Finite Group Databasebeta

← #30639855

Subgroups — #30639855

Order 16256.

638 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 73 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, complemented
2C2×#16861179C211yesmaximal, characteristic, complemented
3C2×#16861177C211yesmaximal, characteristic, complemented
4C2×#16861123C211yesmaximal, characteristic, complemented
5#22549731C211yesmaximal, characteristic, complemented
6#22549642C211yesmaximal, characteristic, complemented
7C2×#99×C127C211yesmaximal, characteristic, complemented
8C24×D127C2211yescharacteristic
9C22×D508—21no—
10C22×C22⋊D127—21no—
11C22×C22⋊D127C2211yescharacteristic
12#16861192—21no—
13#16861190—21no—
14#16861181—22no—
15#16861179—22no—
16#16861179—22no—
17#16861178—21no—
18#16861177—22no—
19#16861177—21no—
20#16861177—22no—
21C2×C22⋊Dic127C2211yescharacteristic
22C2×C22⋊Dic127C2211yescharacteristic
23C2×(C2×D127)⋊C4—21no—
24C2×(C2×D127)⋊C4—22no—
25C2×(C2×D127)⋊C4—21no—
26#16861123—22no—
27C22×D4×C127C2211yescharacteristic
28C2×C22⋊C4×C127C2211yescharacteristic
29C2×C22⋊C4×C127C2211yescharacteristic
30#99×C127—22no—
31C23×D127—21no—
32C23×D127—21no—
33C23×D127—21no—
34C23×D127—21no—
35C23×D127—21no—
36C23×D127—21no—
37C23×D127—44no—
38C23×D127D411yescharacteristic, complemented
39C23×D127D412yescomplemented
40C2×D508—24no—
41C2×D508—44no—
42C2×C22⋊D127—24no—
43C2×C22⋊D127—22no—
44C2×C22⋊D127—21no—
45C2×C22⋊D127—21no—
46C2×C22⋊D127—21no—
47C2×C22⋊D127—21no—
48C2×C22⋊D127—21no—
49C2×C22⋊D127—21no—
50C2×C22⋊D127—44no—
51C22×Dic127—21no—
52C22×Dic127D411yescharacteristic, complemented
53C22×Dic127D412yescomplemented
54Dic127⋊C4—22no—
55C22⋊Dic127—22no—
56C22⋊Dic127—21no—
57C22⋊Dic127—21no—
58C22⋊Dic127—21no—
59C22⋊Dic127D412yescomplemented
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—24no—
62(C2×D127)⋊C4—24no—
63(C2×D127)⋊C4—44no—
64C24×C127D411yescharacteristic, complemented
65C24×C127C2311yescharacteristic
66C2×D4×C127—21no—
67C2×D4×C127—21no—
68C2×D4×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127—21no—
72C22×C4×C127—21no—
73C22×C4×C127—21no—
74C22×C4×C127D411yescharacteristic, complemented
75C22⋊C4×C127—21no—
76C22⋊C4×C127—21no—
77C22⋊C4×C127—21no—
78C22⋊C4×C127D412yescomplemented
79C22×D127—24no—
80C22×D127—24no—
81C22×D127—24no—
82C22×D127—22no—
83C22×D127—22no—
84C22×D127—22no—
85C22×D127—21no—
86C22×D127—21no—
87C22×D127—21no—
88C22×D127—44no—
89C22×D127—44no—
90C22×D127—44no—
91C22×D127—44no—
92C22×D127—42no—
93C22×D127—42no—
94C22×D127—42no—
95C22×D127—41no—
96C22×D127—41no—
97C22×D127—41no—
98C22×D127—41no—
99C22×D127—42no—
100D508—48no—
101C22⋊D127—48no—
102C22⋊D127—41no—
103C22⋊D127—41no—
104C22⋊D127—41no—
105C22⋊D127—41no—
106C22⋊D127—82no—
107C2×Dic127—21no—
108C2×Dic127—22no—
109C2×Dic127—22no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C2×Dic127—42no—
