Finite Group Databasebeta

← #34080625

Subgroups — #34080625

Order 19328.

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×#18012772C211yesmaximal, characteristic, complemented
2C22×#14558993C211yesmaximal, characteristic, complemented
3D4×Dic302C211yesmaximal, characteristic, complemented
4#23752858C211yesmaximal, characteristic
5#23752606C211yesmaximal, characteristic, complemented
6#23752458C211yesmaximal, characteristic, complemented
7#23752453C211yesmaximal, characteristic, complemented
8#23752435C211yesmaximal, characteristic, complemented
9#23752418C211yesmaximal, characteristic, complemented
10#23752376C211yesmaximal, characteristic, complemented
11#23752370C211yesmaximal, characteristic, complemented
12#23752365C211yesmaximal, characteristic, complemented
13#23752353C211yesmaximal, characteristic, complemented
14#23752314C211yesmaximal, characteristic, complemented
15#435×C151C211yesmaximal, characteristic, complemented
16C2×#14559004C2211yescharacteristic
17C22×Dic302C2211yescharacteristic
18C22×Dic302C2211yescharacteristic, complemented
19C22×D604C2211yescharacteristic
20#18012772—24no—
21C2×Dic604C2211yescharacteristic, complemented
22C2×Dic604C2211yescharacteristic, complemented
23C2×Dic604C2211yescharacteristic, complemented
24C2×#14558993—24no—
25C2×#14558993C2211yescharacteristic
26C2×#14558993C2211yescharacteristic
27C2×#14558993C2211yescharacteristic
28C2×#14558993C2211yescharacteristic
29C2×#14558993C2211yescharacteristic
30C2×#14558993C2211yescharacteristic
31#18012739C2211yescharacteristic, complemented
32#18012738—21no—
33#18012727C2211yescharacteristic
34#18012724—21no—
35#18012718C2211yescharacteristic
36#18012718C2211yescharacteristic
37#18012718C2211yescharacteristic, complemented
38#18012716—21no—
39#18012716—21no—
40#18012713C2211yescharacteristic
41#18012712—21no—
42#18012707—21no—
43C4×Dic302C2211yescharacteristic, complemented
44C4×D604C2211yescharacteristic
45D4×Dic151—21no—
46C22⋊C4×D151—21no—
47#18012673C2211yescharacteristic, complemented
48#18012664C2211yescharacteristic, complemented
49#18012664C2211yescharacteristic, complemented
50#18012664C2211yescharacteristic, complemented
51#18012664C2211yescharacteristic, complemented
52#18012661C2211yescharacteristic
53#18012661C2211yescharacteristic
54C2×M16×C151C2211yescharacteristic, complemented
55C22×C8×C151C2211yescharacteristic
56C4×D4×C151C2211yescharacteristic
57C4⋊C8×C151C2211yescharacteristic, complemented
58C22⋊C8×C151C2211yescharacteristic
59C22⋊C8×C151C2211yescharacteristic, complemented
60#97×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62#14559004—22no—
63#14559004—41no—
64C2×Dic302—22no—
65C2×Dic302—22no—
66C2×Dic302—21no—
67C2×Dic302—21no—
68C2×Dic302—41no—
69C2×Dic302—41no—
70C2×Dic302C2311yescharacteristic
71C2×Dic302C2311yescharacteristic
