Finite Group Databasebeta

← #30639980

Subgroups — #30639980

Order 16256.

382 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 69 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
1#22550435C211yesmaximal, characteristic
2C2×#16861123C211yesmaximal, characteristic, complemented
3#22549756C211yesmaximal, characteristic, complemented
4#22549753C211yesmaximal, characteristic, complemented
5#22549751C211yesmaximal, characteristic, complemented
6#22549642C211yesmaximal, characteristic, complemented
7C2×#99×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127C2211yescharacteristic
9C23×Dic127C2211yescharacteristic
10#16861192—21no—
11#16861190—21no—
12#16861181—21no—
13#16861181—21no—
14#16861178—21no—
15#16861177—21no—
16C2×C4⋊Dic127—21no—
17C2×C22⋊Dic127—21no—
18C2×C22⋊Dic127C2211yescharacteristic
19C2×C22⋊Dic127C2211yescharacteristic
20C2×(C2×D127)⋊C4—21no—
21#16861123—22no—
22C127⋊#95—21no—
23C127⋊#95—21no—
24C127⋊#95—21no—
25C22×D4×C127C2211yescharacteristic
26C2×C22⋊C4×C127C2211yescharacteristic
27C2×C22⋊C4×C127C2211yescharacteristic
28#99×C127—22no—
29C23×D127D411yescharacteristic
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35C2×C22⋊D127—21no—
36C2×C22⋊D127—21no—
37C2×C22⋊D127—21no—
38C22×Dic127—21no—
39C22×Dic127—21no—
40C22×Dic127—21no—
41C22×Dic127—21no—
42C22×Dic127—42no—
43C22×Dic127D411yescharacteristic
44C22×Dic127D411yescharacteristic
45C22×Dic127D411yescharacteristic
46C4⋊Dic127—42no—
47Dic127⋊C4—21no—
48Dic127⋊C4—21no—
49C22⋊Dic127—21no—
50C22⋊Dic127—21no—
51C22⋊Dic127—21no—
52C22⋊Dic127—21no—
53C22⋊Dic127—21no—
54C22⋊Dic127—21no—
55C22⋊Dic127—42no—
56(C2×D127)⋊C4—21no—
57(C2×D127)⋊C4—21no—
58(C2×D127)⋊C4—42no—
59C24×C127D411yescharacteristic
60C24×C127C2311yescharacteristic
61C2×D4×C127—21no—
62C2×D4×C127—21no—
63C2×D4×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127—21no—
66C2×D4×C127—21no—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127D411yescharacteristic
70C22⋊C4×C127—21no—
71C22⋊C4×C127—21no—
72C22⋊C4×C127—21no—
73C22⋊C4×C127D412yescomplemented
74C22×D127—21no—
75C22×D127—21no—
76C22×D127—21no—
77C22×D127—41no—
78C22×D127—41no—
79C22⋊D127—41no—
80C22⋊D127—41no—
81C22⋊D127—41no—
82C22⋊D127—41no—
83C22⋊D127—81no—
84C22⋊D127—81no—
85C2×Dic127—22no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127—21no—
89C2×Dic127—21no—
90C2×Dic127—21no—
91C2×Dic127—21no—
92C2×Dic127—21no—
93C2×Dic127—41no—
94C2×Dic127—41no—
95C2×Dic127—41no—
96C2×Dic127—41no—
97C2×Dic127—41no—
98C2×Dic127—41no—
99C2×Dic127—41no—
100C2×Dic127—41no—
101C2×Dic127—41no—
102C23×C127—21no—
103C23×C127—21no—
104C23×C127—21no—
105C23×C127—41no—
106C23×C127—41no—
107C23×C127—41no—
108C23×C127—41no—
109C23×C127C2×D411yescharacteristic
110C23×C127C2×D411yescharacteristic
111C23×C127C2×D411yescharacteristic
112C23×C127C2×D411yescharacteristic
113C23×C127#4111yescharacteristic
114C23×C127#4111yescharacteristic
115C23×C127#4111yescharacteristic
116D4×C127—41no—
117D4×C127—41no—
118D4×C127—41no—
119D4×C127—41no—
120C2×C4×C127—21no—
121C2×C4×C127—22no—
122C2×C4×C127—21no—
123C2×C4×C127—21no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C22×C127—21no—
130C22×C127—21no—
131C22×C127—21no—
132C22×C127—21no—
133C22×C127—21no—
134C22×C127—21no—
135C22×C127—41no—
136C22×C127—41no—
137C22×C127—41no—
138C22×C127—41no—
139C22×C127—42no—
140C22×C127—41no—
141C22×C127—41no—
142C22×C127—41no—
143C22×C127—81no—
144C22×C127#12011yescharacteristic
145C22×C127#12111yescharacteristic
146C22×C127#12111yescharacteristic
147C22×C127#12111yescharacteristic
148C22×C127#12311yescharacteristic
149C22×C127#12311yescharacteristic
150C22×C127#12411yescharacteristic
151C2×D127—42no—
152C2×D127—41no—
153C2×D127—41no—
154C2×D127—41no—
155C2×D127—81no—
156C4×C127—42no—
157C4×C127—41no—
158C4×C127—81no—
159Dic127—41no—
160Dic127—42no—
161Dic127—41no—
