Finite Group Databasebeta

← #30648398

Subgroups — #30648398

Order 16256.

484 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861230C211yesmaximal, characteristic, complemented
2C2×#16861217C211yesmaximal, characteristic, complemented
3#22550456C211yesmaximal, characteristic, complemented
4#22550455C211yesmaximal, characteristic, complemented
5#22550123C211yesmaximal, characteristic, complemented
6#22550122C211yesmaximal, characteristic, complemented
7#22550119C211yesmaximal, characteristic, complemented
8#22550116C211yesmaximal, characteristic, complemented
9#22550115C211yesmaximal, characteristic, complemented
10#22550111C211yesmaximal, characteristic, complemented
11#22550110C211yesmaximal, characteristic, complemented
12#107×D127C211yesmaximal, characteristic, complemented
13#22549876C211yesmaximal, characteristic, complemented
14#22549874C211yesmaximal, characteristic, complemented
15#468×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C2×#14523247C2211yescharacteristic
19#16861230—24no—
20C2×#14523238C2211yescharacteristic
21C2×#14523237C2211yescharacteristic
22C2×#14523233C2211yescharacteristic
23#16861217—24no—
24C2×#14523231C2211yescharacteristic
25C2×#14523231C2211yescharacteristic
26C22×C127⋊C8C2211yescharacteristic
27C2×C8×D127C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic
29#16861189C2211yescharacteristic
30#16861188C2211yescharacteristic
31#16861187—21no—
32#16861186—21no—
33#16861185C2211yescharacteristic
34#16861184—21no—
35#16861184C2211yescharacteristic
36#16861183C2211yescharacteristic
37#16861182—21no—
38#16861182—21no—
39#16861182—21no—
40#16861181—21no—
41#16861180—21no—
42#16861180C2211yescharacteristic
43C4⋊C4×D127C2211yescharacteristic
44C4⋊C4×D127C2211yescharacteristic
45#16861143C2211yescharacteristic
46#16861142C2211yescharacteristic
47#16861142C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861133C2211yescharacteristic
50#16861132C2211yescharacteristic
51#16861131C2211yescharacteristic
52#16861130C2211yescharacteristic
53#16861129C2211yescharacteristic
54C2×SD16×C127C2211yescharacteristic
55C2×M16×C127C2211yescharacteristic
56C22⋊Q8×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic
58#107×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60#102×C127C2211yescharacteristic
61#14523249—22no—
62#14523249—22no—
63#14523248—22no—
64#14523248—22no—
65#14523247—22no—
66#14523247—22no—
67C2×Dic254D411yescharacteristic
68C2×Dic254C2311yescharacteristic
69C2×D508—21no—
70C2×D508—21no—
71C2×D508D411yescharacteristic
72C2×D508C2311yescharacteristic
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127D411yescharacteristic
77C2×C22⋊D127D411yescharacteristic
78C22×Dic127—21no—
79C2×C4×D127—21no—
80C2×C4×D127—21no—
81C2×C4×D127—21no—
82C2×C4×D127C2311yescharacteristic
83C2×C4×D127C2311yescharacteristic
84#14523238—22no—
85#14523237—22no—
86#14523233—22no—
87#14523231—22no—
88#14523231—22no—
89C2×C127⋊C8—22no—
90C2×C127⋊C8C2311yescharacteristic
91C2×C127⋊C8C2311yescharacteristic
92C8×D127—22no—
93C8×D127D414yescomplemented
94C4⋊Dic127—21no—
95C4⋊Dic127C2311yescharacteristic
96C4⋊Dic127C2311yescharacteristic
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100Dic127⋊C4—21no—