72C2×Dic302C2311yescharacteristic
73C2×Dic302C2311yescharacteristic
74C2×D604—22no—
75C2×D604—21no—
76C2×D604—41no—
77C2×D604C2311yescharacteristic
78C2×D604C2311yescharacteristic
79C2×C22⋊D151—21no—
80C2×C22⋊D151—21no—
81C22×Dic151—21no—
82C22×Dic151—21no—
83C2×C4×D151—21no—
84C2×C4×D151—21no—
85Dic604—22no—
86Dic604—22no—
87Dic604—41no—
88#14558993—22no—
89#14558993—22no—
90#14558993—22no—
91#14558993—22no—
92#14558993—41no—
93#14558993D414yescomplemented
94C4⋊Dic151—21no—
95C4⋊Dic151—21no—
96C4⋊Dic151C2311yescharacteristic
97C4⋊Dic151C2311yescharacteristic
98Dic151⋊C4—21no—
99Dic151⋊C4—21no—
100Dic151⋊C4—21no—
101Dic151⋊C4—21no—
102C22⋊Dic151—21no—
103C22⋊Dic151—21no—
104C22⋊Dic151—21no—
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4—21no—
107(C2×D151)⋊C4—21no—
108(C2×D151)⋊C4—21no—
109C4×Dic151—21no—
110C2×D4×C151D411yescharacteristic
111C22×C4×C151C2311yescharacteristic
112C22×C4×C151C2311yescharacteristic
113M16×C151—22no—
114C2×C8×C151—22no—
115C2×C8×C151C2311yescharacteristic
116C2×C8×C151C2311yescharacteristic
117C2×C8×C151C2311yescharacteristic
118C2×C8×C151C2311yescharacteristic
119C4⋊C4×C151D411yescharacteristic
120C22⋊C4×C151D411yescharacteristic
121C22⋊C4×C151D411yescharacteristic
122C42×C151C2311yescharacteristic
123C22×D151—21no—
124C22×D151—21no—
125C22×D151—42no—
126C22×D151—42no—
127C22×D151—81no—
128Dic302—21no—
129Dic302—21no—
130Dic302—21no—
131Dic302—21no—
132Dic302—21no—
133Dic302—21no—
134Dic302—21no—
135Dic302—42no—
136Dic302—42no—
137Dic302C2×D412yes—
138D604—21no—
139D604—21no—
140D604—21no—
141D604—42no—
142D604C2×D412yes—
143C22⋊D151—42no—
144C22⋊D151—81no—
145C2×Dic151—21no—
146C2×Dic151—21no—
147C2×Dic151—21no—
148C2×Dic151—21no—
149C2×Dic151—21no—
150C2×Dic151—21no—
151C2×Dic151—42no—
152C2×Dic151—42no—
153C4×D151—42no—
154C4×D151—81no—
155C23×C151C2×D411yescharacteristic
156C23×C151C2×D411yescharacteristic
157D4×C151—22no—
158C2×C4×C151—21no—
159C2×C4×C151—21no—
160C2×C4×C151—21no—
161C2×C4×C151—21no—
162C2×C4×C151C2×D411yescharacteristic
163C2×C4×C151C2×D411yescharacteristic
164C2×C4×C151C2×D411yescharacteristic
165C2×C4×C151C2×D411yescharacteristic
166C2×C4×C151C2411yescharacteristic
167C8×C151—21no—
168C8×C151—21no—
169C8×C151—21no—
170C8×C151C2×D412yes—
171C22×C151—21no—
172C22×C151—21no—
173C22×C151—21no—
174C22×C151—21no—
175C22×C151—21no—
176C22×C151#13712yescomplemented
177C22×C151C22×D411yescharacteristic
178C2×D151—22no—
179C2×D151—41no—
180C2×D151—41no—
181C2×D151—41no—
182C2×D151—41no—
183C2×D151—41no—
184C2×D151—84no—
185C4×C151—21no—
186C4×C151—21no—
187C4×C151—21no—
188C4×C151—21no—
189C4×C151C22×D411yescharacteristic
190C4×C151H(5,𝔽2)11yescharacteristic
191Dic151—22no—
192Dic151—41no—
193Dic151—41no—
194Dic151—41no—
195Dic151—41no—
196Dic151—41no—
197C2×C151—22no—
198C2×C151—41no—
199C2×C151D4211yescharacteristic
