Finite Group Databasebeta

← #30641986

Subgroups — #30641986

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×D4×Dic127C211yesmaximal, characteristic
2C2×#16861165C211yesmaximal, characteristic, complemented
3#22549818C211yesmaximal, characteristic, complemented
4#22549812C211yesmaximal, characteristic, complemented
5#22549809C211yesmaximal, characteristic, complemented
6#22549804C211yesmaximal, characteristic, complemented
7#22549804C211yesmaximal, characteristic, complemented
8#22549801C211yesmaximal, characteristic, complemented
9#22549790C211yesmaximal, characteristic
10C4⋊C4×Dic127C211yesmaximal, characteristic, complemented
11#22549727C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
15#394×C127C211yesmaximal, characteristic
16C23×Dic127C2211yescharacteristic
17C23×Dic127C2211yescharacteristic
18D4×Dic127—24no—
19#16861165—24no—
20C2×C4⋊Dic127C2211yescharacteristic
21C2×C4⋊Dic127C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×C22⋊Dic127C2211yescharacteristic
27C2×C22⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127C2211yescharacteristic
29C2×C22⋊Dic127C2211yescharacteristic
30C2×C22⋊Dic127C2211yescharacteristic
31C2×C22⋊Dic127C2211yescharacteristic
32C2×C4×Dic127C2211yescharacteristic
33C2×C4×Dic127C2211yescharacteristic
34C2×C4×Dic127C2211yescharacteristic
35C2×C4×Dic127C2211yescharacteristic
36C2×C4×Dic127C2211yescharacteristic
37C127⋊#95C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C22×D4×C127C2211yescharacteristic
47C2×C4⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50C2×C22⋊C4×C127C2211yescharacteristic
51C2×C22⋊C4×C127C2211yescharacteristic
52#95×C127C2211yescharacteristic
53C22×Dic127—22no—
54C22×Dic127—22no—
55C22×Dic127—22no—
56C22×Dic127—22no—
57C22×Dic127—22no—
58C22×Dic127D414yes—
59C22×Dic127C2311yescharacteristic
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C4⋊Dic127—22no—
68C4⋊Dic127—22no—
69Dic127⋊C4—22no—
70Dic127⋊C4—22no—
71Dic127⋊C4—22no—
72Dic127⋊C4—22no—
73C22⋊Dic127—22no—
74C22⋊Dic127—22no—
75C22⋊Dic127—22no—
76C22⋊Dic127—22no—
77C22⋊Dic127—22no—
78C22⋊Dic127—22no—
79C4×Dic127—22no—
80C4×Dic127—22no—
81C4×Dic127—22no—
82C4×Dic127—22no—
83C4×Dic127—22no—
84C24×C127C2311yescharacteristic
85C24×C127C2311yescharacteristic
86C2×D4×C127—22no—
87C2×D4×C127—22no—
88C2×D4×C127—22no—
89C22×C4×C127C2311yescharacteristic
90C22×C4×C127C2311yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C22×C4×C127C2311yescharacteristic
93C22×C4×C127C2311yescharacteristic
94C4⋊C4×C127—22no—
95C22⋊C4×C127—22no—
96C22⋊C4×C127—22no—
97C22⋊C4×C127—22no—
98C22⋊C4×C127—22no—
99C2×Dic127—21no—
100C2×Dic127—21no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C2×Dic127—28no—
104C2×Dic127—21no—
105C2×Dic127—21no—
106C2×Dic127—21no—
107C2×Dic127—21no—
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—44no—
120C2×Dic127C2×D412yes—
121C2×Dic127C2×D412yes—
122C2×Dic127#4112yes—
123C2×Dic127#4112yes—
124C2×Dic127#4112yes—
125C23×C127—21no—
126C23×C127—21no—
127C23×C127—21no—
128C23×C127—21no—
129C23×C127—21no—
130C23×C127—21no—
131C23×C127—21no—
132C23×C127—21no—
133C23×C127—21no—
134C23×C127—21no—
135C23×C127—21no—
136C23×C127—21no—
137C23×C127C2×D412yes—
138C23×C127#4112yes—
139C23×C127C2411yescharacteristic
140D4×C127—28no—
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—21no—
153C2×C4×C127—21no—
154C2×C4×C127#4112yes—
155C2×C4×C127#4112yes—
156C22×C127—22no—
157C22×C127—22no—
158C22×C127—22no—
159C22×C127—22no—
160C22×C127—22no—
161C22×C127—22no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127C22×D411yescharacteristic
171C22×C127C2×#4111yescharacteristic
172C22×C127C2×#4111yescharacteristic
173C22×C127C2×#4111yescharacteristic
174C22×C127C2×#4111yescharacteristic
175C22×C127C2×#4111yescharacteristic
176C22×C127C2×#4111yescharacteristic
177C4×C127—22no—
178C4×C127—22no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127—41no—
