Finite Group Databasebeta

← #34080634

Subgroups — #34080634

Order 19328.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18012766C211yesmaximal, characteristic, complemented
2C2×#18012762C211yesmaximal, characteristic, complemented
3#23752865C211yesmaximal, characteristic, complemented
4#23752855C211yesmaximal, characteristic, complemented
5#23752467C211yesmaximal, characteristic, complemented
6#23752438C211yesmaximal, characteristic, complemented
7#23752424C211yesmaximal, characteristic, complemented
8#23752414C211yesmaximal, characteristic, complemented
9#23752373C211yesmaximal, characteristic, complemented
10#23752373C211yesmaximal, characteristic, complemented
11#23752361C211yesmaximal, characteristic, complemented
12#23752356C211yesmaximal, characteristic, complemented
13#23752325C211yesmaximal, characteristic, complemented
14#23752285C211yesmaximal, characteristic, complemented
15#497×C151C211yesmaximal, characteristic, complemented
16C2×#14559005C2211yescharacteristic
17C2×#14559005C2211yescharacteristic
18C2×#14559005C2211yescharacteristic, complemented
19C2×D4×D151C2211yescharacteristic
20#18012766—24no—
21#18012762—24no—
22C2×#14558991C2211yescharacteristic
23C2×#14558991C2211yescharacteristic
24C2×#14558991C2211yescharacteristic, complemented
25C2×D1208C2211yescharacteristic
26C2×#14558989C2211yescharacteristic
27C2×#14558987C2211yescharacteristic
28C2×C8×D151C2211yescharacteristic, complemented
29#18012724—21no—
30#18012724—21no—
31#18012721—21no—
32#18012715C2211yescharacteristic
33#18012715C2211yescharacteristic
34#18012715C2211yescharacteristic, complemented
35#18012714—21no—
36#18012712—21no—
37#18012710C2211yescharacteristic
38#18012709—21no—
39C4×D604C2211yescharacteristic
40C4×D604C2211yescharacteristic
41C4×D604C2211yescharacteristic, complemented
42C4⋊C4×D151—21no—
43C22⋊C4×D151—21no—
44#18012671C2211yescharacteristic, complemented
45C4⋊(C151⋊C8)C2211yescharacteristic, complemented
46#18012663C2211yescharacteristic, complemented
47#18012663C2211yescharacteristic, complemented
48#18012663C2211yescharacteristic, complemented
49#18012661C2211yescharacteristic
50#18012661C2211yescharacteristic, complemented
51#18012660C2211yescharacteristic
52(C2×D151)⋊C8C2211yescharacteristic
53(C2×D151)⋊C8C2211yescharacteristic, complemented
54C2×Q16×C151C2211yescharacteristic, complemented
55C2×Q16×C151C2211yescharacteristic, complemented
56C2×SD16×C151C2211yescharacteristic
57C2×SD16×C151C2211yescharacteristic
58C4⋊Q8×C151C2211yescharacteristic, complemented
59#124×C151C2211yescharacteristic
60#97×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62#14559005—22no—
63#14559005—22no—
64#14559005—22no—
65#14559005—41no—
66#14559005—41no—
67#14559005—41no—
68C2×D604—21no—
69C2×D604—21no—
70C2×D604—21no—
71C2×D604D411yescharacteristic
72C2×D604C2311yescharacteristic
73C2×D604C2311yescharacteristic
74C2×C22⋊D151—21no—
75C2×C22⋊D151—21no—
76D4×D151—22no—
77D4×D151—41no—
78C2×C4×D151—21no—
79C2×C4×D151—21no—
80C2×C4×D151—21no—
81C2×C4×D151—21no—
82C2×C4×D151—21no—
83C2×C4×D151C2311yescharacteristic
84C2×C4×D151C2311yescharacteristic
85#14558991—22no—
86#14558991—22no—
87#14558991—22no—
88D1208D414yescomplemented
89#14558989—22no—
90#14558987—22no—
91C2×C151⋊C8C2311yescharacteristic
92C2×C151⋊C8C2311yescharacteristic
93C8×D151—22no—
94C4⋊Dic151D411yescharacteristic
95C4⋊Dic151C2311yescharacteristic
96C4⋊Dic151C2311yescharacteristic
97Dic151⋊C4—21no—
98Dic151⋊C4—21no—
99Dic151⋊C4—21no—
100C22⋊Dic151—21no—
101(C2×D151)⋊C4—21no—
102(C2×D151)⋊C4—21no—
103(C2×D151)⋊C4—21no—
104(C2×D151)⋊C4—21no—
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4D411yescharacteristic
107(C2×D151)⋊C4D411yescharacteristic