200C2×C151C2×#13711yescharacteristic
201C2×C151#57711yescharacteristic
202D151—42no—
203D151—81no—
204C151#326611yescharacteristic, complemented, Sylow
205#3266—1511nomaximal, Sylow
206C2×#137—1511no—
207C22×SD16—1511no—
208D4×Q8—1511no—
209#546—1511no—
210#497—1511no—
211#468—1511no—
212#467—1511no—
213#464—1511no—
214#461—1511no—
215#452—1511no—
216#451—1511no—
217#450—1511no—
218#448—1511no—
219#440—1511no—
220#435D15111yescharacteristic, complemented
221C2×#41—1511no—
222C22×Q8—1511no—
223C22×Q8—1511no—
224C22×D4—1511no—
225#137—3024no—
226C2×Q16—1511no—
227C2×Q16—1511no—
228C2×Q16—1511no—
229C2×SD16—3024no—
230C2×SD16—1511no—
231C2×SD16—1511no—
232C2×SD16—1511no—
233C2×SD16—1511no—
234C2×SD16—1511no—
235C2×SD16—1511no—
236C2×M16C2×D15111yescharacteristic, complemented
237C22×C8C2×D15111yescharacteristic
238C4⋊Q8—3021no—
239C4⋊Q8—1511no—
240#124—1511no—
241#123—3021no—
242C22⋊Q8—3021no—
243C22⋊Q8—3021no—
244C22⋊Q8—1511no—
245C22⋊Q8—1511no—
246C22⋊Q8—1511no—
247#121—3021no—
248#121—1511no—
249#120—3021no—
250C4×Q8—1511no—
251C4×D4—3021no—
252C4×D4—1511no—
253C4×D4C2×D15111yescharacteristic
254C2×C22⋊C4—3021no—
255#107—1511no—
256C4⋊C8C2×D15111yescharacteristic, complemented
257#103—1511no—
258#103—1511no—
259#103—1511no—
260#103—1511no—
261#102—1511no—
262#102—1511no—
263C22⋊C8C2×D15111yescharacteristic
264C22⋊C8C2×D15111yescharacteristic, complemented
265#97C2×D15111yescharacteristic, complemented
266C24—3021no—
267#41—3022no—
268#41—6041no—
269C2×Q8—3022no—
270C2×Q8—3022no—
271C2×Q8—3021no—
272C2×Q8—3021no—
273C2×Q8—6041no—
274C2×Q8—6041no—
275C2×Q8—1511no—
276C2×Q8—1511no—
277C2×Q8—1511no—
278C2×Q8—1511no—
279C2×D4—3022no—
280C2×D4—3021no—
281C2×D4—3021no—
282C2×D4—3021no—
283C2×D4—6041no—
284C2×D4—1511no—
285C2×D4—1511no—
286C2×D4D60411yescharacteristic
287C22×C4—3021no—
288C22×C4—3021no—
289C22×C4—3021no—
290C22×C4—3021no—
291C22×C4C22×D15111yescharacteristic
292C22×C4C22×D15111yescharacteristic
293Q16—3022no—
294Q16—3022no—
295Q16—6041no—
296SD16—3022no—
297SD16—3022no—
298SD16—3022no—
299SD16—3022no—
300SD16—6041no—
301SD16—1514no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D15111yescharacteristic
305C2×C8C22×D15111yescharacteristic
306C2×C8C22×D15111yescharacteristic
307C2×C8C22×D15111yescharacteristic
308C4⋊C4—3021no—
309C4⋊C4—3021no—
310C4⋊C4—3021no—
311C4⋊C4—3021no—
312C4⋊C4—3021no—
313C4⋊C4—3021no—
314C4⋊C4—1511no—
315C4⋊C4—1511no—
316C4⋊C4D60411yescharacteristic
317C22⋊C4—3021no—
318C22⋊C4—3021no—
319C22⋊C4—3021no—
320C22⋊C4—3021no—
321C22⋊C4—3021no—
322C22⋊C4—3021no—
323C22⋊C4—3021no—
324C22⋊C4D60411yescharacteristic
325C22⋊C4D60411yescharacteristic
326C42—3021no—
327C42C22×D15111yescharacteristic
328C23—3021no—
329C23—3021no—
330C23—6042no—
331C23—6042no—
332C23—12081no—
333C23C2×D60411yescharacteristic
