Finite Group Databasebeta

← #34080629

Subgroups — #34080629

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×#18012758C211yesmaximal, characteristic, complemented
2#23752861C211yesmaximal, characteristic, complemented
3#23752856C211yesmaximal, characteristic, complemented
4#23752603C211yesmaximal, characteristic, complemented
5#23752470C211yesmaximal, characteristic, complemented
6#23752438C211yesmaximal, characteristic, complemented
7#23752425C211yesmaximal, characteristic, complemented
8#23752415C211yesmaximal, characteristic, complemented
9#23752374C211yesmaximal, characteristic, complemented
10#23752371C211yesmaximal, characteristic, complemented
11#23752363C211yesmaximal, characteristic, complemented
12#23752354C211yesmaximal, characteristic, complemented
13#23752324C211yesmaximal, characteristic, complemented
14#23752281C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C151C211yesmaximal, characteristic, complemented
16C2×#14559003C2211yescharacteristic
17C2×#14559003C2211yescharacteristic, complemented
18#18012758—24no—
19C2×Dic604C2211yescharacteristic, complemented
20C2×#14558994C2211yescharacteristic, complemented
21C2×#14558992C2211yescharacteristic
22C2×#14558992C2211yescharacteristic
23C2×#14558992C2211yescharacteristic, complemented
24C2×C8×D151C2211yescharacteristic, complemented
25#18012737C2211yescharacteristic, complemented
26#18012726C2211yescharacteristic
27#18012722—21no—
28#18012720—21no—
29#18012720—21no—
30#18012718C2211yescharacteristic
31#18012718C2211yescharacteristic
32#18012715C2211yescharacteristic, complemented
33#18012711—21no—
34#18012711—21no—
35#18012710C2211yescharacteristic
36#18012708—21no—
37D4×Dic151C2211yescharacteristic
38D4×Dic151C2211yescharacteristic, complemented
39C4×C22⋊D151C2211yescharacteristic
40C2×Dic151⋊C4—21no—
41C2×C22⋊Dic151—21no—
42#18012671C2211yescharacteristic, complemented
43Dic151⋊C8C2211yescharacteristic, complemented
44#18012664C2211yescharacteristic, complemented
45#18012664C2211yescharacteristic, complemented
46#18012663C2211yescharacteristic, complemented
47#18012660C2211yescharacteristic
48#18012660C2211yescharacteristic
49#18012660C2211yescharacteristic, complemented
50C22⋊(C151⋊C8)C2211yescharacteristic
51(C2×D151)⋊C8C2211yescharacteristic, complemented
52#18012649C2211yescharacteristic, complemented
53C2×#41×C151C2211yescharacteristic
54C22×D4×C151C2211yescharacteristic
55AGL(1,ℤ/8)×C151—24no—
56C2×SD16×C151C2211yescharacteristic
57C2×SD16×C151C2211yescharacteristic
58C2×D8×C151C2211yescharacteristic
59C2×D8×C151C2211yescharacteristic
60C2×M16×C151C2211yescharacteristic
61#14559003—22no—
62#14559003—22no—
63#14559003—41no—
64#14559003—41no—
65C2×Dic302C2311yescharacteristic
66C2×Dic302C2311yescharacteristic
67C2×C22⋊D151—21no—
68C2×C22⋊D151—21no—
69C2×C22⋊D151—21no—
70C2×C22⋊D151D411yescharacteristic
71C22×Dic151—21no—
72C22×Dic151—21no—
73C22×Dic151—21no—
74C22×Dic151—21no—
75C2×C4×D151C2311yescharacteristic
76Dic604—22no—
77#14558994—41no—
78#14558992—22no—
79#14558992—22no—
80#14558992—41no—
81C2×C151⋊C8C2311yescharacteristic
82C2×C151⋊C8C2311yescharacteristic
83C8×D151—22no—
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⋊Dic151—21no—
95C22⋊Dic151—21no—
96C22⋊Dic151—21no—
97C22⋊Dic151—21no—
98C22⋊Dic151—21no—
99C22⋊Dic151D411yescharacteristic
100(C2×D151)⋊C4—21no—
101(C2×D151)⋊C4D411yescharacteristic
