Finite Group Databasebeta

← #30648418

Subgroups — #30648418

Order 16256.

472 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861241C211yesmaximal, characteristic, complemented
2C2×#16861217C211yesmaximal, characteristic, complemented
3#22550456C211yesmaximal, characteristic
4#22550449C211yesmaximal, characteristic, complemented
5#22550133C211yesmaximal, characteristic, complemented
6#22550125C211yesmaximal, characteristic, complemented
7#22550122C211yesmaximal, characteristic, complemented
8#22550115C211yesmaximal, characteristic, complemented
9#22550114C211yesmaximal, characteristic, complemented
10#22550107C211yesmaximal, characteristic, complemented
11#22550106C211yesmaximal, characteristic, complemented
12#22549915C211yesmaximal, characteristic, complemented
13#22549871C211yesmaximal, characteristic, complemented
14#103×D127C211yesmaximal, characteristic, complemented
15#470×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19#16861241—24no—
20C2×#14523238C2211yescharacteristic
21C2×#14523237C2211yescharacteristic
22C2×#14523235C2211yescharacteristic
23#16861217—24no—
24C2×#14523231C2211yescharacteristic
25C2×#14523231C2211yescharacteristic
26C22×C127⋊C8C2211yescharacteristic
27C2×C8×D127C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic
29#16861208—21no—
30#16861195—21no—
31#16861194—21no—
32#16861191—21no—
33#16861189C2211yescharacteristic
34#16861188C2211yescharacteristic
35#16861187—21no—
36#16861185C2211yescharacteristic
37#16861185C2211yescharacteristic
38#16861184—21no—
39#16861184C2211yescharacteristic
40#16861183C2211yescharacteristic
41#16861182—21no—
42#16861182—21no—
43#16861167C2211yescharacteristic
44C4⋊C4×D127C2211yescharacteristic
45#16861143C2211yescharacteristic
46#16861142C2211yescharacteristic
47#16861140C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861133C2211yescharacteristic
50#16861133C2211yescharacteristic
51#16861132C2211yescharacteristic
52#16861131C2211yescharacteristic
53#16861129C2211yescharacteristic
54C2×Q16×C127C2211yescharacteristic
55C2×M16×C127C2211yescharacteristic
56C22⋊Q8×C127C2211yescharacteristic
57C22⋊Q8×C127C2211yescharacteristic
58#106×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60#103×C127C2211yescharacteristic
61#14523249—22no—
62#14523249—22no—
63#14523248—22no—
64#14523248—22no—
65C2×Dic254—21no—
66C2×Dic254D411yescharacteristic
67C2×Dic254C2311yescharacteristic
68Q8×D127—22no—
69Q8×D127—22no—
70C2×D508—21no—
71C2×D508—21no—
72C2×D508D411yescharacteristic
73C2×D508C2311yescharacteristic
74C2×C22⋊D127—21no—
75C2×C22⋊D127D411yescharacteristic
76C2×C22⋊D127D411yescharacteristic
77C2×C4×D127—21no—
78C2×C4×D127—21no—
79C2×C4×D127—21no—
80C2×C4×D127C2311yescharacteristic
81C2×C4×D127C2311yescharacteristic
82#14523238—22no—
83#14523237—22no—
84#14523235—22no—
85#14523231—22no—
86#14523231D414yescomplemented
87C2×C127⋊C8—22no—
88C2×C127⋊C8C2311yescharacteristic
89C2×C127⋊C8C2311yescharacteristic
90C8×D127—22no—
91C8×D127—22no—
92C4⋊Dic127—21no—
93C4⋊Dic127C2311yescharacteristic
94C4⋊Dic127C2311yescharacteristic
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99C22⋊Dic127—21no—