162Dic127—41no—
163Dic127—81no—
164C2×C127—21no—
165C2×C127—21no—
166C2×C127—21no—
167C2×C127—21no—
168C2×C127—21no—
169C2×C127—21no—
170C2×C127—81no—
171C2×C127#39411yescharacteristic
172C2×C127B(4,𝔽2)11yescharacteristic
173C2×C127#45811yescharacteristic
174D127—81no—
175#2044—1271nomaximal, Sylow
176C127#204411yescharacteristic, complemented, Sylow
177#534—1271no—
178C2×#99—1271no—
179C2×#99D12711yescharacteristic, complemented
180#386—1271no—
181#386—1271no—
182#385—1271no—
183#342—1271no—
184C22×D4—1271no—
185C22×D4C2×D12711yescharacteristic
186C23×C4—1271no—
187#123—2541no—
188#123—2541no—
189#121—2541no—
190#121—2541no—
191#120—2541no—
192#120—2541no—
193C2×C4⋊C4—2541no—
194C2×C22⋊C4—1271no—
195C2×C22⋊C4—1271no—
196C2×C22⋊C4—2541no—
197C2×C22⋊C4—2541no—
198C2×C22⋊C4C2×D12711yescharacteristic
199C2×C22⋊C4C2×D12711yescharacteristic
200#99—22no—
201#99—2542no—
202#95—2541no—
203#95—2541no—
204#95—2541no—
205C24—1271no—
206C24D50811yescharacteristic
207C24C22×D12711yescharacteristic
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—2541no—
215C2×D4—2541no—
216C2×D4—2541no—
217C2×D4—2541no—
218C2×D4—2541no—
219C2×D4—2541no—
220C2×D4—2541no—
221C2×D4—2541no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—1271no—
225C22×C4—1271no—
226C22×C4—1271no—
227C22×C4—5082no—
228C22×C4—2541no—
229C22×C4—2541no—
230C22×C4—2541no—
231C22×C4—2541no—
232C22×C4D50811yescharacteristic
233C4⋊C4—5082no—
234C4⋊C4—2541no—
235C4⋊C4—2541no—
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—21no—
239C22⋊C4—5082no—
240C22⋊C4—5082no—
241C22⋊C4—2541no—
242C22⋊C4—2541no—
243C22⋊C4—2541no—
244C22⋊C4—2541no—
245C22⋊C4—2541no—
246C22⋊C4—2541no—
247C22⋊C4—2541no—
248C22⋊C4—2541no—
249C22⋊C4C22⋊D12712yescomplemented
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—5081no—
258C23—5081no—
259C23—2541no—
260C23—2541no—
261C23—2541no—
262C23D4×D12711yescharacteristic
263C23D4×D12711yescharacteristic
264C23C2×C22⋊D12711yescharacteristic
265C23C2×D50811yescharacteristic
266C23#1452324711yescharacteristic
267C23#1452324711yescharacteristic
268C23#1452324811yescharacteristic
269D4—41no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—10161no—
274D4—10161no—
275D4—5081no—
276D4—5081no—
277D4—5081no—
278D4—5081no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—5081no—
289C2×C4—5081no—
290C2×C4—5081no—
291C2×C4—5081no—
292C2×C4—5081no—
293C2×C4—5081no—
294C2×C4—5081no—
295C2×C4—5081no—
296C2×C4—5081no—
297C2×C4—2542no—
298C2×C4—2542no—
299C2×C4—2542no—
300C2×C4—2541no—
301C2×C4—2541no—
302C2×C4—2541no—
303C2×C4—2541no—
304C2×C4—2541no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22—10161no—
321C22—5082no—
322C22—5081no—
323C22—5081no—
324C22—5081no—
325C22#1686117911yescharacteristic
326C22#1686118011yescharacteristic
327C22#1686118111yescharacteristic
328C22#1686118311yescharacteristic
329C22#1686119111yescharacteristic
330C22#1686119311yescharacteristic
331C22#1686119511yescharacteristic
332C4—42no—
333C4—41no—
334C4—81no—
335C4—10161no—
336C4—5081no—
337C4—5082no—
338C4—5081no—
339C4—5081no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—81no—
347C2—10161no—
348C2#2254979211yescharacteristic
349C2#2255006411yescharacteristic
350C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC24×C127C23
FrattiniC24C22×D127
FittingC2×#99×C127C2
socleC22×C127#120
prime radical#22550435C2
hypercenterC2×#99D127
hypocenterC127#2044
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#16861123C2semidirect
3. #22549756C2semidirect
4. #22549753C2semidirect
5. #22549751C2semidirect
6. #22549642C2semidirect
7. C2×#99×C127C2semidirect
73. C22⋊C4×C127D4semidirect
176. C127#2044semidirect
179. C2×#99D127semidirect
249. C22⋊C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504352
C2×#168611232
#225497562
#225497532
#225497512
#225496422
C2×#99×C1272
#2044127