Finite Group Databasebeta

← #30638659

Subgroups — #30638659

Order 16256.

848 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D127C211yesmaximal, characteristic
2C2×#16861131C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic, complemented
4C2×#16861129C211yesmaximal, characteristic, complemented
5C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
6#22549733C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic, complemented
8C24×D127—21no—
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—21no—
11C2×D4×D127—22no—
12C2×D4×D127—22no—
13C2×D4×D127—22no—
14C2×D4×D127—22no—
15C2×D4×D127—22no—
16C2×D4×D127C414yescomplemented
17C22×C4×D127C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic, complemented
19C2×C4⋊Dic127C2211yescharacteristic, complemented
20C2×(C2×D127)⋊C4—21no—
21#16861131—22no—
22#16861130—22no—
23#16861129—22no—
24(C2×D127)⋊C8—22no—
25C127⋊#95—21no—
26C22×D4×C127C2211yescharacteristic
27C22×C8×C127C2211yescharacteristic, complemented
28C2×C4⋊C4×C127C2211yescharacteristic, complemented
29#102×C127—22no—
30C23×D127—24no—
31C23×D127—24no—
32C23×D127—24no—
33C23×D127—21no—
34C23×D127—42no—
35C23×D127—42no—
36C23×D127—42no—
37C23×D127—42no—
38C23×D127D411yescharacteristic
39C2×D508—21no—
40C2×D508—21no—
41C2×D508—21no—
42C2×D508C2×C412yes—
43C2×D508D412yes—
44C2×D508D412yes—
45C2×C22⋊D127—24no—
46C2×C22⋊D127—24no—
47C2×C22⋊D127—24no—
48D4×D127—28no—
49D4×D127—28no—
50D4×D127—28no—
51D4×D127—28no—
52C22×Dic127—21no—
53C22×Dic127D411yescharacteristic
54C2×C4×D127—21no—
55C2×C4×D127—21no—
56C2×C4×D127—21no—
57C2×C4×D127—21no—
58C2×C4×D127C2×C412yes—
59C2×C4×D127D412yes—
60C2×C127⋊C8—21no—
61C2×C127⋊C8—21no—
62C2×C127⋊C8—21no—
63C4⋊Dic127—21no—
64C4⋊Dic127—21no—
65(C2×D127)⋊C4—42no—
66C24×C127—21no—
67C2×D4×C127—21no—
68C2×D4×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127C2×C412yes—
71C2×D4×C127D412yes—
72C2×D4×C127D412yes—
73C22×C4×C127—21no—
74C22×C4×C127C2311yescharacteristic
75C2×C8×C127—21no—
76C2×C8×C127—21no—
77C2×C8×C127—21no—
78C4⋊C4×C127—21no—
79C4⋊C4×C127—21no—
80C22×D127—21no—
81C22×D127—21no—
82C22×D127—21no—
83C22×D127—21no—
84C22×D127—216no—
85C22×D127—22no—
86C22×D127—22no—
87C22×D127—22no—
88C22×D127—48no—
89C22×D127—48no—
90C22×D127—48no—
91C22×D127—48no—
92C22×D127—48no—
93C22×D127—41no—
94C22×D127—41no—
95C22×D127—41no—
96C22×D127—41no—
97C22×D127—48no—
98C22×D127C22⋊C412yes—
99C22×D127D812yes—
100C22×D127SD1612yes—
101D508—22no—
102D508—22no—
103D508—22no—
104D508C22⋊C414yescomplemented
105C22⋊D127—216no—
106C2×Dic127—21no—
107C2×Dic127—21no—
108C2×Dic127—41no—
109C2×Dic127—41no—
110C2×Dic127—41no—
111C2×Dic127C22⋊C412yes—
112C4×D127—22no—
113C4×D127—22no—
114C4×D127—22no—
115C4×D127—22no—
116C127⋊C8—41no—
117C23×C127—22no—
118C23×C127—22no—
119C23×C127—22no—
120C23×C127—41no—
121C23×C127—41no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127C2×D411yescharacteristic
125D4×C127—22no—
126D4×C127—22no—
127D4×C127—22no—