108C2×Q8×C151C2311yescharacteristic
109C2×Q8×C151C2311yescharacteristic
110C2×Q8×C151C2311yescharacteristic
111C2×D4×C151C2311yescharacteristic
112Q16×C151—22no—
113Q16×C151—41no—
114SD16×C151—22no—
115SD16×C151—41no—
116C2×C8×C151C2311yescharacteristic
117C2×C8×C151C2311yescharacteristic
118C4⋊C4×C151—21no—
119C4⋊C4×C151—21no—
120C22⋊C4×C151—21no—
121C22⋊C4×C151—21no—
122C42×C151C2311yescharacteristic
123C22×D151—21no—
124C22×D151—21no—
125C22×D151—42no—
126C22×D151—42no—
127C22×D151—81no—
128C22×D151C2×D411yescharacteristic
129C22×D151C2×D411yescharacteristic
130D604—21no—
131D604—21no—
132D604—41no—
133D604—42no—
134D604—42no—
135D604—42no—
136D604C2×D412yes—
137D604C2×D412yes—
138C22⋊D151—42no—
139C22⋊D151—81no—
140C2×Dic151—21no—
141C2×Dic151—21no—
142C2×Dic151C2×D411yescharacteristic
143C2×Dic151C2×D411yescharacteristic
144C4×D151—21no—
145C4×D151—21no—
146C4×D151—21no—
147C4×D151—21no—
148C4×D151—42no—
149C4×D151—42no—
150C4×D151—42no—
151C4×D151—42no—
152C4×D151—81no—
153C151⋊C8—21no—
154C151⋊C8—21no—
155C23×C151—21no—
156Q8×C151—21no—
157Q8×C151—21no—
158Q8×C151—21no—
159Q8×C151—21no—
160Q8×C151—21no—
161Q8×C151—21no—
162D4×C151—21no—
163D4×C151—21no—
164C2×C4×C151—21no—
165C2×C4×C151—21no—
166C2×C4×C151—21no—
167C2×C4×C151C2×D411yescharacteristic
168C2×C4×C151C2×D411yescharacteristic
169C2×C4×C151C2411yescharacteristic
170C8×C151—21no—
171C8×C151C2×D412yes—
172C22×C151—41no—
173C22×C151—41no—
174C22×C151—41no—
175C22×C151C22×D411yescharacteristic
176C2×D151—21no—
177C2×D151—21no—
178C2×D151—22no—
179C2×D151—22no—
180C2×D151—21no—
181C2×D151—21no—
182C2×D151—21no—
183C2×D151—21no—
184C2×D151—41no—
185C2×D151—41no—
186C2×D151—41no—
187C2×D151—41no—
188C2×D151—84no—
189C4×C151—21no—
190C4×C151—21no—
191C4×C151—41no—
192C4×C151—41no—
193C4×C151—41no—
194C4×C151C22×D411yescharacteristic
195C4×C151H(5,𝔽2)11yescharacteristic
196Dic151—21no—
197Dic151—21no—
198Dic151—41no—
199Dic151—41no—
200C2×C151—81no—
201C2×C151D4211yescharacteristic
202C2×C151#57511yescharacteristic
203C2×C151#57711yescharacteristic
204D151—41no—
205D151—41no—
206D151—42no—
207D151—42no—
208C151#326711yescharacteristic, complemented, Sylow
209#3267—1511nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1511no—
211C2×#135—1511no—
212#547—1511no—
213#546—1511no—
214#497D15111yescharacteristic, complemented
215#470—1511no—
216#465—1511no—
217#463—1511no—
218#461—1511no—
219#452—1511no—
220#452—1511no—
221#450—1511no—
222#449—1511no—
223#442—1511no—
224#435—1511no—
225C2×#41—1511no—
226C2×#41—1511no—
227C2×#41—1511no—
228C22×D4—1511no—
229AGL(1,ℤ/8)—3024no—
230#135—3024no—
231C2×Q16C2×D15111yescharacteristic, complemented
232C2×Q16C2×D15111yescharacteristic, complemented
233C2×SD16—1511no—
234C2×SD16—1511no—
235C2×SD16—1511no—
236C2×SD16C2×D15111yescharacteristic
237C2×SD16C2×D15111yescharacteristic
238C2×D8—1511no—
239C2×D8—1511no—
240C2×M16—1511no—
241C22×C8—1511no—
242C4⋊Q8C2×D15111yescharacteristic, complemented
243#124C2×D15111yescharacteristic
244#123—3021no—
245#123—3021no—
246#123—3021no—
247C22⋊Q8—1511no—
248C22⋊Q8—1511no—
249C22⋊Q8—1511no—
250#121—3021no—
251#121—3021no—
252#121—1511no—
253#120—3021no—
254C4×D4—1511no—
255C4×D4—1511no—
256C4×D4—1511no—
257C2×C4⋊C4—3021no—
258C2×C22⋊C4—3021no—
259#106—1511no—
260C4⋊C8—1511no—
261#103—1511no—
262#103—1511no—
263#103—1511no—
264#102—1511no—
265#102—1511no—
266#102—1511no—
267C22⋊C8—1511no—
268C22⋊C8—1511no—
269#97C2×D15111yescharacteristic, complemented
270C24—3021no—
271#41—3022no—
272#41—3022no—
273#41—3022no—
274#41—6041no—
275#41—6041no—
276#41—6041no—
277C2×Q8C22×D15111yescharacteristic
278C2×Q8C22×D15111yescharacteristic
279C2×Q8C22×D15111yescharacteristic