100(C2×D127)⋊C4—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106C4×Dic127—21no—
107C2×Q8×C127C2311yescharacteristic
108C2×Q8×C127C2311yescharacteristic
109C22×C4×C127C2311yescharacteristic
110Q16×C127—22no—
111M16×C127—22no—
112C2×C8×C127C2311yescharacteristic
113C2×C8×C127C2311yescharacteristic
114C4⋊C4×C127—21no—
115C4⋊C4×C127—21no—
116C4⋊C4×C127C2311yescharacteristic
117C4⋊C4×C127C2311yescharacteristic
118C22⋊C4×C127—21no—
119C22⋊C4×C127—21no—
120C22×D127—21no—
121C22×D127C2×D411yescharacteristic
122C22×D127C2×D411yescharacteristic
123Dic254—22no—
124Dic254—21no—
125Dic254—21no—
126Dic254—42no—
127D508—22no—
128D508—21no—
129D508—21no—
130D508—42no—
131D508—81no—
132C22⋊D127—22no—
133C22⋊D127—22no—
134C22⋊D127—81no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127C2×D411yescharacteristic
139C2×Dic127C2×D411yescharacteristic
140C4×D127—21no—
141C4×D127—21no—
142C4×D127—42no—
143C4×D127—42no—
144C4×D127—81no—
145C4×D127C2×D412yes—
146C4×D127#4112yes—
147C127⋊C8—21no—
148C127⋊C8C2×D412yes—
149C23×C127C2×D411yescharacteristic
150Q8×C127—21no—
151Q8×C127—21no—
152Q8×C127—21no—
153Q8×C127—21no—
154C2×C4×C127—21no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127C2×D411yescharacteristic
161C2×C4×C127C2411yescharacteristic
162C8×C127—21no—
163C8×C127C2×D412yes—
164C22×C127—21no—
165C22×C127—21no—
166C22×C127—21no—
167C22×C127C22×D411yescharacteristic
168C2×D127—21no—
169C2×D127—21no—
170C2×D127—21no—
171C2×D127—21no—
172C2×D127—21no—
173C2×D127—41no—
174C2×D127—41no—
175C2×D127—41no—
176C2×D127#12112yes—
177C4×C127—21no—
178C4×C127—41no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127C22×D411yescharacteristic
183C4×C127C2×#4111yescharacteristic
184Dic127—21no—
185Dic127—41no—
186Dic127—41no—
187Dic127—41no—
188Dic127#12112yes—
189C2×C127—41no—
190C2×C127C2×#12111yescharacteristic
191C2×C127#57711yescharacteristic
192C2×C127#57811yescharacteristic
193D127—22no—
194D127—41no—
195D127—81no—
196#3063—1271nomaximal, Sylow
197C127#306311yescharacteristic, complemented, Sylow
198C2×#137—1271no—
199C2×#131—1271no—
200#537—1271no—
201#536—1271no—
202#470—1271no—
203#470D12711yescharacteristic, complemented
204#468—1271no—
205#468—1271no—
206#467—1271no—
207#467—1271no—
208#465—1271no—
209#465—1271no—
210#428—1271no—
211#416—1271no—
212C2×#103—1271no—
213C2×#41—1271no—
214C2×#41—1271no—
215C22×Q8—1271no—
216#137—2544no—
217C2×Q16—1271no—
218C2×Q16C2×D12711yescharacteristic
219C2×SD16—1271no—
220C2×SD16—1271no—
221#131—2544no—
222C2×M16—1271no—
223C2×M16—1271no—
224C2×M16C2×D12711yescharacteristic
225C22×C8—1271no—
226C22×C8—1271no—
227C22×C8—1271no—
228C4⋊Q8—2541no—
229#124—2541no—
230#123—2541no—
231#123—2541no—
232C22⋊Q8—1271no—
233C22⋊Q8—1271no—
234C22⋊Q8—1271no—
235C22⋊Q8—1271no—
236C22⋊Q8—2541no—
237C22⋊Q8C2×D12711yescharacteristic
238C22⋊Q8C2×D12711yescharacteristic
239#121—1271no—
240#121—1271no—
241#121—2541no—
242#121—2541no—
243#121—2541no—
244#117—1271no—
245C2×C4⋊C4—1271no—
246#107—1271no—
247#107—1271no—
248#106—1271no—
249#106C2×D12711yescharacteristic
250#103—1271no—
251#103—1271no—
252#103—1271no—
253#103—1271no—
254#103C2×D12711yescharacteristic
255#103C2×D12711yescharacteristic
256#102—1271no—
257#102—1271no—