102C4×Dic151C2311yescharacteristic
103C24×C151—21no—
104#41×C151—22no—
105#41×C151—41no—
106C2×Q8×C151C2311yescharacteristic
107C2×D4×C151—22no—
108C2×D4×C151—21no—
109C2×D4×C151—21no—
110C2×D4×C151—41no—
111C2×D4×C151C2311yescharacteristic
112C2×D4×C151C2311yescharacteristic
113C2×D4×C151C2311yescharacteristic
114C22×C4×C151—21no—
115C22×C4×C151C2311yescharacteristic
116SD16×C151—22no—
117SD16×C151—22no—
118D8×C151—22no—
119D8×C151D414yescomplemented
120M16×C151—22no—
121C2×C8×C151C2311yescharacteristic
122C2×C8×C151C2311yescharacteristic
123C22×D151C2×D411yescharacteristic
124Dic302—21no—
125Dic302—21no—
126Dic302—21no—
127Dic302—21no—
128C22⋊D151—41no—
129C22⋊D151—42no—
130C22⋊D151—42no—
131C22⋊D151—81no—
132C2×Dic151—21no—
133C2×Dic151—21no—
134C2×Dic151—21no—
135C2×Dic151—21no—
136C2×Dic151—42no—
137C2×Dic151—42no—
138C2×Dic151—42no—
139C2×Dic151—81no—
140C2×Dic151C2×D411yescharacteristic
141C2×Dic151C2×D411yescharacteristic
142C2×Dic151C2×D411yescharacteristic
143C4×D151—21no—
144C4×D151—21no—
145C151⋊C8—21no—
146C151⋊C8—21no—
147C23×C151—21no—
148C23×C151—21no—
149C23×C151—21no—
150C23×C151—42no—
151C23×C151—42no—
152C23×C151—81no—
153C23×C151C2×D411yescharacteristic
154Q8×C151—21no—
155Q8×C151—21no—
156D4×C151—21no—
157D4×C151—21no—
158D4×C151—21no—
159D4×C151—21no—
160D4×C151—42no—
161D4×C151—42no—
162D4×C151C2×D412yes—
163D4×C151C2×D412yes—
164C2×C4×C151—21no—
165C2×C4×C151—21no—
166C2×C4×C151—21no—
167C2×C4×C151—42no—
168C2×C4×C151C2×D411yescharacteristic
169C2×C4×C151C2411yescharacteristic
170C8×C151—21no—
171C8×C151C2×D412yes—
172C22×C151—21no—
173C22×C151—22no—
174C22×C151—22no—
175C22×C151—21no—
176C22×C151—21no—
177C22×C151—41no—
178C22×C151—41no—
179C22×C151—41no—
180C22×C151—41no—
181C22×C151—41no—
182C22×C151—41no—
183C22×C151—41no—
184C22×C151—84no—
185C22×C151C22×D411yescharacteristic
186C2×D151—21no—
187C2×D151—21no—
188C2×D151—21no—
189C4×C151—21no—
190C4×C151—41no—
191C4×C151C22×D411yescharacteristic
192C4×C151H(5,𝔽2)11yescharacteristic
193Dic151—21no—
194Dic151—21no—
195Dic151—21no—
196Dic151—41no—
197Dic151—41no—
198Dic151—41no—
199Dic151—41no—
200C2×C151—41no—
201C2×C151—42no—
202C2×C151—42no—
203C2×C151—81no—
204C2×C151D4211yescharacteristic
205C2×C151#57511yescharacteristic
206C2×C151#57711yescharacteristic
207D151—41no—
208C151#326711yescharacteristic, complemented, Sylow
209#3267—1511nomaximal, Sylow
210C2×AGL(1,ℤ/8)D15111yescharacteristic, complemented
211C2×#135—1511no—
212#547—1511no—
213#546—1511no—
214#497—1511no—
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×#41C2×D15111yescharacteristic
228C22×D4C2×D15111yescharacteristic
229AGL(1,ℤ/8)—24no—
230#135—3024no—
231C2×Q16—1511no—
232C2×Q16—1511no—
233C2×SD16—1511no—
234C2×SD16—1511no—
235C2×SD16—1511no—
236C2×SD16C2×D15111yescharacteristic
237C2×SD16C2×D15111yescharacteristic
238C2×D8C2×D15111yescharacteristic
239C2×D8C2×D15111yescharacteristic
240C2×M16C2×D15111yescharacteristic
241C22×C8—1511no—
242C4⋊Q8—1511no—
243#124—1511no—
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#97—1511no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—3022no—
274#41—3022no—
275#41—6041no—
276#41—6041no—
277C2×Q8—1511no—
278C2×Q8—1511no—
279C2×Q8C22×D15111yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—3021no—
285C2×D4—3021no—
286C2×D4—3021no—
287C2×D4—1511no—
288C2×D4C22×D15111yescharacteristic