113C2×Dic127—42no—
114C23×C127—21no—
115C23×C127—21no—
116C23×C127—21no—
117C23×C127—42no—
118C23×C127—42no—
119C23×C127C2×D411yescharacteristic
120C23×C127C2×D411yescharacteristic
121C23×C127C2×D411yescharacteristic
122C23×C127C2×D411yescharacteristic
123C23×C127C2×D411yescharacteristic
124C23×C127C2×D411yescharacteristic
125C23×C127#4111yescharacteristic
126D4×C127—41no—
127D4×C127—41no—
128D4×C127—41no—
129D4×C127—41no—
130C2×C4×C127—22no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—42no—
135C2×C4×C127—42no—
136C2×C4×C127—41no—
137C22×C127—22no—
138C22×C127—22no—
139C22×C127—22no—
140C22×C127—41no—
141C22×C127—41no—
142C22×C127—41no—
143C22×C127—41no—
144C22×C127—42no—
145C22×C127—41no—
146C22×C127—42no—
147C22×C127—81no—
148C22×C127#12011yescharacteristic
149C22×C127#12011yescharacteristic
150C22×C127#12011yescharacteristic
151C22×C127#12111yescharacteristic
152C22×C127#12111yescharacteristic
153C22×C127#12111yescharacteristic
154C22×C127#12711yescharacteristic
155C2×D127—24no—
156C2×D127—24no—
157C2×D127—24no—
158C2×D127—42no—
159C2×D127—42no—
160C2×D127—44no—
161C2×D127—42no—
162C2×D127—44no—
163C2×D127—44no—
164C2×D127—42no—
165C2×D127—44no—
166C2×D127—42no—
167C2×D127—41no—
168C2×D127—42no—
169C2×D127—81no—
170C4×C127—42no—
171C4×C127—41no—
172C4×C127—81no—
173Dic127—42no—
174Dic127—41no—
175Dic127—42no—
176C2×C127—21no—
177C2×C127—21no—
178C2×C127—21no—
179C2×C127—21no—
180C2×C127—21no—
181C2×C127—21no—
182C2×C127—81no—
183C2×C127#39211yescharacteristic
184C2×C127B(4,𝔽2)12yes—
185D127—44no—
186D127—81no—
187#2021—1271nomaximal, Sylow
188C127#202111yescharacteristic, complemented, Sylow
189#534—1271no—
190C2×#120—1271no—
191C2×#120—1271no—
192C2×#99—1271no—
193C2×#99D12711yescharacteristic, complemented
194#379—1271no—
195#342—1271no—
196C25—1271no—
197C22×D4—1271no—
198C22×D4—2541no—
199C22×D4—2541no—
200C22×D4C2×D12711yescharacteristic
201#123—2541no—
202#123—2541no—
203#121—2542no—
204#120—2542no—
205#120—2542no—
206#120—2542no—
207#120—2541no—
208#120—2541no—
209#120—2542no—
210C2×C22⋊C4—1271no—
211C2×C22⋊C4—1271no—
212C2×C22⋊C4—2542no—
213C2×C22⋊C4—2541no—
214C2×C22⋊C4—2541no—
215C2×C22⋊C4C2×D12711yescharacteristic
216C2×C22⋊C4C2×D12711yescharacteristic
217#99—22no—
218#99—2542no—
219C24—1272no—
220C24—1271no—
221C24—5084no—
222C24—2541no—
223C24—2541no—
224C24—2541no—
225C24—2541no—
226C24—2541no—
227C24—2541no—
228C24D50811yescharacteristic, complemented
229C24C22×D12711yescharacteristic
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—5084no—
237C2×D4—5084no—
238C2×D4—2544no—
239C2×D4—2542no—
240C2×D4—2544no—
241C2×D4—2541no—
242C2×D4—2541no—
243C2×D4—2541no—
244C2×D4—2541no—
245C2×D4—2541no—
246C2×D4—2541no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—1272no—
250C22×C4—1271no—
251C22×C4—2541no—
252C22×C4D50811yescharacteristic, complemented
