Finite Group Databasebeta

← #30648283

Subgroups — #30648283

Order 16256.

472 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861241C211yesmaximal, characteristic, complemented
2C2×#16861230C211yesmaximal, characteristic
3#22550446C211yesmaximal, characteristic, complemented
4#22550358C211yesmaximal, characteristic, complemented
5#22550096C211yesmaximal, characteristic, complemented
6#22550090C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550086C211yesmaximal, characteristic, complemented
9#22550084C211yesmaximal, characteristic, complemented
10#22550082C211yesmaximal, characteristic, complemented
11#22550080C211yesmaximal, characteristic, complemented
12#22549901C211yesmaximal, characteristic, complemented
13#22549874C211yesmaximal, characteristic, complemented
14#103×D127C211yesmaximal, characteristic, complemented
15#464×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19#16861241—24no—
20#16861230—24no—
21C2×#14523238C2211yescharacteristic
22C2×#14523238C2211yescharacteristic
23C2×#14523237C2211yescharacteristic
24C2×#14523237C2211yescharacteristic
25C2×#14523235C2211yescharacteristic
26C2×#14523233C2211yescharacteristic
27C2×#14523231C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic
29#16861209C2211yescharacteristic
30#16861207C2211yescharacteristic
31#16861196C2211yescharacteristic
32#16861191—21no—
33#16861191—21no—
34#16861186—21no—
35#16861186—21no—
36Q8×Dic127C2211yescharacteristic
37C4×D508—21no—
38C4×D508C2211yescharacteristic
39#16861171—21no—
40#16861171C2211yescharacteristic
41#16861166—21no—
42#16861166C2211yescharacteristic
43C4⋊C4×D127—21no—
44C4⋊C4×D127C2211yescharacteristic
45C4⋊(C127⋊C8)C2211yescharacteristic
46Dic127⋊C8C2211yescharacteristic
47Dic127⋊C8C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861134C2211yescharacteristic
50#16861133C2211yescharacteristic
51#16861132C2211yescharacteristic
52#16861132C2211yescharacteristic
53#16861130C2211yescharacteristic
54C2×Q16×C127C2211yescharacteristic
55C2×SD16×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic
57C4×Q8×C127C2211yescharacteristic
58C4⋊C8×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60#102×C127C2211yescharacteristic
61#14523249—22no—
62#14523249D414yes—
63#14523247—22no—
64#14523247—22no—
65C2×Dic254—21no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68Q8×D127—22no—
69Q8×D127—22no—
70C2×D508—21no—
71C2×D508C2311yescharacteristic
72C2×C22⋊D127—21no—
73C22×Dic127—21no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127—21no—
79C2×C4×D127C2311yescharacteristic
80#14523238—22no—
81#14523238—22no—
82#14523237—22no—
83#14523237—22no—
84#14523235—22no—
85#14523233—22no—
86#14523231—22no—
87C2×C127⋊C8C2311yescharacteristic
88C2×C127⋊C8C2311yescharacteristic
89C8×D127—22no—
90C4⋊Dic127—21no—
91C4⋊Dic127—21no—
92C4⋊Dic127—21no—
93C4⋊Dic127C2311yescharacteristic
94Dic127⋊C4—21no—
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4D411yescharacteristic
98Dic127⋊C4D411yescharacteristic
99C22⋊Dic127—21no—
100C22⋊Dic127—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4D411yescharacteristic
104(C2×D127)⋊C4D411yescharacteristic
105C4×Dic127—21no—
