Finite Group Databasebeta

← #30650308

Subgroups — #30650308

Order 16256.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861238C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3#22550544C211yesmaximal, characteristic, complemented
4#22550533C211yesmaximal, characteristic, complemented
5#22550264C211yesmaximal, characteristic, complemented
6#22550128C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550097C211yesmaximal, characteristic, complemented
9#22550074C211yesmaximal, characteristic, complemented
10#22550040C211yesmaximal, characteristic, complemented
11#22550018C211yesmaximal, characteristic, complemented
12#22550016C211yesmaximal, characteristic, complemented
13#22549980C211yesmaximal, characteristic, complemented
14#22549944C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19#16861238—24no—
20#16861229—24no—
21C2×Dic508C2211yescharacteristic
22C2×#14523238C2211yescharacteristic
23C2×#14523237C2211yescharacteristic
24C2×#14523236C2211yescharacteristic
25C2×#14523236C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27C2×#14523233C2211yescharacteristic
28C2×#14523231C2211yescharacteristic
29C22×C127⋊C8C2211yescharacteristic
30#16861205—21no—
31#16861205C2211yescharacteristic
32#16861196—21no—
33#16861196C2211yescharacteristic
34#16861188—21no—
35#16861188C2211yescharacteristic
36#16861184C2211yescharacteristic
37#16861182—21no—
38#16861181—21no—
39#16861180—21no—
40#16861180C2211yescharacteristic
41Q8×Dic127C2211yescharacteristic
42#16861174C2211yescharacteristic
43C4×C22⋊D127—21no—
44C4×C22⋊D127C2211yescharacteristic
45(C22⋊D127)⋊C4—21no—
46#16861140C2211yescharacteristic
47Dic127⋊C8C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861132C2211yescharacteristic
50#16861131C2211yescharacteristic
51#16861130C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53#16861119C2211yescharacteristic
54C2×#41×C127C2211yescharacteristic
55C2×SD16×C127C2211yescharacteristic
56C2×D8×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#102×C127C2211yescharacteristic
60C22⋊C8×C127C2211yescharacteristic
61#14523248—22no—
62#14523248—41no—
63#14523247—22no—
64#14523247—41no—
65C2×Dic254—21no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68Q8×D127—22no—
69Q8×D127—41no—
70C2×D508—21no—
71C2×D508C2311yescharacteristic
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77C2×C22⋊D127D411yescharacteristic
78C22×Dic127—21no—
79C2×C4×D127—21no—
80C2×C4×D127—21no—
81C2×C4×D127C2311yescharacteristic
82Dic508—22no—
83#14523238D414yescomplemented
84#14523237—41no—
85#14523236—22no—
86#14523236—22no—
87#14523235—22no—
88#14523233—22no—
89#14523231—22no—
90C2×C127⋊C8—22no—
91C2×C127⋊C8C2311yescharacteristic
92C2×C127⋊C8C2311yescharacteristic
93C4⋊Dic127—21no—
94C4⋊Dic127C2311yescharacteristic
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4D411yescharacteristic
98C22⋊Dic127—21no—
99C22⋊Dic127—21no—
100C22⋊Dic127—21no—
101C22⋊Dic127—21no—
102C22⋊Dic127D411yescharacteristic
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4D411yescharacteristic
105C4×Dic127—21no—
106C4×Dic127—21no—
107C4×Dic127C2311yescharacteristic
108#41×C127—22no—
109#41×C127—41no—
110C2×Q8×C127C2311yescharacteristic
111C2×D4×C127—21no—
112C2×D4×C127—21no—
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115C22×C4×C127—21no—
116C22×C4×C127C2311yescharacteristic
117SD16×C127—22no—
118D8×C127—41no—
119C2×C8×C127C2311yescharacteristic
120C2×C8×C127C2311yescharacteristic
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22×D127—21no—
124C22×D127C2×D411yescharacteristic
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—42no—
129Dic254C2×D412yes—
130D508—21no—
131D508—21no—
132D508—81no—
133C22⋊D127—41no—
134C22⋊D127—42no—
135C22⋊D127—42no—
136C22⋊D127—42no—
137C22⋊D127—42no—
138C22⋊D127—81no—
139C2×Dic127—21no—
140C2×Dic127—21no—
141C2×Dic127—21no—
142C2×Dic127—42no—
143C2×Dic127C2×D411yescharacteristic
144C2×Dic127C2×D411yescharacteristic
145C2×Dic127C2×D411yescharacteristic
146C4×D127—21no—
147C4×D127—21no—
148C4×D127—42no—
149C4×D127—42no—
150C127⋊C8—21no—
151C127⋊C8C2×D412yes—
152C23×C127—21no—
153C23×C127—21no—
154C23×C127C2×D411yescharacteristic
155Q8×C127—21no—
156Q8×C127C2×D412yes—
157D4×C127—21no—
158D4×C127—21no—
159D4×C127—21no—
160D4×C127—21no—
161D4×C127—42no—
162D4×C127—81no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—42no—
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2411yescharacteristic
170C8×C127—21no—
171C8×C127—21no—
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—21no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127C22×D411yescharacteristic
182C2×D127—21no—
183C2×D127—21no—
184C2×D127—21no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—41no—
188C4×C127—21no—