253C4⋊C4—2542no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—1272no—
258C22⋊C4—5084no—
259C22⋊C4—2544no—
260C22⋊C4—2544no—
261C22⋊C4—2542no—
262C22⋊C4—2542no—
263C22⋊C4—2541no—
264C22⋊C4—2541no—
265C22⋊C4—2541no—
266C22⋊C4C22⋊D12712yescomplemented
267C23—21no—
268C23—21no—
269C23—21no—
270C23—42no—
271C23—42no—
272C23—5082no—
273C23—5082no—
274C23—5082no—
275C23—5082no—
276C23—5084no—
277C23—5084no—
278C23—5084no—
279C23—5084no—
280C23—5081no—
281C23—5081no—
282C23—5081no—
283C23—5081no—
284C23—2544no—
285C23—2544no—
286C23—2544no—
287C23—2541no—
288C23—2541no—
289C23—2541no—
290C23—2542no—
291C23—2542no—
292C23—2542no—
293C23D4×D12711yescharacteristic
294C23D4×D12711yescharacteristic
295C23D4×D12711yescharacteristic
296C23D4×D12711yescharacteristic
297C23C2×C22⋊D12711yescharacteristic
298C23C2×D50811yescharacteristic
299C23#1452324811yescharacteristic
300D4—41no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—10162no—
305D4—5088no—
306D4—5088no—
307D4—5081no—
308D4—5081no—
309D4—5081no—
310D4—5081no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—5082no—
319C2×C4—5082no—
320C2×C4—2541no—
321C2×C4—2542no—
322C2×C4—2542no—
323C2×C4—2541no—
324C2×C4—2541no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22—41no—
334C22—42no—
335C22—81no—
336C22—10161no—
337C22—5082no—
338C22—5082no—
339C22—5084no—
340C22—5082no—
341C22—5084no—
342C22—5084no—
343C22—5082no—
344C22—5081no—
345C22—5082no—
346C22—5082no—
347C22—5084no—
348C22—2544no—
349C22—2544no—
350C22—2544no—
351C22#1686117711yescharacteristic
352C22#1686117911yescharacteristic
353C22#1686117911yescharacteristic
354C22#1686118211yescharacteristic
355C22#1686118211yescharacteristic
356C22#1686118311yescharacteristic
357C22#1686120511yescharacteristic
358C4—42no—
359C4—41no—
360C4—81no—
361C4—5082no—
362C4—5081no—
363C4—5082no—
364C2—21no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—81no—
371C2—10161no—
372C2—5084no—
373C2#2254978511yescharacteristic
374C2#2255006412yes—
subgroupG/H
centerC22#16861179
commutator, cosocleC24×C127C23
FrattiniC24C22×D127
FittingC2×#99×C127C2
socleC22×C127#120
hypercenterC2×#99D127
hypocenterC127#2021
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
1. #22550435C2semidirect
2. C2×#16861179C2semidirect
3. C2×#16861177C2semidirect
4. C2×#16861123C2semidirect
5. #22549731C2semidirect
6. #22549642C2semidirect
7. C2×#99×C127C2semidirect
38. C23×D127D4semidirect
39. C23×D127D4semidirect
52. C22×Dic127D4semidirect
53. C22×Dic127D4semidirect
59. C22⋊Dic127D4semidirect
64. C24×C127D4semidirect
74. C22×C4×C127D4semidirect
78. C22⋊C4×C127D4semidirect
188. C127#2021semidirect
193. C2×#99D127semidirect
228. C24D508semidirect
252. C22×C4D508semidirect
266. 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×#168611792
C2×#168611772
C2×#168611232
#225497312
#225496422
C2×#99×C1272
#2021127