Finite Group Databasebeta

← #30642017

Subgroups — #30642017

Order 16256.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic254C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D127C211yesmaximal, characteristic
3#22549818C211yesmaximal, characteristic, complemented
4#22549817C211yesmaximal, characteristic, complemented
5#22549805C211yesmaximal, characteristic, complemented
6#22549804C211yesmaximal, characteristic, complemented
7#22549792C211yesmaximal, characteristic
8#22549791C211yesmaximal, characteristic
9#22549789C211yesmaximal, characteristic, complemented
10#22549788C211yesmaximal, characteristic, complemented
11C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
12C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
14#22549722C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
16C22×Dic254C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19C4×Dic254—24no—
20C4×C22⋊D127—24no—
21C2×C4⋊Dic127C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×C22⋊Dic127C2211yescharacteristic
26C2×C22⋊Dic127C2211yescharacteristic
27C2×C22⋊Dic127C2211yescharacteristic
28C2×(C2×D127)⋊C4C2211yescharacteristic
29C2×(C2×D127)⋊C4C2211yescharacteristic
30C2×(C2×D127)⋊C4C2211yescharacteristic
31C2×(C2×D127)⋊C4C2211yescharacteristic
32C2×(C2×D127)⋊C4C2211yescharacteristic
33C2×C4×Dic127C2211yescharacteristic
34C2×C4×Dic127C2211yescharacteristic
35C2×C4×Dic127C2211yescharacteristic
36C127⋊#95C2211yescharacteristic
37C127⋊#95C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C23×C4×C127C2211yescharacteristic
47C2×C22⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C42×C127C2211yescharacteristic
50C2×C42×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52#95×C127C2211yescharacteristic
53C23×D127C2311yescharacteristic
54C2×Dic254—22no—
55C2×Dic254—22no—
56C2×Dic254—22no—
57C2×C22⋊D127—22no—
58C2×C22⋊D127—22no—
59C2×C22⋊D127—22no—
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C2×C4×D127—22no—
68C2×C4×D127—22no—
69C2×C4×D127—22no—
70C4⋊Dic127—22no—
71Dic127⋊C4—22no—
72Dic127⋊C4—22no—
73Dic127⋊C4—22no—
74C22⋊Dic127—22no—
75C22⋊Dic127—22no—
76C22⋊Dic127—22no—
77(C2×D127)⋊C4—22no—
78(C2×D127)⋊C4—22no—
79(C2×D127)⋊C4—22no—
80(C2×D127)⋊C4—22no—
81(C2×D127)⋊C4—22no—
82C4×Dic127—22no—
83C4×Dic127—22no—
84C4×Dic127D414yes—
85C24×C127C2311yescharacteristic
86C22×C4×C127—22no—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127C2311yescharacteristic
90C22×C4×C127C2311yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C22×C4×C127C2311yescharacteristic
93C22×C4×C127C2311yescharacteristic
94C22×C4×C127C2311yescharacteristic
95C22⋊C4×C127—22no—
96C22⋊C4×C127—22no—
97C42×C127—22no—
98C42×C127—22no—
99C22×D127—21no—
100C22×D127—21no—
101C22×D127—21no—
102C22×D127—21no—
103C22×D127—21no—
104C22×D127—21no—
105C22×D127#4112yes—
106Dic254—28no—
107C22⋊D127—28no—
108C2×Dic127—21no—
109C2×Dic127—21no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C2×Dic127—21no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127C2×D412yes—
127C2×Dic127C2×D412yes—
128C2×Dic127#4112yes—
129C4×D127—44no—
130C23×C127—21no—
131C23×C127—21no—
132C23×C127—21no—
133C23×C127—21no—
134C23×C127—21no—
135C23×C127—21no—
136C23×C127#4112yes—
137C23×C127C2411yescharacteristic
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—44no—
153C2×C4×C127C2×D412yes—
154C2×C4×C127#4112yes—
155C2×C4×C127#4112yes—
156C2×C4×C127#4112yes—
157C22×C127—22no—
158C22×C127—22no—
159C22×C127—22no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127C22×D411yescharacteristic
165C22×C127C2×#4111yescharacteristic
166C22×C127C2×#4111yescharacteristic
167C22×C127C2×#4111yescharacteristic
168C22×C127C2×#4111yescharacteristic
169C22×C127C2×#4111yescharacteristic
170C22×C127C2×#4111yescharacteristic
171C2×D127—22no—
172C2×D127—22no—
173C2×D127—22no—
174C2×D127—41no—
175C2×D127—41no—
176C2×D127—41no—
177C2×D127—41no—
178C4×C127—22no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127—41no—
183C4×C127#12414yes—
184Dic127—22no—
185Dic127—22no—
186Dic127—22no—
187Dic127—41no—
188Dic127—41no—
189Dic127—41no—
190Dic127—41no—
191C2×C127—42no—
192C2×C127#52511yescharacteristic
193C2×C127#52511yescharacteristic
194C2×C127C2×#12411yescharacteristic
195C2×C127#52911yescharacteristic
196C2×C127#52911yescharacteristic
197C2×C127#52911yescharacteristic
198C2×C127#52911yescharacteristic
199D127—42no—
200#2398—1271nomaximal, Sylow
201C127#239811yescharacteristic, complemented, Sylow
202C2×C4×Q8—1271no—