101C22⋊Dic127—21no—
102C22⋊Dic127—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107C2×Q8×C127C2311yescharacteristic
108C2×D4×C127—21no—
109C2×D4×C127C2311yescharacteristic
110C22×C4×C127C2311yescharacteristic
111SD16×C127—22no—
112M16×C127—22no—
113C2×C8×C127C2311yescharacteristic
114C2×C8×C127C2311yescharacteristic
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—
126D508—22no—
127D508—21no—
128D508—21no—
129D508—42no—
130D508—81no—
131C22⋊D127—22no—
132C22⋊D127—22no—
133C22⋊D127—42no—
134C22⋊D127—81no—
135C22⋊D127—81no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C2×Dic127—42no—
140C2×Dic127C2×D411yescharacteristic
141C2×Dic127C2×D411yescharacteristic
142C4×D127—21no—
143C4×D127—21no—
144C4×D127—42no—
145C4×D127—81no—
146C4×D127—81no—
147C4×D127C2×D412yes—
148C4×D127#4112yes—
149C127⋊C8—21no—
150C127⋊C8C2×D412yes—
151C23×C127—21no—
152C23×C127C2×D411yescharacteristic
153Q8×C127—21no—
154Q8×C127—21no—
155D4×C127—21no—
156D4×C127—21no—
157D4×C127—81no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127C2×D411yescharacteristic
164C2×C4×C127C2411yescharacteristic
165C8×C127—21no—
166C8×C127C2×D412yes—
167C22×C127—21no—
168C22×C127—21no—
169C22×C127—21no—
170C22×C127—41no—
171C22×C127—41no—
172C22×C127—41no—
173C22×C127C22×D411yescharacteristic
174C2×D127—21no—
175C2×D127—21no—
176C2×D127—21no—
177C2×D127—21no—
178C2×D127—21no—
179C2×D127—41no—
180C2×D127—41no—
181C2×D127—41no—
182C2×D127#12112yes—
183C4×C127—21no—
184C4×C127—41no—
185C4×C127—41no—
186C4×C127—41no—
187C4×C127C22×D411yescharacteristic
188C4×C127C2×#4111yescharacteristic
189Dic127—21no—
190Dic127—41no—
191Dic127—41no—
192Dic127—41no—
193Dic127#12112yes—
194C2×C127—41no—
195C2×C127—81no—
196C2×C127C2×#12111yescharacteristic
197C2×C127#57611yescharacteristic
198C2×C127#57811yescharacteristic
199D127—22no—
200D127—41no—
201D127—81no—
202#3061—1271nomaximal, Sylow
203C127#306111yescharacteristic, complemented, Sylow
204C2×#135—1271no—
205C2×#131—1271no—
206#537—1271no—
207#537—1271no—
208#468—1271no—
209#468—1271no—
210#468—1271no—
211#468D12711yescharacteristic, complemented
212#467—1271no—
213#467—1271no—
214#466—1271no—
215#466—1271no—
216C2×#107—1271no—
217#417—1271no—
218#417—1271no—
219C2×#41—1271no—
220C2×#41—1271no—
221C2×#41—1271no—
222#135—2544no—
223C2×Q16—1271no—
224C2×SD16—1271no—
225C2×SD16C2×D12711yescharacteristic
226C2×D8—1271no—
227#131—2544no—
228C2×M16—1271no—
229C2×M16—1271no—
230C2×M16C2×D12711yescharacteristic
231C22×C8—1271no—
232C22×C8—1271no—
233C22×C8—1271no—
234C22⋊Q8—1271no—
235C22⋊Q8—1271no—
236C22⋊Q8—1271no—
237C22⋊Q8—2541no—
238C22⋊Q8—2541no—
239C22⋊Q8C2×D12711yescharacteristic
240#121—1271no—
241#121—1271no—
242#121—1271no—
243#121—2541no—
244#121—2541no—
245#121—2541no—
246#121—2541no—
247#121—2541no—
248#121—2541no—
249#121C2×D12711yescharacteristic
250C2×C4⋊C4—1271no—
251C2×C4⋊C4—1271no—
252#107—1271no—
253#107—1271no—
254#107—1271no—
255#107C2×D12711yescharacteristic
256#103—1271no—
257#103—1271no—
258#103—1271no—
259#103C2×D12711yescharacteristic
260#102—1271no—
261#102—1271no—
262#102—1271no—
263#102C2×D12711yescharacteristic
264#41—2542no—
265#41—2542no—
266#41—2542no—
267#41—2542no—
268#41—2542no—
