Finite Group Databasebeta

← #30641409

Subgroups — #30641409

Order 16256.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861186C211yesmaximal, characteristic
2C2×C4×Dic254C211yesmaximal, characteristic, complemented
3C2×#16861165C211yesmaximal, characteristic
4#22549780C211yesmaximal, characteristic, complemented
5#22549773C211yesmaximal, characteristic, complemented
6#22549772C211yesmaximal, characteristic, complemented
7#22549771C211yesmaximal, characteristic, complemented
8#22549769C211yesmaximal, characteristic
9#22549764C211yesmaximal, characteristic
10#22549764C211yesmaximal, characteristic
11#22549761C211yesmaximal, characteristic, complemented
12#22549760C211yesmaximal, characteristic, complemented
13#22549734C211yesmaximal, characteristic
14#22549727C211yesmaximal, characteristic, complemented
15#388×C127C211yesmaximal, characteristic
16C22×Dic254C2211yescharacteristic
17C23×Dic127C2211yescharacteristic
18#16861186C418yescomplemented
19C4×Dic254—24no—
20#16861165—24no—
21C2×C4⋊Dic127C2211yescharacteristic
22C2×C4⋊Dic127C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×Dic127⋊C4C2211yescharacteristic
27C2×Dic127⋊C4C2211yescharacteristic
28C2×Dic127⋊C4C2211yescharacteristic
29C2×Dic127⋊C4C2211yescharacteristic
30C2×Dic127⋊C4C2211yescharacteristic
31C2×C22⋊Dic127C2211yescharacteristic
32C2×C22⋊Dic127C2211yescharacteristic
33C2×C22⋊Dic127C2211yescharacteristic
34C2×C4×Dic127C2211yescharacteristic
35C2×C4×Dic127C2211yescharacteristic
36C2×C4×Dic127C2211yescharacteristic
37C2×C4×Dic127C2211yescharacteristic
38C2×C4×Dic127C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C127⋊#95C2211yescharacteristic
47C2×C4⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50C2×C22⋊C4×C127C2211yescharacteristic
51C2×C42×C127C2211yescharacteristic
52#95×C127C2211yescharacteristic
53#95×C127C2211yescharacteristic
54C2×Dic254—22no—
55C2×Dic254—22no—
56C2×Dic254C2×C414yes—
57C22×Dic127—22no—
58C22×Dic127—22no—
59C22×Dic127C2×C414yes—
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C4⋊Dic127—22no—
69C4⋊Dic127C2×C414yes—
70Dic127⋊C4—22no—
71Dic127⋊C4—22no—
72Dic127⋊C4—22no—
73Dic127⋊C4—22no—
74Dic127⋊C4—22no—
75Dic127⋊C4—41no—
76Dic127⋊C4C2×C414yes—
77Dic127⋊C4C2×C414yes—
78C22⋊Dic127—22no—
79C22⋊Dic127—41no—
80C22⋊Dic127C2×C414yes—
81C4×Dic127—22no—
82C4×Dic127—22no—
83C4×Dic127—41no—
84C4×Dic127—41no—
85C4×Dic127D414yes—
86C24×C127C2311yescharacteristic
87C22×C4×C127C2311yescharacteristic
88C22×C4×C127C2311yescharacteristic
89C22×C4×C127C2311yescharacteristic
90C22×C4×C127C2311yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C22×C4×C127C2311yescharacteristic
93C4⋊C4×C127—41no—
94C22⋊C4×C127—22no—
95C22⋊C4×C127—41no—
96C22⋊C4×C127C2×C414yes—
97C42×C127—22no—
98Dic254—28no—
99C2×Dic127—21no—
100C2×Dic127—21no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C2×Dic127—21no—
104C2×Dic127—21no—
105C2×Dic127—21no—
106C2×Dic127—21no—
107C2×Dic127—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×Dic127C22×C412yes—
118C2×Dic127C22×C412yes—
119C2×Dic127C22×C412yes—
120C2×Dic127C22×C412yes—
121C2×Dic127C2×D412yes—
122C2×Dic127C2×D412yes—
123C2×Dic127#4112yes—
124C23×C127—21no—
125C23×C127—21no—
126C23×C127—21no—
127C23×C127—21no—
128C23×C127—21no—
129C23×C127—21no—
130C23×C127C22×C412yes—
131C23×C127C2411yescharacteristic
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
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×C127C22×C412yes—
148C2×C4×C127C22×C412yes—
149C2×C4×C127C2×D412yes—
150C22×C127—22no—
151C22×C127—22no—
152C22×C127—22no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127C23×C411yescharacteristic
158C22×C127C22×D411yescharacteristic
159C22×C127C2×#4111yescharacteristic
160C22×C127H(5,𝔽2)11yescharacteristic
161C22×C1272−1+411yescharacteristic
162C22×C1272−1+411yescharacteristic
163C22×C1272−1+411yescharacteristic
164C4×C127—22no—
165C4×C127—22no—
166C4×C127—22no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127—41no—
170Dic127—22no—
171Dic127—22no—
172Dic127—22no—
173Dic127—22no—
174Dic127—41no—
175Dic127—41no—
176Dic127—41no—
177Dic127C4×D414yes—
178C2×C127—42no—
179C2×C127C2×C4×D411yescharacteristic