280C2×D4—3021no—
281C2×D4—3021no—
282C2×D4—3021no—
283C2×D4—3022no—
284C2×D4—3021no—
285C2×D4—3021no—
286C2×D4—6041no—
287C2×D4—1511no—
288C2×D4—1511no—
289C2×D4—1511no—
290C2×D4C22×D15111yescharacteristic
291C22×C4—3021no—
292C22×C4—3021no—
293C22×C4—3021no—
294C22×C4—3021no—
295C22×C4—3021no—
296C22×C4—1511no—
297C22×C4—1511no—
298Q16—22no—
299Q16—41no—
300SD16—22no—
301SD16—41no—
302SD16—3022no—
303SD16—3022no—
304SD16—3022no—
305D8—3022no—
306D8—1514no—
307M16—3022no—
308C2×C8—3022no—
309C2×C8—1511no—
310C2×C8—1511no—
311C2×C8C22×D15111yescharacteristic
312C2×C8C22×D15111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—3021no—
316C4⋊C4—3021no—
317C4⋊C4—3021no—
318C4⋊C4—1511no—
319C4⋊C4—1511no—
320C4⋊C4—1511no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—3021no—
324C22⋊C4—3021no—
325C22⋊C4—3021no—
326C22⋊C4—3021no—
327C22⋊C4—3021no—
328C22⋊C4—3021no—
329C22⋊C4—1511no—
330C22⋊C4—1511no—
331C42C22×D15111yescharacteristic
332C23—21no—
333C23—3021no—
334C23—3021no—
335C23—6042no—
336C23—6042no—
337C23—1511no—
338C23—1511no—
339C23—12081no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—3021no—
349D4—3021no—
350D4—6041no—
351D4—6042no—
352D4—6042no—
353D4—6042no—
354D4—6042no—
355D4—1512no—
356D4—1512no—
357D4—12081no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—3021no—
362C2×C4—3021no—
363C2×C4—3021no—
364C2×C4—3021no—
365C2×C4—3021no—
366C2×C4—3021no—
367C2×C4—6042no—
368C2×C4—6042no—
369C2×C4—6042no—
370C2×C4—6042no—
371C2×C4—1511no—
372C2×C4—1511no—
373C2×C4—12081no—
374C2×C4D4×D15111yescharacteristic
375C2×C4D4×D15111yescharacteristic
376C2×C4C23×D15111yescharacteristic
377C8—21no—
378C8—3021no—
379C8—3021no—
380C8D4×D15112yes—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—3021no—
385C22—3021no—
386C22—3021no—
387C22—3021no—
388C22—3021no—
389C22—3021no—
390C22—3022no—
391C22—3022no—
392C22—6041no—
393C22—6041no—
394C22—6041no—
395C22—6041no—
396C22—12084no—
397C22C2×D4×D15111yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—3021no—
404C4—3021no—
405C4—6041no—
406C4—6041no—
407C4C2×D4×D15111yescharacteristic
408C4#1801278411yescharacteristic
409C2—81no—
410C2—6041no—
411C2—6041no—
412C2—6042no—
413C2—6042no—
414C2#2375284711yescharacteristic
415C2#2375301111yescharacteristic
416C2#2375302111yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#497×C151C2
socleC22×C151C22×D4
hypercenter#497D151
hypocenterC151#3267
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#18012766C2semidirect
2. C2×#18012762C2semidirect
3. #23752865C2semidirect
4. #23752855C2semidirect
5. #23752467C2semidirect
6. #23752438C2semidirect
7. #23752424C2semidirect
8. #23752414C2semidirect
9. #23752373C2semidirect
10. #23752373C2semidirect
11. #23752361C2semidirect
12. #23752356C2semidirect
13. #23752325C2semidirect
14. #23752285C2semidirect
15. #497×C151C2semidirect
18. C2×#14559005C22semidirect
24. C2×#14558991C22semidirect
28. C2×C8×D151C22semidirect
34. #18012715C22semidirect
41. C4×D604C22semidirect
44. #18012671C22semidirect
45. C4⋊(C151⋊C8)C22semidirect
46. #18012663C22semidirect
47. #18012663C22semidirect
48. #18012663C22semidirect
50. #18012661C22semidirect
53. (C2×D151)⋊C8C22semidirect
54. C2×Q16×C151C22semidirect
55. C2×Q16×C151C22semidirect
58. C4⋊Q8×C151C22semidirect
60. #97×C151C22semidirect
88. D1208D4semidirect
208. C151#3267semidirect
214. #497D151semidirect
231. C2×Q16C2×D151semidirect
232. C2×Q16C2×D151semidirect
242. C4⋊Q8C2×D151semidirect
269. #97C2×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127662
C2×#180127622
#237528652
#237528552
#237524672
#237524382
#237524242
#237524142
#237523732
#237523612
#237523562
#237523252
#237522852
#497×C1512
#3267151