106C4×Dic127C2311yescharacteristic
107C2×Q8×C127C2311yescharacteristic
108C2×Q8×C127C2311yescharacteristic
109C2×D4×C127C2311yescharacteristic
110Q16×C127—22no—
111SD16×C127—22no—
112C2×C8×C127C2311yescharacteristic
113C2×C8×C127C2311yescharacteristic
114C4⋊C4×C127—21no—
115C4⋊C4×C127C2311yescharacteristic
116C22⋊C4×C127—21no—
117C22⋊C4×C127—21no—
118C42×C127—21no—
119C42×C127C2311yescharacteristic
120C22×D127—21no—
121C22×D127C2×D411yescharacteristic
122Dic254—21no—
123Dic254—21no—
124Dic254—21no—
125Dic254—21no—
126Dic254—42no—
127D508—24no—
128D508—21no—
129D508C2×D412yes—
130C22⋊D127—42no—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—42no—
135C2×Dic127C2×D411yescharacteristic
136C2×Dic127C2×D411yescharacteristic
137C2×Dic127C2×D411yescharacteristic
138C4×D127—24no—
139C4×D127—42no—
140C4×D127—81no—
141C4×D127—81no—
142C4×D127—81no—
143C4×D127C2×D412yes—
144C4×D127#4112yes—
145C127⋊C8—21no—
146C127⋊C8—21no—
147C23×C127—21no—
148Q8×C127—21no—
149Q8×C127—21no—
150Q8×C127—21no—
151Q8×C127C2×D412yes—
152D4×C127—21no—
153D4×C127—21no—
154C2×C4×C127—21no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127C2×D411yescharacteristic
158C2×C4×C127C2×D411yescharacteristic
159C2×C4×C127C2411yescharacteristic
160C8×C127—21no—
161C8×C127—21no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127C22×D411yescharacteristic
166C2×D127—21no—
167C2×D127—21no—
168C2×D127—22no—
169C2×D127—41no—
170C2×D127—41no—
171C2×D127#12112yes—
172C4×C127—22no—
173C4×C127—21no—
174C4×C127—21no—
175C4×C127—41no—
176C4×C127—41no—
177C4×C127C22×D411yescharacteristic
178C4×C127C2×#4111yescharacteristic
179Dic127—21no—
180Dic127—21no—
181Dic127—41no—
182Dic127—41no—
183Dic127—41no—
184Dic127#12112yes—
185C2×C127—81no—
186C2×C127C2×#12111yescharacteristic
187C2×C127#57711yescharacteristic
188C2×C127#57811yescharacteristic
189D127—22no—
190D127—42no—
191#3046—1271nomaximal, Sylow
192C127#304611yescharacteristic, complemented, Sylow
193C2×#137—1271no—
194C2×#135—1271no—
195#536—1271no—
196#520—1271no—
197#464—1271no—
198#464D12711yescharacteristic, complemented
199#463—1271no—
200#463—1271no—
201#462—1271no—
202#462—1271no—
203#461—1271no—
204#461—1271no—
205#424—1271no—
206#417—1271no—
207C2×#103—1271no—
208C2×#41—1271no—
209C2×#41—1271no—
210C22×Q8—1271no—
211#137—2544no—
212#135—2544no—
213C2×Q16—1271no—
214C2×Q16—1271no—
215C2×Q16C2×D12711yescharacteristic
216C2×SD16—1271no—
217C2×SD16—1271no—
218C2×SD16—1271no—
219C2×SD16C2×D12711yescharacteristic
220C2×D8—1271no—
221C2×M16—1271no—
222C22×C8—1271no—
223C4⋊Q8—1271no—
224C4⋊Q8—1271no—
225#124—1271no—
226#124C2×D12711yescharacteristic
227#123—2541no—
228#123—2541no—
229C22⋊Q8—2541no—
230C22⋊Q8—2541no—
231C4×Q8—1271no—
232C4×Q8C2×D12711yescharacteristic
233C4×D4—1271no—
234C4×D4—1271no—
235C4×D4—2541no—
236C4×D4—2541no—
237#117—1271no—
238#117—2541no—
239C2×C4⋊C4—1271no—
240C2×C4⋊C4—2541no—
241C4⋊C8—1271no—
242C4⋊C8—1271no—
243C4⋊C8—1271no—
244C4⋊C8C2×D12711yescharacteristic
245#103—1271no—
246#103—1271no—
247#103—1271no—
248#103—1271no—
249#103—1271no—
250#103C2×D12711yescharacteristic
251#102—1271no—
252#102C2×D12711yescharacteristic
