
#237523612
#237523522
#237523492
C2×(C2×D151)⋊C82
#465×C1512
#3187151