189C4×C127—22no—
190C4×C127—41no—
191C4×C127C22×D411yescharacteristic
192C4×C127H(5,𝔽2)11yescharacteristic
193Dic127—21no—
194Dic127—22no—
195Dic127—21no—
196Dic127—21no—
197Dic127—41no—
198Dic127—41no—
199C2×C127—41no—
200C2×C127—81no—
201C2×C127—81no—
202C2×C127D4211yescharacteristic
203C2×C127#57511yescharacteristic
204C2×C127#57711yescharacteristic
205D127—41no—
206D127—81no—
207#3271—1271nomaximal, Sylow
208C127#327111yescharacteristic, complemented, Sylow
209C2×#137—1271no—
210C2×#135—1271no—
211#550—1271no—
212#548—1271no—
213#496—1271no—
214#469—1271no—
215#465—1271no—
216#464—1271no—
217#460—1271no—
218#452—1271no—
219#449—1271no—
220#449D12711yescharacteristic, complemented
221#448—1271no—
222#441—1271no—
223#435—1271no—
224C2×#41—1271no—
225C2×#41—1271no—
226C2×#41C2×D12711yescharacteristic
227C22×Q8—1271no—
228#137—2544no—
229#135—2544no—
230C2×Q16—1271no—
231C2×Q16—1271no—
232C2×SD16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16C2×D12711yescharacteristic
237C2×D8—1271no—
238C2×D8C2×D12711yescharacteristic
239C2×M16—1271no—
240C22×C8—1271no—
241#127—1271no—
242#127—2541no—
243#124—1271no—
244#124—2541no—
245C22⋊Q8—1271no—
246C22⋊Q8—2541no—
247#121—1271no—
248#121—1271no—
249#121—2541no—
250#121—2541no—
251#121—2541no—
252#121C2×D12711yescharacteristic
253C4×Q8—1271no—
254C4×Q8—1271no—
255C4×D4—1271no—
256C4×D4—2541no—
257C4×D4—2541no—
258#106—1271no—
259C4⋊C8—1271no—
260#103—1271no—
261#103—1271no—
262#103C2×D12711yescharacteristic
263#102—1271no—
264#102—1271no—
265#102C2×D12711yescharacteristic
266C22⋊C8—1271no—
267C22⋊C8C2×D12711yescharacteristic
268#97—1271no—
269#41—22no—
270#41—41no—
271#41—5081no—
272#41—5081no—
273#41—2542no—
274#41—2542no—
275C2×Q8—1271no—
276C2×Q8—1271no—
277C2×Q8—5081no—
278C2×Q8—2541no—
279C2×Q8—2542no—
280C2×Q8C22×D12711yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—1271no—
284C2×D4—1271no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4—2541no—
290C2×D4—2541no—
291C2×D4C22×D12711yescharacteristic
292C2×D4C22×D12711yescharacteristic
293C22×C4—21no—
294C22×C4—1271no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4—2541no—
298C22×C4C22×D12711yescharacteristic
299Q16—1274no—
300Q16—2542no—
301SD16—22no—
302SD16—5081no—
303SD16—2542no—
304SD16—2542no—
305SD16—2542no—
306D8—41no—
307D8—2542no—
308M16—2542no—
309C2×C8—1271no—
310C2×C8—1271no—
311C2×C8—2542no—
312C2×C8C22×D12711yescharacteristic
313C2×C8C22×D12711yescharacteristic
314C4⋊C4—1271no—
315C4⋊C4—1271no—
316C4⋊C4—2541no—
317C4⋊C4—2541no—
318C4⋊C4—2541no—
319C4⋊C4C22×D12711yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—1271no—
322C22⋊C4—1271no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C42—1271no—
329C42—2541no—
330C42—2541no—
331C23—21no—
332C23—21no—
333C23—1271no—
334C23—2541no—
335C23D4×D12711yescharacteristic
336Q8—21no—
337Q8—1272no—
338Q8—5082no—
339Q8—2541no—
340Q8—2541no—
341Q8—2541no—
342Q8D4×D12712yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—42no—
348D4—81no—
349D4—10161no—
350D4—10161no—
351D4—5082no—
352D4—5082no—
353D4—5081no—
354D4—5082no—
355D4—5082no—
356D4—2541no—
357D4—2541no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—1271no—
364C2×C4—1271no—
365C2×C4—1271no—
366C2×C4—5082no—
367C2×C4—5082no—
368C2×C4—5082no—
369C2×C4—2541no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4—2541no—
374C2×C4D4×D12711yescharacteristic
375C2×C4C23×D12711yescharacteristic
376C8—21no—
377C8—21no—
378C8—1272no—
379C8—2541no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—5081no—
390C22—5081no—
391C22—5081no—
392C22—2541no—
393C22—2541no—
394C22—2541no—
395C22C2×D4×D12711yescharacteristic
396C4—21no—
397C4—22no—
398C4—41no—
399C4—5081no—
400C4—5081no—
401C4—2541no—
402C4—2541no—
403C4—2541no—
404C4—2542no—
405C4C2×D4×D12711yescharacteristic
406C4#1686125411yescharacteristic
407C2—41no—
408C2—81no—
409C2—81no—
410C2—10161no—
411C2—5081no—
412C2#2255050811yescharacteristic
413C2#2255067311yescharacteristic
414C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3271
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#16861238C2semidirect
2. C2×#16861229C2semidirect
3. #22550544C2semidirect
4. #22550533C2semidirect
5. #22550264C2semidirect
6. #22550128C2semidirect
7. #22550103C2semidirect
8. #22550097C2semidirect
9. #22550074C2semidirect
10. #22550040C2semidirect
11. #22550018C2semidirect
12. #22550016C2semidirect
13. #22549980C2semidirect
14. #22549944C2semidirect
15. #449×C127C2semidirect
83. #14523238D4semidirect
208. C127#3271semidirect
220. #449D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
C2×#168612292
#225505442
#225505332
#225502642
#225501282
#225501032
#225500972
#225500742
#225500402
#225500182
#225500162
#225499802
#225499442
#449×C1272
#3271127