Finite Group Databasebeta

← #34080614

Subgroups — #34080614

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×SD16×D151C211yesmaximal, characteristic, complemented
2Q8×C22⋊D151C211yesmaximal, characteristic, complemented
3#23752850C211yesmaximal, characteristic
4#23752605C211yesmaximal, characteristic, complemented
5#23752458C211yesmaximal, characteristic, complemented
6#23752450C211yesmaximal, characteristic, complemented
7#23752436C211yesmaximal, characteristic, complemented
8#23752415C211yesmaximal, characteristic, complemented
9#23752374C211yesmaximal, characteristic, complemented
10#23752366C211yesmaximal, characteristic, complemented
11#23752361C211yesmaximal, characteristic, complemented
12#23752351C211yesmaximal, characteristic, complemented
13#23752313C211yesmaximal, characteristic, complemented
14#23752281C211yesmaximal, characteristic, complemented
15C2×#137×C151C211yesmaximal, characteristic, complemented
16C2×Q8×D151C2211yescharacteristic
17C2×D4×D151C2211yescharacteristic
18C2×#14558994C2211yescharacteristic, complemented
19C2×#14558993C2211yescharacteristic
20C2×#14558992C2211yescharacteristic
21C2×#14558991C2211yescharacteristic
22C2×#14558991C2211yescharacteristic
23SD16×D151—24no—
24C2×C8×D151C2211yescharacteristic
25#18012737—21no—
26#18012737C2211yescharacteristic, complemented
27#18012728C2211yescharacteristic
28#18012722—21no—
29#18012718—21no—
30#18012718C2211yescharacteristic, complemented
31#18012715—21no—
32#18012715C2211yescharacteristic
33#18012715C2211yescharacteristic
34#18012713C2211yescharacteristic
35#18012711—21no—
36#18012707—21no—
37Q8×Dic151C2211yescharacteristic, complemented
38D4×Dic151C2211yescharacteristic
39C4×C22⋊D151—21no—
40C4×C22⋊D151C2211yescharacteristic
41C2×(C2×D151)⋊C4—21no—
42#18012673C2211yescharacteristic, complemented
43Dic151⋊C8C2211yescharacteristic, complemented
44#18012664C2211yescharacteristic, complemented
45#18012663C2211yescharacteristic, complemented
46#18012663C2211yescharacteristic, complemented
47#18012663C2211yescharacteristic, complemented
48#18012661C2211yescharacteristic
49#18012660C2211yescharacteristic
50C22⋊(C151⋊C8)C2211yescharacteristic, complemented
51(C2×D151)⋊C8C2211yescharacteristic
52#18012649C2211yescharacteristic, complemented
53C2×#41×C151C2211yescharacteristic
54C22×Q8×C151C2211yescharacteristic, complemented
55#137×C151—24no—
56C2×Q16×C151C2211yescharacteristic, complemented
57C2×Q16×C151C2211yescharacteristic, complemented
58C2×SD16×C151C2211yescharacteristic
59C2×SD16×C151C2211yescharacteristic
60C2×M16×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62C2×Dic302—21no—
63C2×Dic302C2311yescharacteristic
64Q8×D151—22no—
65Q8×D151—41no—
66C2×D604C2311yescharacteristic
67C2×C22⋊D151—21no—
68C2×C22⋊D151—21no—
69C2×C22⋊D151D411yescharacteristic
70D4×D151—22no—
71D4×D151—41no—
72C22×Dic151—21no—
73C2×C4×D151—21no—
74C2×C4×D151C2311yescharacteristic
75#14558994—41no—
76#14558993—22no—
77#14558992—22no—
78#14558991—22no—
79#14558991—41no—
80C2×C151⋊C8C2311yescharacteristic
81C2×C151⋊C8C2311yescharacteristic
82C8×D151—22no—
83C4⋊Dic151—21no—
84C4⋊Dic151C2311yescharacteristic
85C4⋊Dic151C2311yescharacteristic
86Dic151⋊C4—21no—
87Dic151⋊C4—21no—
88Dic151⋊C4—21no—
89Dic151⋊C4—21no—
90Dic151⋊C4—21no—
91Dic151⋊C4D411yescharacteristic
92C22⋊Dic151—21no—
93C22⋊Dic151—21no—
94C22⋊Dic151D411yescharacteristic
95(C2×D151)⋊C4—21no—
96(C2×D151)⋊C4—21no—
97(C2×D151)⋊C4—21no—
98(C2×D151)⋊C4—21no—
99(C2×D151)⋊C4—21no—