269#41—2542no—
270C2×Q8—1271no—
271C2×Q8—1271no—
272C2×Q8C22×D12711yescharacteristic
273C2×D4—21no—
274C2×D4—1271no—
275C2×D4—1271no—
276C2×D4—1271no—
277C2×D4—1271no—
278C2×D4—2541no—
279C2×D4—2541no—
280C2×D4—2541no—
281C2×D4—2541no—
282C2×D4—2541no—
283C2×D4C22×D12711yescharacteristic
284C22×C4—1271no—
285C22×C4—1271no—
286C22×C4—2541no—
287C22×C4—2541no—
288C22×C4—2541no—
289C22×C4—2541no—
290C22×C4C22×D12711yescharacteristic
291Q16—2542no—
292SD16—22no—
293SD16—2542no—
294D8—2542no—
295M16—22no—
296M16—2542no—
297M16—2542no—
298C2×C8—1271no—
299C2×C8—1274no—
300C2×C8—1271no—
301C2×C8—2542no—
302C2×C8—2542no—
303C2×C8C22×D12711yescharacteristic
304C2×C8C22×D12711yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—1271no—
307C4⋊C4—1271no—
308C4⋊C4—2541no—
309C4⋊C4—2541no—
310C4⋊C4—2541no—
311C4⋊C4—2541no—
312C4⋊C4—2541no—
313C4⋊C4C22×D12711yescharacteristic
314C4⋊C4C22×D12711yescharacteristic
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—2541no—
318C22⋊C4—2541no—
319C22⋊C4—2541no—
320C22⋊C4—2541no—
321C22⋊C4—2541no—
322C22⋊C4—2541no—
323C23—21no—
324C23—1271no—
325C23—1271no—
326C23—2541no—
327C23D4×D12711yescharacteristic
328Q8—21no—
329Q8—21no—
330Q8—2542no—
331Q8—2541no—
332Q8—2541no—
333D4—21no—
334D4—21no—
335D4—81no—
336D4—10161no—
337D4—10161no—
338D4—10161no—
339D4—5082no—
340D4—5082no—
341D4—2542no—
342D4—2542no—
343D4—2542no—
344D4—2541no—
345D4—2541no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—1271no—
352C2×C4—1272no—
353C2×C4—1272no—
354C2×C4—1271no—
355C2×C4—10161no—
356C2×C4—10161no—
357C2×C4—5082no—
358C2×C4—5082no—
359C2×C4—2541no—
360C2×C4—2541no—
361C2×C4—2541no—
362C2×C4—2541no—
363C2×C4—2541no—
364C2×C4D4×D12711yescharacteristic
365C2×C4C23×D12711yescharacteristic
366C8—21no—
367C8—1272no—
368C8—2541no—
369C8D4×D12712yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—1272no—
377C22—5081no—
378C22—5081no—
379C22—5081no—
380C22—2541no—
381C22—2541no—
382C22—2541no—
383C22—2541no—
384C22—2541no—
385C22C2×D4×D12711yescharacteristic
386C4—21no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—1272no—
391C4—5081no—
392C4—5081no—
393C4—5081no—
394C4—2541no—
395C4C2×D4×D12711yescharacteristic
396C4#41×D12711yescharacteristic
397C2—41no—
398C2—81no—
399C2—10161no—
400C2—5081no—
401C2—2542no—
402C2#121×D12711yescharacteristic
403C2#2255068011yescharacteristic
404C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#468×C127C2
socleC22×C127C22×D4
hypercenter#468D127
hypocenterC127#3061
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#16861230C2semidirect
2. C2×#16861217C2semidirect
3. #22550456C2semidirect
4. #22550455C2semidirect
5. #22550123C2semidirect
6. #22550122C2semidirect
7. #22550119C2semidirect
8. #22550116C2semidirect
9. #22550115C2semidirect
10. #22550111C2semidirect
11. #22550110C2semidirect
12. #107×D127C2semidirect
13. #22549876C2semidirect
14. #22549874C2semidirect
15. #468×C127C2semidirect
93. C8×D127D4semidirect
203. C127#3061semidirect
211. #468D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612302
C2×#168612172
#225504562
#225504552
#225501232
#225501222
#225501192
#225501162
#225501152
#225501112
#225501102
#107×D1272
#225498762
#225498742
#468×C1272
#3061127