Finite Group Databasebeta

← #34080624

Subgroups — #34080624

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×#18012768C211yesmaximal, characteristic, complemented
2C22×#14558992C211yesmaximal, characteristic, complemented
3#23752875C211yesmaximal, characteristic, complemented
4#23752856C211yesmaximal, characteristic
5#23752603C211yesmaximal, characteristic, complemented
6#23752457C211yesmaximal, characteristic, complemented
7#23752451C211yesmaximal, characteristic, complemented
8#23752433C211yesmaximal, characteristic, complemented
9#23752415C211yesmaximal, characteristic, complemented
10#23752375C211yesmaximal, characteristic, complemented
11#23752370C211yesmaximal, characteristic, complemented
12#23752349C211yesmaximal, characteristic, complemented
13#23752309C211yesmaximal, characteristic, complemented
14#23752280C211yesmaximal, characteristic, complemented
15#450×C151C211yesmaximal, characteristic, complemented
16C2×#14559003C2211yescharacteristic
17C22×Dic302C2211yescharacteristic
18C2×Q8×D151C2211yescharacteristic, complemented
19#18012768—24no—
20C2×Dic604C2211yescharacteristic, complemented
21C2×Dic604C2211yescharacteristic, complemented
22C2×#14558994C2211yescharacteristic, complemented
23C2×#14558992—24no—
24C2×#14558992C2211yescharacteristic
25C2×#14558992C2211yescharacteristic
26C2×#14558992C2211yescharacteristic
27C2×#14558992C2211yescharacteristic
28C2×#14558987C2211yescharacteristic, complemented
29C22×C151⋊C8C2211yescharacteristic
30#18012738—21no—
31#18012737C2211yescharacteristic, complemented
32#18012726C2211yescharacteristic
33#18012720—21no—
34#18012719C2211yescharacteristic, complemented
35#18012718C2211yescharacteristic
36#18012717—21no—
37#18012716—21no—
38#18012711—21no—
39#18012710C2211yescharacteristic
40#18012708—21no—
41#18012704C2211yescharacteristic, complemented
42D4×Dic151C2211yescharacteristic
43C4×C22⋊D151C2211yescharacteristic
44(C22⋊D151)⋊C4—21no—
45C2×C22⋊Dic151—21no—
46#18012672C2211yescharacteristic, complemented
47Dic151⋊C8C2211yescharacteristic, complemented
48#18012664C2211yescharacteristic, complemented
49#18012664C2211yescharacteristic, complemented
50#18012663C2211yescharacteristic, complemented
51#18012662C2211yescharacteristic, complemented
52(C2×D151)⋊C8C2211yescharacteristic, complemented
53#18012649C2211yescharacteristic, complemented
54C22×D4×C151C2211yescharacteristic
55C2×SD16×C151C2211yescharacteristic
56C2×SD16×C151C2211yescharacteristic
57C22⋊Q8×C151C2211yescharacteristic
58#102×C151C2211yescharacteristic
59#102×C151C2211yescharacteristic
60C22⋊C8×C151C2211yescharacteristic
61#14559003—22no—
62#14559003—41no—
63C2×Dic302—21no—
64C2×Dic302—22no—
65C2×Dic302—21no—
66C2×Dic302—41no—
67C2×Dic302C2311yescharacteristic
68C2×Dic302C2311yescharacteristic
69C2×Dic302C2311yescharacteristic
70Q8×D151—22no—
71Q8×D151—41no—
72C2×C22⋊D151—21no—
73C2×C22⋊D151—21no—
74C2×C22⋊D151D411yescharacteristic
75C22×Dic151—21no—
76C22×Dic151—21no—
77C22×Dic151—21no—
78C2×C4×D151—21no—
79C2×C4×D151C2311yescharacteristic
80Dic604—22no—
81Dic604—41no—
82#14558994—22no—
83#14558992—22no—
84#14558992—22no—
85#14558992—22no—
86#14558992D414yescomplemented
87#14558987—22no—
88C2×C151⋊C8—22no—
89C2×C151⋊C8C2311yescharacteristic
90C2×C151⋊C8C2311yescharacteristic
91C4⋊Dic151—21no—
92C4⋊Dic151C2311yescharacteristic