289C2×D4C22×D15111yescharacteristic
290C2×D4C22×D15111yescharacteristic
291C22×C4—21no—
292C22×C4—3021no—
293C22×C4—3021no—
294C22×C4—3021no—
295C22×C4—3021no—
296C22×C4—1511no—
297C22×C4C22×D15111yescharacteristic
298Q16—3022no—
299Q16—6041no—
300SD16—22no—
301SD16—22no—
302SD16—3022no—
303SD16—3022no—
304SD16—6041no—
305D8—22no—
306D8C22⋊D15114yescomplemented
307M16—22no—
308C2×C8—3022no—
309C2×C8—1511no—
310C2×C8—1511no—
311C2×C8C22×D15111yescharacteristic
312C2×C8C22×D15111yescharacteristic
313C4⋊C4—3021no—
314C4⋊C4—3021no—
315C4⋊C4—3021no—
316C4⋊C4—3021no—
317C4⋊C4—3021no—
318C4⋊C4—1511no—
319C4⋊C4—1511no—
320C4⋊C4—1511no—
321C22⋊C4—3021no—
322C22⋊C4—3021no—
323C22⋊C4—3021no—
324C22⋊C4—3021no—
325C22⋊C4—3021no—
326C22⋊C4—3021no—
327C22⋊C4—3021no—
328C22⋊C4—3021no—
329C22⋊C4—1511no—
330C22⋊C4—1511no—
331C42—1511no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1511no—
339C23D4×D15111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3021no—
343Q8—3021no—
344Q8—3021no—
345Q8—3021no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—6041no—
353D4—6042no—
354D4—6042no—
355D4—12081no—
356D4C2×C22⋊D15112yes—
357D4C2×C22⋊D15112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—3021no—
363C2×C4—3021no—
364C2×C4—3021no—
365C2×C4—3021no—
366C2×C4—3021no—
367C2×C4—3021no—
368C2×C4—6042no—
369C2×C4—6042no—
370C2×C4—6042no—
371C2×C4—1511no—
372C2×C4—1511no—
373C2×C4—1511no—
374C2×C4—12081no—
375C2×C4D4×D15111yescharacteristic
376C2×C4C23×D15111yescharacteristic
377C8—21no—
378C8—3021no—
379C8—3021no—
380C8C2×C22⋊D15112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—3021no—
395C22—3021no—
396C22—3021no—
397C22C2×D4×D15111yescharacteristic
398C4—21no—
399C4—41no—
400C4—3021no—
401C4—3021no—
402C4—3021no—
403C4—6041no—
404C4—6041no—
405C4—6041no—
406C4—6041no—
407C4C22×C22⋊D15111yescharacteristic
408C4#1801278411yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—6041no—
414C2D4×C22⋊D15111yescharacteristic
415C2#2375300811yescharacteristic
416C2#2375302211yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
FittingC2×AGL(1,ℤ/8)×C151C2
socleC22×C151C22×D4
hypercenterC2×AGL(1,ℤ/8)D151
hypocenterC151#3267
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#18012758C2semidirect
2. #23752861C2semidirect
3. #23752856C2semidirect
4. #23752603C2semidirect
5. #23752470C2semidirect
6. #23752438C2semidirect
7. #23752425C2semidirect
8. #23752415C2semidirect
9. #23752374C2semidirect
10. #23752371C2semidirect
11. #23752363C2semidirect
12. #23752354C2semidirect
13. #23752324C2semidirect
14. #23752281C2semidirect
15. C2×AGL(1,ℤ/8)×C151C2semidirect
17. C2×#14559003C22semidirect
19. C2×Dic604C22semidirect
20. C2×#14558994C22semidirect
23. C2×#14558992C22semidirect
24. C2×C8×D151C22semidirect
25. #18012737C22semidirect
32. #18012715C22semidirect
38. D4×Dic151C22semidirect
42. #18012671C22semidirect
43. Dic151⋊C8C22semidirect
44. #18012664C22semidirect
45. #18012664C22semidirect
46. #18012663C22semidirect
49. #18012660C22semidirect
51. (C2×D151)⋊C8C22semidirect
52. #18012649C22semidirect
119. D8×C151D4semidirect
208. C151#3267semidirect
210. C2×AGL(1,ℤ/8)D151semidirect
306. D8C22⋊D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127582
#237528612
#237528562
#237526032
#237524702
#237524382
#237524252
#237524152
#237523742
#237523712
#237523632
#237523542
#237523242
#237522812
C2×AGL(1,ℤ/8)×C1512
#3267151