128D4×C127C22⋊C414yescomplemented
129C2×C4×C127—41no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
132C2×C4×C127C22×C411yescharacteristic
133C2×C4×C127C2×D411yescharacteristic
134C2×C4×C127C2×D411yescharacteristic
135C2×C4×C127C2×D411yescharacteristic
136C2×C4×C127C2×D411yescharacteristic
137C2×C4×C127C2×D411yescharacteristic
138C8×C127—41no—
139C22×C127—24no—
140C22×C127—42no—
141C22×C127—42no—
142C22×C127—42no—
143C22×C127—42no—
144C22×C127—42no—
145C22×C127—42no—
146C22×C127C2×C22⋊C411yescharacteristic
147C22×C127#12011yescharacteristic
148C22×C127#12011yescharacteristic
149C22×C127C2×D811yescharacteristic
150C22×C127C2×SD1611yescharacteristic
151C22×C127AGL(1,ℤ/8)11yescharacteristic
152C22×C127AGL(1,ℤ/8)11yescharacteristic
153C2×D127—22no—
154C2×D127—22no—
155C2×D127—22no—
156C2×D127—22no—
157C2×D127—22no—
158C2×D127—22no—
159C2×D127—24no—
160C2×D127—432no—
161C2×D127—42no—
162C2×D127—42no—
163C2×D127—42no—
164C2×D127—42no—
165C2×D127—42no—
166C2×D127—42no—
167C2×D127#10214yescomplemented
168C4×C127—81no—
169C4×C127C2×C22⋊C411yescharacteristic
170C4×C127C2×C22⋊C411yescharacteristic
171C4×C127#12011yescharacteristic
172C4×C127#12011yescharacteristic
173Dic127—22no—
174Dic127—81no—
175C2×C127—44no—
176C2×C127#37911yescharacteristic
177C2×C127C2×#10211yescharacteristic
178C2×C127#41811yescharacteristic
179C2×C127#44711yescharacteristic
180C2×C127#44711yescharacteristic
181C2×C127#45011yescharacteristic
182C2×C127#45011yescharacteristic
183D127—24no—
184D127—44no—
185#1786—1271nomaximal, Sylow
186C127#178611yescharacteristic, complemented, Sylow
187C23×D4—1271no—
188C2×#102—1271no—
189C2×#102—1271no—
190C2×#102—1271no—
191C2×#102D12711yescharacteristic, complemented
192C2×C22⋊C8—1271no—
193#380—1271no—
194C25—2541no—
195C22×D4—1274no—
196C22×D4—1271no—
197C22×D4—2542no—
198C22×D4—2541no—
199C22×D4—2542no—
200C22×D4—2542no—
201C22×D4—2542no—
202C22×D4—2542no—
203C22×D4C2×D12711yescharacteristic
204C23×C4—1271no—
205C22×C8—1271no—
206C22×C8C2×D12711yescharacteristic, complemented
207C2×C4⋊C4—1271no—
208C2×C4⋊C4C2×D12711yescharacteristic, complemented
209C2×C22⋊C4—2541no—
210#102—22no—
211#102—2542no—
212#102—2542no—
213#102—2542no—
214C22⋊C8—2542no—
215#95—2541no—
216C24—21no—
217C24—1271no—
218C24—5082no—
219C24—5082no—
220C24—5082no—
221C24—5082no—
222C24—2544no—
223C24—2544no—
224C24—2544no—
225C24—2541no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—1272no—
230C2×D4—1272no—
231C2×D4—1272no—
232C2×D4—2544no—
233C2×D4—2548no—
234C2×D4—2548no—
235C2×D4—2548no—
236C2×D4—2544no—
237C2×D4—2544no—
238C2×D4—2548no—
239C2×D4—2541no—
240C2×D4—2541no—
241C2×D4—2541no—
242C2×D4C4×D12712yes—
243C2×D4C22⋊D12712yes—
244C2×D4D50812yes—
245C22×C4—21no—
246C22×C4—1271no—
247C22×C4—1272no—
248C22×C4—1272no—
249C22×C4—2541no—
250C22×C4—2541no—
251C22×C4—2541no—
252C22×C4—2541no—
253C22×C4—2541no—
254C22×C4C22×D12711yescharacteristic
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—2541no—