93Dic151⋊C4—21no—
94Dic151⋊C4—21no—
95Dic151⋊C4—21no—
96Dic151⋊C4—21no—
97Dic151⋊C4D411yescharacteristic
98C22⋊Dic151—21no—
99C22⋊Dic151—21no—
100C22⋊Dic151—21no—
101C22⋊Dic151—21no—
102C22⋊Dic151—21no—
103C22⋊Dic151D411yescharacteristic
104(C2×D151)⋊C4—21no—
105(C2×D151)⋊C4D411yescharacteristic
106C4×Dic151—21no—
107C4×Dic151C2311yescharacteristic
108C24×C151—21no—
109C2×Q8×C151C2311yescharacteristic
110C2×D4×C151—22no—
111C2×D4×C151—21no—
112C2×D4×C151—41no—
113C2×D4×C151C2311yescharacteristic
114C2×D4×C151C2311yescharacteristic
115C22×C4×C151C2311yescharacteristic
116SD16×C151—22no—
117SD16×C151—41no—
118C2×C8×C151C2311yescharacteristic
119C2×C8×C151C2311yescharacteristic
120C4⋊C4×C151—21no—
121C4⋊C4×C151C2311yescharacteristic
122C22⋊C4×C151—21no—
123C22×D151C2×D411yescharacteristic
124Dic302—21no—
125Dic302—21no—
126Dic302—21no—
127Dic302—21no—
128Dic302—21no—
129Dic302—42no—
130Dic302—42no—
131Dic302C2×D412yes—
132C22⋊D151—22no—
133C22⋊D151—42no—
134C22⋊D151—81no—
135C2×Dic151—21no—
136C2×Dic151—21no—
137C2×Dic151—21no—
138C2×Dic151—21no—
139C2×Dic151—42no—
140C2×Dic151—42no—
141C2×Dic151—81no—
142C2×Dic151C2×D411yescharacteristic
143C2×Dic151C2×D411yescharacteristic
144C2×Dic151C2×D411yescharacteristic
145C4×D151—21no—
146C4×D151—21no—
147C4×D151—42no—
148C151⋊C8—21no—
149C151⋊C8C2×D412yes—
150C23×C151—21no—
151C23×C151—21no—
152C23×C151—42no—
153C23×C151—42no—
154C23×C151—81no—
155C23×C151C2×D411yescharacteristic
156Q8×C151—21no—
157Q8×C151—21no—
158D4×C151—21no—
159D4×C151—21no—
160D4×C151—21no—
161D4×C151—42no—
162D4×C151C2×D412yes—
163C2×C4×C151—21no—
164C2×C4×C151—21no—
165C2×C4×C151—21no—
166C2×C4×C151—21no—
167C2×C4×C151C2×D411yescharacteristic
168C2×C4×C151C2411yescharacteristic
169C8×C151—21no—
170C8×C151—21no—
171C22×C151—21no—
172C22×C151—21no—
173C22×C151—22no—
174C22×C151—41no—
175C22×C151—41no—
176C22×C151—41no—
177C22×C151—41no—
178C22×C151—41no—
179C22×C151—84no—
180C22×C151#13712yescomplemented
181C22×C151C22×D411yescharacteristic
182C2×D151—21no—
183C2×D151—21no—
184C2×D151—21no—
185C4×C151—21no—
186C4×C151—41no—
187C4×C151—41no—
188C4×C151C22×D411yescharacteristic
189C4×C151H(5,𝔽2)11yescharacteristic
190Dic151—21no—
191Dic151—22no—
192Dic151—21no—
193Dic151—21no—
194Dic151—41no—
195Dic151—41no—
196Dic151—41no—
197C2×C151—22no—
198C2×C151—42no—
199C2×C151—81no—
200C2×C151D4211yescharacteristic
201C2×C151C2×#13711yescharacteristic
202C2×C151#57711yescharacteristic
203D151—41no—
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#450D15111yescharacteristic, complemented
218#448—1511no—
219#440—1511no—
220#435—1511no—
221C2×#41—1511no—
222C22×Q8—1511no—
223C22×Q8—1511no—
224C22×D4C2×D15111yescharacteristic
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×SD16C2×D15111yescharacteristic
235C2×SD16C2×D15111yescharacteristic
236C2×M16—1511no—
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⋊Q8C2×D15111yescharacteristic
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#102C2×D15111yescharacteristic
262#102C2×D15111yescharacteristic
263C22⋊C8—1511no—
264C22⋊C8C2×D15111yescharacteristic
265#97—1511no—
266C24—21no—
267#41—3022no—
268#41—6041no—
269C2×Q8—3021no—
270C2×Q8—3022no—
271C2×Q8—3021no—
272C2×Q8—3022no—
273C2×Q8—6041no—
274C2×Q8—6041no—
275C2×Q8—1511no—
276C2×Q8—1511no—
277C2×Q8—1511no—