100(C2×D151)⋊C4D411yescharacteristic
101C4×Dic151—21no—
102C4×Dic151C2311yescharacteristic
103#41×C151—22no—
104#41×C151—41no—
105C2×Q8×C151—21no—
106C2×Q8×C151—22no—
107C2×Q8×C151—41no—
108C2×Q8×C151C2311yescharacteristic
109C2×Q8×C151C2311yescharacteristic
110C2×Q8×C151C2311yescharacteristic
111C2×D4×C151—21no—
112C2×D4×C151C2311yescharacteristic
113C22×C4×C151—21no—
114C22×C4×C151—21no—
115C22×C4×C151C2311yescharacteristic
116Q16×C151—22no—
117Q16×C151—22no—
118SD16×C151—22no—
119SD16×C151D414yescomplemented
120M16×C151—22no—
121C2×C8×C151C2311yescharacteristic
122C2×C8×C151C2311yescharacteristic
123C22×D151—21no—
124C22×D151—42no—
125C22×D151—42no—
126C22×D151—81no—
127C22×D151C2×D411yescharacteristic
128Dic302—21no—
129Dic302—21no—
130Dic302—42no—
131D604—21no—
132D604—21no—
133C22⋊D151—22no—
134C22⋊D151—42no—
135C22⋊D151—81no—
136C2×Dic151—21no—
137C2×Dic151—21no—
138C2×Dic151—21no—
139C2×Dic151—81no—
140C2×Dic151C2×D411yescharacteristic
141C2×Dic151C2×D411yescharacteristic
142C2×Dic151C2×D411yescharacteristic
143C4×D151—21no—
144C4×D151—21no—
145C4×D151—42no—
146C151⋊C8—21no—
147C151⋊C8—21no—
148C23×C151—21no—
149C23×C151C2×D411yescharacteristic
150Q8×C151—21no—
151Q8×C151—21no—
152Q8×C151—21no—
153Q8×C151—21no—
154Q8×C151—21no—
155Q8×C151—42no—
156Q8×C151C2×D412yes—
157D4×C151—21no—
158D4×C151—42no—
159D4×C151C2×D412yes—
160C2×C4×C151—21no—
161C2×C4×C151—21no—
162C2×C4×C151—21no—
163C2×C4×C151—21no—
164C2×C4×C151—21no—
165C2×C4×C151—42no—
166C2×C4×C151—42no—
167C2×C4×C151C2×D411yescharacteristic
168C2×C4×C151C2411yescharacteristic
169C8×C151—21no—
170C8×C151C2×D412yes—
171C22×C151—21no—
172C22×C151—22no—
173C22×C151—21no—
174C22×C151—21no—
175C22×C151—41no—
176C22×C151—41no—
177C22×C151C22×D411yescharacteristic
178C2×D151—21no—
179C2×D151—21no—
180C2×D151—41no—
181C2×D151—41no—
182C2×D151—41no—
183C2×D151—84no—
184C2×D151#13712yescomplemented
185C4×C151—21no—
186C4×C151—22no—
187C4×C151—41no—
188C4×C151—41no—
189C4×C151C22×D411yescharacteristic
190C4×C151H(5,𝔽2)11yescharacteristic
191Dic151—21no—
192Dic151—21no—
193Dic151—21no—
194Dic151—41no—
195Dic151—41no—
196Dic151—41no—
197C2×C151—42no—
198C2×C151—41no—
199C2×C151D4211yescharacteristic
200C2×C151C2×#13711yescharacteristic
201C2×C151#57711yescharacteristic
202D151—22no—
203D151—81no—
204C151#326611yescharacteristic, complemented, Sylow
205#3266—1511nomaximal, Sylow
206C2×#137D15111yescharacteristic, complemented
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#435—1511no—
221C2×#41C2×D15111yescharacteristic
222C22×Q8—1511no—
223C22×Q8C2×D15111yescharacteristic, complemented
224C22×D4—1511no—
225#137—24no—
226C2×Q16—1511no—
227C2×Q16C2×D15111yescharacteristic, complemented
228C2×Q16C2×D15111yescharacteristic, complemented
229C2×SD16—3024no—
230C2×SD16—1511no—
231C2×SD16—1511no—
232C2×SD16—1511no—
233C2×SD16—1511no—
234C2×SD16C2×D15111yescharacteristic
235C2×SD16C2×D15111yescharacteristic
236C2×M16C2×D15111yescharacteristic, complemented
237C22×C8—1511no—
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×D4—1511no—
254C2×C22⋊C4—3021no—
255#107—1511no—
256C4⋊C8—1511no—
257#103—1511no—
258#103—1511no—
259#103—1511no—
260#103—1511no—
261#102—1511no—
262#102—1511no—
263C22⋊C8—1511no—
264C22⋊C8—1511no—
265#97—1511no—
266C24—3021no—
267#41—22no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—3021no—
273C2×Q8—3022no—
274C2×Q8—6041no—
275C2×Q8—1511no—
276C2×Q8C22×D15111yescharacteristic
277C2×Q8C22×D15111yescharacteristic
