Finite Group Databasebeta

← #30641993

Subgroups — #30641993

Order 16256.

560 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D508C211yesmaximal, characteristic
2C2×#16861166C211yesmaximal, characteristic, complemented
3#22549818C211yesmaximal, characteristic, complemented
4#22549811C211yesmaximal, characteristic, complemented
5#22549809C211yesmaximal, characteristic, complemented
6#22549805C211yesmaximal, characteristic, complemented
7#22549805C211yesmaximal, characteristic, complemented
8#22549802C211yesmaximal, characteristic, complemented
9#22549794C211yesmaximal, characteristic
10#22549793C211yesmaximal, characteristic
11C4×Dic127⋊C4C211yesmaximal, characteristic, complemented
12C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
13C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
14C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
15C4×C4⋊C4×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic
17C22×C4×D127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19C4×D508—24no—
20#16861166—24no—
21C2×C4⋊Dic127C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×(C2×D127)⋊C4C2211yescharacteristic
27C2×(C2×D127)⋊C4C2211yescharacteristic
28C2×(C2×D127)⋊C4C2211yescharacteristic
29C2×(C2×D127)⋊C4C2211yescharacteristic
30C2×(C2×D127)⋊C4C2211yescharacteristic
31C2×(C2×D127)⋊C4C2211yescharacteristic
32C2×(C2×D127)⋊C4C2211yescharacteristic
33C2×(C2×D127)⋊C4C2211yescharacteristic
34C2×(C2×D127)⋊C4C2211yescharacteristic
35C2×(C2×D127)⋊C4C2211yescharacteristic
36C2×C4×Dic127C2211yescharacteristic
37C2×C4×Dic127C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C2×C4⋊C4×C127C2211yescharacteristic
47C2×C4⋊C4×C127C2211yescharacteristic
48C2×C42×C127C2211yescharacteristic
49C2×C42×C127C2211yescharacteristic
50C2×C42×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52#95×C127C2211yescharacteristic
53C23×D127C2311yescharacteristic
54C23×D127C2311yescharacteristic
55C2×D508—22no—
56C2×D508—22no—
57C2×D508—22no—
58C22×Dic127C2311yescharacteristic
59C22×Dic127C2311yescharacteristic
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C2×C4×D127—22no—
65C2×C4×D127—22no—
66C2×C4×D127—22no—
67C2×C4×D127—22no—
68C2×C4×D127—22no—
69C2×C4×D127D414yes—
70C4⋊Dic127—22no—
71Dic127⋊C4—22no—
72Dic127⋊C4—22no—
73Dic127⋊C4—22no—
74Dic127⋊C4—22no—
75(C2×D127)⋊C4—22no—
76(C2×D127)⋊C4—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—
82(C2×D127)⋊C4—22no—
83(C2×D127)⋊C4—22no—
84(C2×D127)⋊C4—22no—
85C4×Dic127—22no—
86C4×Dic127—22no—
87C22×C4×C127C2311yescharacteristic
88C22×C4×C127C2311yescharacteristic
89C22×C4×C127C2311yescharacteristic
90C22×C4×C127C2311yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C22×C4×C127C2311yescharacteristic
93C22×C4×C127C2311yescharacteristic
94C4⋊C4×C127—22no—
95C4⋊C4×C127—22no—
96C42×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—21no—
106C22×D127—21no—
107C22×D127—21no—
108C22×D127—21no—
109C22×D127—21no—
110C22×D127—21no—
111C22×D127C2×D412yes—
112C22×D127#4112yes—
113D508—28no—
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×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127C2×D412yes—
131C2×Dic127#4112yes—
132C4×D127—28no—
133C4×D127—44no—
134C23×C127C2411yescharacteristic
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
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×C127C2×D412yes—
152C2×C4×C127#4112yes—
153C2×C4×C127#4112yes—
154C2×C4×C127#4112yes—
155C2×C4×C127#4112yes—
156C22×C127C22×D411yescharacteristic
157C22×C127C2×#4111yescharacteristic
158C22×C127C2×#4111yescharacteristic
159C22×C127C2×#4111yescharacteristic
160C22×C127C2×#4111yescharacteristic
161C22×C127C2×#4111yescharacteristic
162C22×C127C2×#4111yescharacteristic
163C2×D127—22no—
164C2×D127—22no—
165C2×D127—22no—
166C2×D127—22no—
167C2×D127—22no—
168C2×D127—22no—
169C2×D127—41no—
170C2×D127—41no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—41no—
175C2×D127—41no—
176C2×D127—41no—
177C4×C127—22no—
178C4×C127—22no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127—41no—
183C4×C127#12314yes—
184Dic127—22no—
185Dic127—22no—
186Dic127—41no—
187Dic127—41no—
188Dic127—41no—
189Dic127—41no—
190C2×C127C2×#12311yescharacteristic
191C2×C127#52511yescharacteristic
192C2×C127#52911yescharacteristic
193C2×C127#52911yescharacteristic
194C2×C127#52911yescharacteristic