278C2×Q8C22×D15111yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—3021no—
283C2×D4—3021no—
284C2×D4—1511no—
285C2×D4C22×D15111yescharacteristic
286C2×D4C22×D15111yescharacteristic
287C22×C4—3021no—
288C22×C4—3021no—
289C22×C4—3021no—
290C22×C4—3021no—
291C22×C4—1511no—
292C22×C4C22×D15111yescharacteristic
293Q16—3022no—
294Q16—3022no—
295Q16—6041no—
296SD16—22no—
297SD16—41no—
298SD16—3022no—
299SD16—3022no—
300SD16—3022no—
301SD16—1514no—
302M16—3022no—
303C2×C8—3022no—
304C2×C8—1511no—
305C2×C8—1511no—
306C2×C8C22×D15111yescharacteristic
307C2×C8C22×D15111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—3021no—
310C4⋊C4—3021no—
311C4⋊C4—3021no—
312C4⋊C4—3021no—
313C4⋊C4—3021no—
314C4⋊C4—1511no—
315C4⋊C4—1511no—
316C4⋊C4C22×D15111yescharacteristic
317C22⋊C4—21no—
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—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1511no—
334C23D4×D15111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—3021no—
338Q8—3021no—
339Q8—3021no—
340Q8—3021no—
341Q8—3021no—
342Q8—6042no—
343Q8—6042no—
344Q8—1512no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—3022no—
350D4—6042no—
351D4—12081no—
352D4D4×D15112yes—
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—1511no—
367C2×C4—1511no—
368C2×C4—1511no—
369C2×C4—12081no—
370C2×C4D4×D15111yescharacteristic
371C2×C4C23×D15111yescharacteristic
372C8—21no—
373C8—21no—
374C8—3021no—
375C8—1512no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—3021no—
386C22—3021no—
387C22—3021no—
388C22#1801276812yescomplemented
389C22C2×D4×D15111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—3021no—
394C4—3021no—
395C4—3021no—
396C4—3022no—
397C4—6041no—
398C4—6041no—
399C4—6041no—
400C4C2×D4×D15111yescharacteristic
401C4#1801278411yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—6041no—
406C2#2375284411yescharacteristic
407C2C2×#1801276811yescharacteristic
408C2#2375302211yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#450×C151C2
socleC22×C151C22×D4
prime radical#23752856C2
hypercenter#450D151
hypocenterC151#3266
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#18012768C2semidirect
2. C22×#14558992C2semidirect
3. #23752875C2semidirect
5. #23752603C2semidirect
6. #23752457C2semidirect
7. #23752451C2semidirect
8. #23752433C2semidirect
9. #23752415C2semidirect
10. #23752375C2semidirect
11. #23752370C2semidirect
12. #23752349C2semidirect
13. #23752309C2semidirect
14. #23752280C2semidirect
15. #450×C151C2semidirect
18. C2×Q8×D151C22semidirect
20. C2×Dic604C22semidirect
21. C2×Dic604C22semidirect
22. C2×#14558994C22semidirect
28. C2×#14558987C22semidirect
31. #18012737C22semidirect
34. #18012719C22semidirect
41. #18012704C22semidirect
46. #18012672C22semidirect
47. Dic151⋊C8C22semidirect
48. #18012664C22semidirect
49. #18012664C22semidirect
50. #18012663C22semidirect
51. #18012662C22semidirect
52. (C2×D151)⋊C8C22semidirect
53. #18012649C22semidirect
86. #14558992D4semidirect
180. C22×C151#137semidirect
204. C151#3266semidirect
217. #450D151semidirect
388. C22#18012768semidirect

Maximal subgroups

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

subgroupindex
C2×#180127682
C22×#145589922
#237528752
#237528562
#237526032
#237524572
#237524512
#237524332
#237524152
#237523752
#237523702
#237523492
#237523092
#237522802
#450×C1512
#3266151