253#41—1274no—
254#41—2542no—
255#41—2542no—
256#41—2542no—
257C2×Q8—1271no—
258C2×Q8—1271no—
259C2×Q8—2541no—
260C2×Q8—2542no—
261C2×Q8—2542no—
262C2×Q8C22×D12711yescharacteristic
263C2×Q8C22×D12711yescharacteristic
264C2×D4—1271no—
265C2×D4—2541no—
266C2×D4—2541no—
267C2×D4C22×D12711yescharacteristic
268C22×C4—1271no—
269C22×C4—2541no—
270C22×C4—2541no—
271C22×C4—2541no—
272C22×C4—2541no—
273C22×C4—2541no—
274C22×C4—2541no—
275Q16—22no—
276Q16—2542no—
277Q16—2542no—
278SD16—22no—
279SD16—2542no—
280SD16—2542no—
281SD16—2542no—
282D8—2542no—
283M16—2542no—
284C2×C8—1271no—
285C2×C8—1271no—
286C2×C8—2542no—
287C2×C8C22×D12711yescharacteristic
288C2×C8C22×D12711yescharacteristic
289C4⋊C4—21no—
290C4⋊C4—1271no—
291C4⋊C4—1271no—
292C4⋊C4—1271no—
293C4⋊C4—2541no—
294C4⋊C4—2541no—
295C4⋊C4—2541no—
296C4⋊C4—2541no—
297C4⋊C4—2541no—
298C4⋊C4—2541no—
299C4⋊C4C22×D12711yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—1271no—
303C22⋊C4—1271no—
304C22⋊C4—2541no—
305C22⋊C4—2541no—
306C22⋊C4—2541no—
307C22⋊C4—2541no—
308C42—21no—
309C42—1271no—
310C42—2541no—
311C42C22×D12711yescharacteristic
312C23—21no—
313C23—1271no—
314C23—2541no—
315Q8—21no—
316Q8—21no—
317Q8—21no—
318Q8—5082no—
319Q8—2541no—
320Q8—2541no—
321Q8—2541no—
322Q8—2541no—
323Q8D4×D12712yes—
324D4—21no—
325D4—21no—
326D4—1272no—
327D4—5082no—
328D4—2544no—
329D4—2541no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—1271no—
334C2×C4—1272no—
335C2×C4—1272no—
336C2×C4—1271no—
337C2×C4—1271no—
338C2×C4—10161no—
339C2×C4—10161no—
340C2×C4—10161no—
341C2×C4—5082no—
342C2×C4—5082no—
343C2×C4—2544no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4—2541no—
347C2×C4D4×D12711yescharacteristic
348C2×C4D4×D12711yescharacteristic
349C2×C4C23×D12711yescharacteristic
350C8—21no—
351C8—21no—
352C8—2541no—
353C8—2541no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—1272no—
358C22—5081no—
359C22—5081no—
360C22—2542no—
361C22—2541no—
362C22—2541no—
363C22C2×D4×D12711yescharacteristic
364C4—22no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—1272no—
370C4—5081no—
371C4—5081no—
372C4—5081no—
373C4—2541no—
374C4—2541no—
375C4C2×D4×D12711yescharacteristic
376C4#41×D12711yescharacteristic
377C2—81no—
378C2—5082no—
379C2—2542no—
380C2#121×D12711yescharacteristic
381C2#2255068311yescharacteristic
382C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#464×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861230C2
hypercenter#464D127
hypocenterC127#3046
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#16861241C2semidirect
3. #22550446C2semidirect
4. #22550358C2semidirect
5. #22550096C2semidirect
6. #22550090C2semidirect
7. #22550088C2semidirect
8. #22550086C2semidirect
9. #22550084C2semidirect
10. #22550082C2semidirect
11. #22550080C2semidirect
12. #22549901C2semidirect
13. #22549874C2semidirect
14. #103×D127C2semidirect
15. #464×C127C2semidirect
192. C127#3046semidirect
198. #464D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612412
C2×#168612302
#225504462
#225503582
#225500962
#225500902
#225500882
#225500862
#225500842
#225500822
#225500802
#225499012
#225498742
#103×D1272
#464×C1272
#3046127