195C2×C127#52911yescharacteristic
196C2×C127#53211yescharacteristic
197D127—42no—
198D127—42no—
199#2395—1271nomaximal, Sylow
200C127#239511yescharacteristic, complemented, Sylow
201C2×C4×D4—1271no—
202C2×#117—1271no—
203#400—1271no—
204#399—1271no—
205#398—1271no—
206#397—1271no—
207#397—1271no—
208#397—1271no—
209#394—1271no—
210#394—1271no—
211C4×C4⋊C4—1271no—
212C4×C4⋊C4D12711yescharacteristic, complemented
213C4×C22⋊C4—1271no—
214C4×C22⋊C4—1271no—
215C4×C22⋊C4—1271no—
216C22×D4—1271no—
217C23×C4—1271no—
218C23×C4—1271no—
219C4×D4—2544no—
220#117—2544no—
221C2×C4⋊C4—1271no—
222C2×C4⋊C4—1271no—
223C2×C4⋊C4—1271no—
224C2×C4⋊C4—1271no—
225C2×C4⋊C4—1271no—
226C2×C4⋊C4C2×D12711yescharacteristic
227C2×C4⋊C4C2×D12711yescharacteristic
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⋊C4—1271no—
236C2×C22⋊C4—1271no—
237C2×C22⋊C4—1271no—
238C2×C42—1271no—
239C2×C42—1271no—
240C2×C42C2×D12711yescharacteristic
241C2×C42C2×D12711yescharacteristic
242C2×C42C2×D12711yescharacteristic
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#95C2×D12711yescharacteristic
252#95C2×D12711yescharacteristic
253C24—1271no—
254C24—1271no—
255C2×D4—2542no—
256C2×D4—2542no—
257C2×D4—2542no—
258C22×C4—1271no—
259C22×C4—1271no—
260C22×C4—1274no—
261C22×C4—1271no—
262C22×C4—1271no—
263C22×C4—1271no—
264C22×C4—1271no—
265C22×C4—2542no—
266C22×C4—2542no—
267C22×C4—2542no—
268C22×C4—2542no—
269C22×C4—2542no—
270C22×C4C22×D12711yescharacteristic
271C22×C4C22×D12711yescharacteristic
272C22×C4C22×D12711yescharacteristic
273C22×C4C22×D12711yescharacteristic
274C22×C4C22×D12711yescharacteristic
275C22×C4C22×D12711yescharacteristic
276C22×C4C22×D12711yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—2542no—
280C4⋊C4—2542no—
281C4⋊C4—2542no—
282C4⋊C4—2542no—
283C4⋊C4—2542no—
284C22⋊C4—2542no—
285C22⋊C4—2542no—
286C22⋊C4—2542no—
287C22⋊C4—2542no—
288C22⋊C4—2542no—
289C22⋊C4—2542no—
290C22⋊C4—2542no—
291C22⋊C4—2542no—
292C22⋊C4—2542no—
293C22⋊C4—2542no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—2542no—
298C42—2542no—
299C23—1272no—
300C23—1272no—
301C23—2541no—
302C23—2541no—
303C23—2541no—
304C23—2541no—
305C23—2541no—
306C23—2541no—
307C23—2541no—
308C23—2541no—
309C23—2541no—
310C23—2541no—
311C23—2541no—
312C23—2541no—
313C23C23×D12711yescharacteristic
314D4—2548no—
315C2×C4—21no—
316C2×C4—21no—
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—1272no—
332C2×C4—1272no—
333C2×C4—5084no—
334C2×C4—2541no—
335C2×C4—2541no—
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—2548no—
351C2×C4D4×D12712yes—
352C2×C4#1452324812yes—
353C2×C4#1452324812yes—
354C2×C4#1452324812yes—
355C2×C4#1452324912yes—
356C22—5081no—
357C22—5081no—
358C22—5081no—
359C22—5081no—
360C22—5081no—
361C22—5081no—
362C22—5081no—
363C22—5081no—
364C22—2542no—
365C22—2542no—
366C22—2542no—
367C22—2542no—
368C22—2542no—
369C22—2542no—
370C22C2×D4×D12711yescharacteristic
371C22#41×D12711yescharacteristic
372C22#41×D12711yescharacteristic
373C22C2×#1452324811yescharacteristic
374C22C2×#1452324811yescharacteristic
375C22C2×#1452324811yescharacteristic
376C22C2×#1452324911yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—5081no—
384C4—5081no—
385C4—5081no—
386C4—5081no—
387C4—2542no—
388C4—2542no—
389C4#1686119414yes—
390C2—5082no—
391C2—5082no—
392C2C2×#1686119411yescharacteristic
393C2#2255038511yescharacteristic
394C2#2255040411yescharacteristic
395C2#2255041011yescharacteristic
396C2#2255041311yescharacteristic
397C2#2255041311yescharacteristic
398C2#2255042311yescharacteristic
subgroupG/H
centerC22×C4C22×D127
commutator, socle, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC4×C4⋊C4×C127C2
prime radicalC22×D508C22
hypercenterC4×C4⋊C4D127
hypocenterC127#2395
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16861166C2semidirect
3. #22549818C2semidirect
4. #22549811C2semidirect
5. #22549809C2semidirect
6. #22549805C2semidirect
7. #22549805C2semidirect
8. #22549802C2semidirect
11. C4×Dic127⋊C4C2semidirect
12. C4×(C2×D127)⋊C4C2semidirect
13. C4×(C2×D127)⋊C4C2semidirect
14. C4×(C2×D127)⋊C4C2semidirect
15. C4×C4⋊C4×C127C2semidirect
200. C127#2395semidirect
212. C4×C4⋊C4D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D5082
C2×#168611662
#225498182
#225498112
#225498092
#225498052
#225498022
#225497942
#225497932
C4×Dic127⋊C42
C4×(C2×D127)⋊C42
C4×C4⋊C4×C1272
#2395127