258#41—2542no—
259#41—2542no—
260#41—2542no—
261#41—2542no—
262C2×Q8—1271no—
263C2×Q8—1271no—
264C2×Q8—2541no—
265C2×Q8—2542no—
266C2×Q8—2542no—
267C2×Q8C22×D12711yescharacteristic
268C2×Q8C22×D12711yescharacteristic
269C2×D4—1271no—
270C2×D4—1271no—
271C2×D4—1271no—
272C2×D4—1271no—
273C2×D4—2541no—
274C2×D4—2541no—
275C2×D4—2541no—
276C22×C4—1271no—
277C22×C4—1271no—
278C22×C4—2541no—
279C22×C4—2541no—
280C22×C4—2541no—
281C22×C4C22×D12711yescharacteristic
282Q16—22no—
283Q16—2542no—
284SD16—2542no—
285SD16—2542no—
286M16—22no—
287M16—1274no—
288M16—2542no—
289C2×C8—1271no—
290C2×C8—1271no—
291C2×C8—2542no—
292C2×C8—2542no—
293C2×C8—2542no—
294C2×C8C22×D12711yescharacteristic
295C2×C8C22×D12711yescharacteristic
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—1271no—
299C4⋊C4—1271no—
300C4⋊C4—2541no—
301C4⋊C4—2541no—
302C4⋊C4—2541no—
303C4⋊C4—2541no—
304C4⋊C4—2541no—
305C4⋊C4C22×D12711yescharacteristic
306C4⋊C4C22×D12711yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—2541no—
310C22⋊C4—2541no—
311C22⋊C4—2541no—
312C22⋊C4—2541no—
313C22⋊C4—2541no—
314C22⋊C4—2541no—
315C22⋊C4—2541no—
316C42—2541no—
317C23—1271no—
318C23—1271no—
319C23—2541no—
320C23D4×D12711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—5082no—
326Q8—2542no—
327Q8—2541no—
328Q8—2541no—
329D4—10161no—
330D4—10161no—
331D4—5082no—
332D4—2542no—
333D4—2542no—
334D4—2542no—
335D4—2541no—
336D4—2541no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—1271no—
344C2×C4—1272no—
345C2×C4—1272no—
346C2×C4—1271no—
347C2×C4—10161no—
348C2×C4—5082no—
349C2×C4—5082no—
350C2×C4—2541no—
351C2×C4—2541no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4—2541no—
355C2×C4D4×D12711yescharacteristic
356C2×C4C23×D12711yescharacteristic
357C8—21no—
358C8—1272no—
359C8—2541no—
360C8D4×D12712yes—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—1272no—
365C22—5081no—
366C22—5081no—
367C22—5081no—
368C22—2541no—
369C22—2541no—
370C22—2541no—
371C22—2541no—
372C22—2541no—
373C22C2×D4×D12711yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—1272no—
380C4—5081no—
381C4—5081no—
382C4—5081no—
383C4—2541no—
384C4C2×D4×D12711yescharacteristic
385C4#41×D12711yescharacteristic
386C2—41no—
387C2—10161no—
388C2—5081no—
389C2—2542no—
390C2#121×D12711yescharacteristic
391C2#2255068511yescharacteristic
392C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#470×C127C2
socleC22×C127C22×D4
prime radical#22550456C2
hypercenter#470D127
hypocenterC127#3063
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#16861241C2semidirect
2. C2×#16861217C2semidirect
4. #22550449C2semidirect
5. #22550133C2semidirect
6. #22550125C2semidirect
7. #22550122C2semidirect
8. #22550115C2semidirect
9. #22550114C2semidirect
10. #22550107C2semidirect
11. #22550106C2semidirect
12. #22549915C2semidirect
13. #22549871C2semidirect
14. #103×D127C2semidirect
15. #470×C127C2semidirect
86. #14523231D4semidirect
197. C127#3063semidirect
203. #470D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612412
C2×#168612172
#225504562
#225504492
#225501332
#225501252
#225501222
#225501152
#225501142
#225501072
#225501062
#225499152
#225498712
#103×D1272
#470×C1272
#3063127