334C23C2×D60411yescharacteristic
335Q8—3021no—
336Q8—3021no—
337Q8—3021no—
338Q8—3021no—
339Q8—3021no—
340Q8—3021no—
341Q8—3021no—
342Q8—6042no—
343Q8—6042no—
344Q8—1512no—
345D4—22no—
346D4—3021no—
347D4—3021no—
348D4—3021no—
349D4—6042no—
350D4—6042no—
351D4—1512no—
352D4—12081no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—3021no—
358C2×C4—3021no—
359C2×C4—3021no—
360C2×C4—3021no—
361C2×C4—3021no—
362C2×C4—3021no—
363C2×C4—6042no—
364C2×C4—6042no—
365C2×C4—6042no—
366C2×C4—12081no—
367C2×C4C2×D60411yescharacteristic
368C2×C4C2×D60411yescharacteristic
369C2×C4C2×D60411yescharacteristic
370C2×C4C2×D60411yescharacteristic
371C2×C4C23×D15111yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D15112yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—3022no—
382C22—6041no—
383C22—6041no—
384C22—6041no—
385C22—6041no—
386C22—6041no—
387C22—12084no—
388C22#1801277212yescomplemented
389C22C22×D60411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—3022no—
395C4—6041no—
396C4—6041no—
397C4—6041no—
398C4—6041no—
399C4—6041no—
400C4C2×D4×D15111yescharacteristic
401C4#1801278511yescharacteristic
402C2—22no—
403C2—41no—
404C2—6042no—
405C2—12081no—
406C2D4×D60411yescharacteristic
407C2C2×#1801277211yescharacteristic
408C2#2375302411yescharacteristic
subgroupG/H
centerC22C22×D604
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#435×C151C2
socleC22×C151C22×D4
prime radical#23752858C2
hypercenter#435D151
hypocenterC151#3266
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#18012772C2semidirect
2. C22×#14558993C2semidirect
3. D4×Dic302C2semidirect
5. #23752606C2semidirect
6. #23752458C2semidirect
7. #23752453C2semidirect
8. #23752435C2semidirect
9. #23752418C2semidirect
10. #23752376C2semidirect
11. #23752370C2semidirect
12. #23752365C2semidirect
13. #23752353C2semidirect
14. #23752314C2semidirect
15. #435×C151C2semidirect
18. C22×Dic302C22semidirect
21. C2×Dic604C22semidirect
22. C2×Dic604C22semidirect
23. C2×Dic604C22semidirect
31. #18012739C22semidirect
37. #18012718C22semidirect
43. C4×Dic302C22semidirect
47. #18012673C22semidirect
48. #18012664C22semidirect
49. #18012664C22semidirect
50. #18012664C22semidirect
51. #18012664C22semidirect
54. C2×M16×C151C22semidirect
57. C4⋊C8×C151C22semidirect
59. C22⋊C8×C151C22semidirect
60. #97×C151C22semidirect
93. #14558993D4semidirect
176. C22×C151#137semidirect
204. C151#3266semidirect
220. #435D151semidirect
236. C2×M16C2×D151semidirect
256. C4⋊C8C2×D151semidirect
264. C22⋊C8C2×D151semidirect
265. #97C2×D151semidirect
388. C22#18012772semidirect

Maximal subgroups

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

subgroupindex
C2×#180127722
C22×#145589932
D4×Dic3022
#237528582
#237526062
#237524582
#237524532
#237524352
#237524182
#237523762
#237523702
#237523652
#237523532
#237523142
#435×C1512
#3266151