203C2×C4×D4—1271no—
204#400—1271no—
205#400—1271no—
206#397—1271no—
207#397—1271no—
208#394—1271no—
209#394—1271no—
210#393—1271no—
211#393—1271no—
212C4×C22⋊C4—1271no—
213C4×C22⋊C4—1271no—
214C4×C22⋊C4—1271no—
215C4×C22⋊C4D12711yescharacteristic, complemented
216#376—1271no—
217C22×Q8—1271no—
218C22×D4—1271no—
219C23×C4—1271no—
220C23×C4C2×D12711yescharacteristic
221C4×Q8—2544no—
222C4×D4—2544no—
223C2×C4⋊C4—1271no—
224C2×C4⋊C4—1271no—
225C2×C4⋊C4—1271no—
226C2×C4⋊C4—1271no—
227C2×C22⋊C4—1271no—
228C2×C22⋊C4—1271no—
229C2×C22⋊C4—1271no—
230C2×C22⋊C4—1271no—
231C2×C22⋊C4—1271no—
232C2×C22⋊C4—1271no—
233C2×C22⋊C4—1271no—
234C2×C22⋊C4—1271no—
235C2×C22⋊C4C2×D12711yescharacteristic
236C2×C22⋊C4C2×D12711yescharacteristic
237C2×C42—1271no—
238C2×C42—1271no—
239C2×C42—1271no—
240C2×C42C2×D12711yescharacteristic
241C2×C42C2×D12711yescharacteristic
242#95—1271no—
243#95—1271no—
244#95—1271no—
245#95—1271no—
246#95—1271no—
247#95—1271no—
248#95—1271no—
249#95—1271no—
250#95—1271no—
251#95—1271no—
252#95C2×D12711yescharacteristic
253#95C2×D12711yescharacteristic
254C24—1271no—
255C24C22×D12711yescharacteristic
256C2×Q8—2542no—
257C2×Q8—2542no—
258C2×Q8—2542no—
259C2×D4—2542no—
260C2×D4—2542no—
261C2×D4—2542no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—1271no—
266C22×C4—1271no—
267C22×C4—1271no—
268C22×C4—1271no—
269C22×C4—1271no—
270C22×C4—1271no—
271C22×C4—1271no—
272C22×C4—2542no—
273C22×C4—2542no—
274C22×C4—2542no—
275C22×C4C22×D12711yescharacteristic
276C22×C4C22×D12711yescharacteristic
277C22×C4C22×D12711yescharacteristic
278C22×C4C22×D12711yescharacteristic
279C22×C4C22×D12711yescharacteristic
280C22×C4C22×D12711yescharacteristic
281C4⋊C4—2542no—
282C4⋊C4—2542no—
283C4⋊C4—2542no—
284C4⋊C4—2542no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—2542no—
288C22⋊C4—2542no—
289C22⋊C4—2542no—
290C22⋊C4—2542no—
291C22⋊C4—2542no—
292C22⋊C4—2542no—
293C22⋊C4—2542no—
294C22⋊C4—2542no—
295C42—22no—
296C42—22no—
297C42—1274no—
298C42—2542no—
299C42—2542no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—1272no—
307C23—2541no—
308C23—2541no—
309C23—2541no—
310C23—2541no—
311C23—2541no—
312C23—2541no—
313C23#1452324812yes—
314C23C23×D12711yescharacteristic
315Q8—2548no—
316D4—2548no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—1272no—
333C2×C4—1272no—
334C2×C4—1272no—
335C2×C4—5084no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4—2541no—
339C2×C4—2541no—
340C2×C4—2541no—
341C2×C4—2541no—
342C2×C4—2541no—
343C2×C4—2541no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4—2541no—
347C2×C4—2541no—
348C2×C4—2541no—
349C2×C4—2541no—
350C2×C4—2541no—
351C2×C4—2541no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4D4×D12712yes—
355C2×C4#1452324712yes—
356C2×C4#1452324812yes—
357C2×C4#1452324812yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—5081no—
366C22—5081no—
367C22—5081no—
368C22—5081no—
369C22—2542no—
370C22—2542no—
371C22—2542no—
372C22C2×D4×D12711yescharacteristic
373C22#41×D12711yescharacteristic
374C22#41×D12711yescharacteristic
375C22C2×#1452324711yescharacteristic
376C22C2×#1452324811yescharacteristic
377C22C2×#1452324811yescharacteristic
378C22C2×#1452324811yescharacteristic
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—5081no—
385C4—5081no—
386C4—5081no—
387C4—5081no—
388C4—2542no—
389C4—2542no—
390C4—2542no—
391C4#1686119514yes—
392C2—42no—
393C2—5082no—
394C2#2255038511yescharacteristic
395C2#2255038511yescharacteristic
396C2C2×#1686119511yescharacteristic
397C2#2255040411yescharacteristic
398C2#2255040611yescharacteristic
399C2#2255040811yescharacteristic
400C2#2255041011yescharacteristic
subgroupG/H
centerC22×C4C22×D127
commutator, socle, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC4×C22⋊C4×C127C2
prime radicalC22×C22⋊D127C22
hypercenterC4×C22⋊C4D127
hypocenterC127#2398
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic254C2semidirect
3. #22549818C2semidirect
4. #22549817C2semidirect
5. #22549805C2semidirect
6. #22549804C2semidirect
9. #22549789C2semidirect
10. #22549788C2semidirect
11. C4×(C2×D127)⋊C4C2semidirect
12. C4×(C2×D127)⋊C4C2semidirect
13. C4×C22⋊Dic127C2semidirect
14. #22549722C2semidirect
15. C4×C22⋊C4×C127C2semidirect
201. C127#2398semidirect
215. C4×C22⋊C4D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic2542
C2×C4×C22⋊D1272
#225498182
#225498172
#225498052
#225498042
#225497922
#225497912
#225497892
#225497882
C4×(C2×D127)⋊C42
C4×C22⋊Dic1272
#225497222
C4×C22⋊C4×C1272
#2398127