259C2×C8—2541no—
260C2×C8—2541no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C22⋊C4—5082no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—1272no—
274C23—1272no—
275C23—1272no—
276C23—5088no—
277C23—5088no—
278C23—5088no—
279C23—5088no—
280C23—5088no—
281C23—5088no—
282C23—5081no—
283C23—5081no—
284C23—5081no—
285C23—5081no—
286C23—25416no—
287C23—2542no—
288C23—2542no—
289C23—2542no—
290C23—2541no—
291C23—2541no—
292C23—2541no—
293C23—2541no—
294C23D4×D12711yescharacteristic
295D4—22no—
296D4—22no—
297D4—22no—
298D4—1274no—
299D4—25416no—
300D4—2542no—
301D4—2542no—
302D4—2542no—
303D4(C2×D127)⋊C414yescomplemented
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—1272no—
308C2×C4—5081no—
309C2×C4—5081no—
310C2×C4—5081no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4—2542no—
314C2×C4—2542no—
315C2×C4—2542no—
316C2×C4—2542no—
317C2×C4C2×C4×D12711yescharacteristic
318C2×C4D4×D12711yescharacteristic
319C2×C4D4×D12711yescharacteristic
320C2×C4D4×D12711yescharacteristic
321C2×C4C2×C22⋊D12711yescharacteristic
322C2×C4C2×D50811yescharacteristic
323C8—41no—
324C8—5081no—
325C22—24no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—1274no—
333C22—50832no—
334C22—5082no—
335C22—5082no—
336C22—5082no—
337C22—5082no—
338C22—5082no—
339C22—5082no—
340C22—2544no—
341C22—2542no—
342C22—2542no—
343C22—2542no—
344C22—2542no—
345C22—2542no—
346C22—2542no—
347C22C22⋊C4×D12711yescharacteristic
348C22#1686117711yescharacteristic
349C22#1686117911yescharacteristic
350C22D8×D12711yescharacteristic
351C22SD16×D12711yescharacteristic
352C22#1686123311yescharacteristic
353C22#1686123611yescharacteristic
354C4—81no—
355C4—10161no—
356C4—2542no—
357C4C22⋊C4×D12711yescharacteristic
358C4C2×(C2×D127)⋊C411yescharacteristic
359C4#1686117711yescharacteristic
360C4#1686117911yescharacteristic
361C2—44no—
362C2—5084no—
363C2—2544no—
364C2#2254973111yescharacteristic
365C2#102×D12711yescharacteristic
366C2#2254987911yescharacteristic
367C2#2255000811yescharacteristic
368C2#2255001111yescharacteristic
369C2#2255002511yescharacteristic
370C2#2255002811yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×D127C2
cosocleC22×C4×C127C23
hypercenterC2×#102D127
hypocenterC127#1786
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#16861131C2semidirect
3. C2×#16861130C2semidirect
4. C2×#16861129C2semidirect
5. C2×(C2×D127)⋊C8C2semidirect
6. #22549733C2semidirect
7. C2×#102×C127C2semidirect
16. C2×D4×D127C4semidirect
18. C22×C127⋊C8C22semidirect
19. C2×C4⋊Dic127C22semidirect
27. C22×C8×C127C22semidirect
28. C2×C4⋊C4×C127C22semidirect
104. D508C22⋊C4semidirect
128. D4×C127C22⋊C4semidirect
167. C2×D127#102semidirect
186. C127#1786semidirect
191. C2×#102D127semidirect
206. C22×C8C2×D127semidirect
208. C2×C4⋊C4C2×D127semidirect
303. D4(C2×D127)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1272
C2×#168611312
C2×#168611302
C2×#168611292
C2×(C2×D127)⋊C82
#225497332
C2×#102×C1272
#1786127