180C2×C127#51911yescharacteristic
181C2×C127#52011yescharacteristic
182C2×C127#53611yescharacteristic
183C2×C127#53711yescharacteristic
184C2×C127#54111yescharacteristic
185C2×C127#54211yescharacteristic
186#2323—1271nomaximal, Sylow
187C127#232311yescharacteristic, complemented, Sylow
188C2×C22⋊Q8—1271no—
189C2×C4×Q8—1271no—
190C2×#117—1271no—
191#391—1271no—
192#389—1271no—
193#389—1271no—
194#389—1271no—
195#388—1271no—
196#388—1271no—
197#388—1271no—
198#388D12711yescharacteristic
199#387—1271no—
200#387—1271no—
201#380—1271no—
202C4×C4⋊C4—1271no—
203C22×Q8—1271no—
204C23×C4—1271no—
205C22⋊Q8—1278no—
206C4×Q8—2544no—
207#117—2544no—
208C2×C4⋊C4—1271no—
209C2×C4⋊C4—1271no—
210C2×C4⋊C4—1271no—
211C2×C4⋊C4—1271no—
212C2×C4⋊C4—1271no—
213C2×C4⋊C4—1271no—
214C2×C4⋊C4—1271no—
215C2×C4⋊C4—1271no—
216C2×C4⋊C4—1271no—
217C2×C4⋊C4—1271no—
218C2×C4⋊C4C2×D12711yescharacteristic
219C2×C22⋊C4—1271no—
220C2×C22⋊C4—1271no—
221C2×C22⋊C4—1271no—
222C2×C22⋊C4C2×D12711yescharacteristic
223C2×C22⋊C4C2×D12711yescharacteristic
224C2×C22⋊C4C2×D12711yescharacteristic
225C2×C42—1271no—
226C2×C42—1271no—
227C2×C42—1271no—
228C2×C42—1271no—
229C2×C42—1271no—
230C2×C42C2×D12711yescharacteristic
231#95—1271no—
232#95—1271no—
233#95—1271no—
234#95—1271no—
235#95—1271no—
236#95—1271no—
237#95—1271no—
238#95—1271no—
239#95C2×D12711yescharacteristic
240#95C2×D12711yescharacteristic
241C24C22×D12711yescharacteristic
242C2×Q8—1274no—
243C2×Q8—2542no—
244C2×Q8—2542no—
245C22×C4—1271no—
246C22×C4—1274no—
247C22×C4—1271no—
248C22×C4—1271no—
249C22×C4—1271no—
250C22×C4—1271no—
251C22×C4—1271no—
252C22×C4—1271no—
253C22×C4—1271no—
254C22×C4—2542no—
255C22×C4—2542no—
256C22×C4C22×D12711yescharacteristic
257C22×C4C22×D12711yescharacteristic
258C22×C4C22×D12711yescharacteristic
259C22×C4C22×D12711yescharacteristic
260C22×C4C22×D12711yescharacteristic
261C22×C4C22×D12711yescharacteristic
262C4⋊C4—41no—
263C4⋊C4—1274no—
264C4⋊C4—1274no—
265C4⋊C4—1274no—
266C4⋊C4—5081no—
267C4⋊C4—2542no—
268C4⋊C4—2542no—
269C4⋊C4—2542no—
270C4⋊C4—2542no—
271C4⋊C4—2542no—
272C4⋊C4—2542no—
273C22⋊C4—22no—
274C22⋊C4—41no—
275C22⋊C4—1274no—
276C22⋊C4—5081no—
277C22⋊C4—2542no—
278C22⋊C4C4×D12714yes—
279C42—22no—
280C42—1274no—
281C42—5081no—
282C42—5081no—
283C42—2542no—
284C42—2542no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23C2×C4×D12712yes—
292C23C23×D12711yescharacteristic
293Q8—2548no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—1272no—
310C2×C4—1272no—
311C2×C4—1272no—
312C2×C4—1272no—
313C2×C4—1272no—
314C2×C4—1272no—
315C2×C4—1272no—
316C2×C4—2548no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4—2541no—
332C2×C4—2541no—
333C2×C4—2541no—
334C2×C4C2×C4×D12712yes—
335C2×C4C2×C4×D12712yes—
336C2×C4D4×D12712yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22C22×C4×D12711yescharacteristic
345C22C2×D4×D12711yescharacteristic
346C22#41×D12711yescharacteristic
347C22#1686125411yescharacteristic
348C22#1686125611yescharacteristic
349C22#1686125611yescharacteristic
350C22#1686125611yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—1274no—
358C4—5081no—
359C4—5081no—
360C4—5081no—
361C4—2542no—
362C4—2542no—
363C4—2542no—
364C4—2542no—
365C2—42no—
366C2C4×D4×D12711yescharacteristic
367C2#2255035311yescharacteristic
368C2#2255035511yescharacteristic
369C2#2255044811yescharacteristic
370C2#2255045511yescharacteristic
371C2#2255048311yescharacteristic
372C2#2255049011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#388×C127C2
socle, cosocleC23×C127C24
prime radicalC24×C127C23
hypercenter#388D127
hypocenterC127#2323
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic254C2semidirect
4. #22549780C2semidirect
5. #22549773C2semidirect
6. #22549772C2semidirect
7. #22549771C2semidirect
11. #22549761C2semidirect
12. #22549760C2semidirect
14. #22549727C2semidirect
18. #16861186C4semidirect
187. C127#2323semidirect

Maximal subgroups

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

subgroupindex
C2×#168611862
C2×C4×Dic2542
C2×#168611652
#225497802
#225497732
#225497722
#225497712
#225497692
#225497642
#225497612
#225497602
#225497342
#225497272
#388×C1272
#2323127