182Dic127—22no—
183Dic127—22no—
184Dic127—41no—
185Dic127—41no—
186Dic127—41no—
187Dic127—41no—
188Dic127—41no—
189Dic127#12314yes—
190C2×C127—42no—
191C2×C127—42no—
192C2×C127C2×#12311yescharacteristic
193C2×C127#52511yescharacteristic
194C2×C127#52911yescharacteristic
195C2×C127#52911yescharacteristic
196C2×C127#52911yescharacteristic
197C2×C127#52911yescharacteristic
198C2×C127#53211yescharacteristic
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#394D12711yescharacteristic
211C4×C4⋊C4—1271no—
212C4×C4⋊C4—1271no—
213C4×C22⋊C4—1271no—
214C4×C22⋊C4—1271no—
215C4×C22⋊C4—1271no—
216C22×D4C2×D12711yescharacteristic
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⋊C4—1271no—
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⋊C4C2×D12711yescharacteristic
235C2×C22⋊C4C2×D12711yescharacteristic
236C2×C22⋊C4C2×D12711yescharacteristic
237C2×C22⋊C4C2×D12711yescharacteristic
238C2×C42—1271no—
239C2×C42—1271no—
240C2×C42—1271no—
241C2×C42—1271no—
242C2×C42—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
253C24C22×D12711yescharacteristic
254C24C22×D12711yescharacteristic
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C22×C4—1271no—
259C22×C4—1274no—
260C22×C4—1271no—
261C22×C4—1271no—
262C22×C4—1271no—
263C22×C4—1271no—
264C22×C4—1271no—
265C22×C4—1271no—
266C22×C4—1271no—
267C22×C4—2542no—
268C22×C4—2542no—
269C22×C4—2542no—
270C22×C4—2542no—
271C22×C4—2542no—
272C22×C4C22×D12711yescharacteristic
273C22×C4C22×D12711yescharacteristic
274C22×C4C22×D12711yescharacteristic
275C22×C4C22×D12711yescharacteristic
276C22×C4C22×D12711yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—2542no—
279C4⋊C4—2542no—
280C4⋊C4—2542no—
281C4⋊C4—2542no—
282C4⋊C4—2542no—
283C4⋊C4—2542no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—2542no—
289C22⋊C4—2542no—
290C22⋊C4—2542no—
291C22⋊C4—2542no—
292C22⋊C4—2542no—
293C22⋊C4—2542no—
294C42—2542no—
295C42—2542no—
296C42—2542no—
297C42—2542no—
298C42—2542no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23D4×D12712yes—
312C23#1452324712yes—
313C23C23×D12711yescharacteristic
314D4—28no—
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—1272no—
329C2×C4—1272no—
330C2×C4—1272no—
331C2×C4—1272no—
332C2×C4—1272no—
333C2×C4—5084no—
334C2×C4—2548no—
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—2541no—
351C2×C4—2541no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4#1452324712yes—
355C2×C4#1452324712yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22C2×D4×D12711yescharacteristic
371C22#41×D12711yescharacteristic
372C22#41×D12711yescharacteristic
373C22#41×D12711yescharacteristic
374C22C2×#1452324711yescharacteristic
375C22C2×#1452324711yescharacteristic
376C22C2×#1452324711yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—1274no—
383C4—5081no—
384C4—5081no—
385C4—5081no—
386C4—5081no—
387C4—5081no—
388C4—2542no—
389C4—2542no—
390C2—42no—
391C2—42no—
392C2#123×D12711yescharacteristic
393C2#2255038211yescharacteristic
394C2#2255040311yescharacteristic
395C2#2255040511yescharacteristic
396C2#2255040711yescharacteristic
397C2#2255040711yescharacteristic
398C2#2255042111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#394×C127C2
prime radicalC22×D4×C127C22
hypercenter#394D127
hypocenterC127#2395
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#16861165C2semidirect
3. #22549818C2semidirect
4. #22549812C2semidirect
5. #22549809C2semidirect
6. #22549804C2semidirect
7. #22549804C2semidirect
8. #22549801C2semidirect
10. C4⋊C4×Dic127C2semidirect
11. #22549727C2semidirect
12. C22⋊C4×Dic127C2semidirect
13. C22⋊C4×Dic127C2semidirect
14. C22⋊C4×Dic127C2semidirect
200. C127#2395semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1272
C2×#168611652
#225498182
#225498122
#225498092
#225498042
#225498012
#225497902
C4⋊C4×Dic1272
#225497272
C22⋊C4×Dic1272
#394×C1272
#2395127