278C2×Q8C22×D15111yescharacteristic
279C2×D4—21no—
280C2×D4—3022no—
281C2×D4—3021no—
282C2×D4—3021no—
283C2×D4—6041no—
284C2×D4—1511no—
285C2×D4—1511no—
286C2×D4C22×D15111yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—3021no—
290C22×C4—3021no—
291C22×C4—1511no—
292C22×C4C22×D15111yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—6041no—
296SD16—22no—
297SD16—3022no—
298SD16—3022no—
299SD16—3022no—
300SD16—6041no—
301SD16C22⋊D15114yescomplemented
302M16—22no—
303C2×C8—3022no—
304C2×C8—1511no—
305C2×C8—1511no—
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⋊C4—1511no—
317C22⋊C4—3021no—
318C22⋊C4—3021no—
319C22⋊C4—3021no—
320C22⋊C4—3021no—
321C22⋊C4—3021no—
322C22⋊C4—3021no—
323C22⋊C4—3021no—
324C22⋊C4—1511no—
325C22⋊C4—1511no—
326C42—3021no—
327C42—1511no—
328C23—21no—
329C23—3021no—
330C23—6042no—
331C23—6042no—
332C23—1511no—
333C23—12081no—
334C23D4×D15111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—3021no—
342Q8—3021no—
343Q8—6042no—
344Q8C2×C22⋊D15112yes—
345D4—21no—
346D4—42no—
347D4—3022no—
348D4—3021no—
349D4—3021no—
350D4—6042no—
351D4—12081no—
352D4C2×C22⋊D15112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—3021no—
361C2×C4—3021no—
362C2×C4—3021no—
363C2×C4—3021no—
364C2×C4—3021no—
365C2×C4—6042no—
366C2×C4—1511no—
367C2×C4—1511no—
368C2×C4—1511no—
369C2×C4—12081no—
370C2×C4D4×D15111yescharacteristic
371C2×C4C23×D15111yescharacteristic
372C8—21no—
373C8—3021no—
374C8—3021no—
375C8C2×C22⋊D15112yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—3021no—
383C22—3021no—
384C22—6041no—
385C22—6041no—
386C22—6041no—
387C22—1512no—
388C22—12084no—
389C22C2×D4×D15111yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—3021no—
395C4—3021no—
396C4—3021no—
397C4—6041no—
398C4—6041no—
399C4—6041no—
400C4C22×C22⋊D15111yescharacteristic
401C4#1801278411yescharacteristic
402C2—41no—
403C2—42no—
404C2—3022no—
405C2—12081no—
406C2D4×C22⋊D15111yescharacteristic
407C2#137×D15111yescharacteristic
408C2#2375302111yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
FittingC2×#137×C151C2
socleC22×C151C22×D4
prime radical#23752850C2
hypercenterC2×#137D151
hypocenterC151#3266
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D151C2semidirect
2. Q8×C22⋊D151C2semidirect
4. #23752605C2semidirect
5. #23752458C2semidirect
6. #23752450C2semidirect
7. #23752436C2semidirect
8. #23752415C2semidirect
9. #23752374C2semidirect
10. #23752366C2semidirect
11. #23752361C2semidirect
12. #23752351C2semidirect
13. #23752313C2semidirect
14. #23752281C2semidirect
15. C2×#137×C151C2semidirect
18. C2×#14558994C22semidirect
26. #18012737C22semidirect
30. #18012718C22semidirect
37. Q8×Dic151C22semidirect
42. #18012673C22semidirect
43. Dic151⋊C8C22semidirect
44. #18012664C22semidirect
45. #18012663C22semidirect
46. #18012663C22semidirect
47. #18012663C22semidirect
50. C22⋊(C151⋊C8)C22semidirect
52. #18012649C22semidirect
54. C22×Q8×C151C22semidirect
56. C2×Q16×C151C22semidirect
57. C2×Q16×C151C22semidirect
60. C2×M16×C151C22semidirect
119. SD16×C151D4semidirect
184. C2×D151#137semidirect
204. C151#3266semidirect
206. C2×#137D151semidirect
223. C22×Q8C2×D151semidirect
227. C2×Q16C2×D151semidirect
228. C2×Q16C2×D151semidirect
236. C2×M16C2×D151semidirect
301. SD16C22⋊D151semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1512
Q8×C22⋊D1512
#237528502
#237526052
#237524582
#237524502
#237524362
#237524152
#237523742
#237523662
#237523612
#237523512
#237523132
#237522812
C2×#137×C1512
#3266151