Finite Group Databasebeta

← #72230814

Subgroups — #72230814

Order 8192.

318 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 96 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#72230812C211yesmaximal, characteristic, complemented
2#72230810C211yesmaximal, characteristic, complemented
3#72230805C211yesmaximal, characteristic
4#72230794C411yescharacteristic
5#72230785C411yescharacteristic
6#72230781C411yescharacteristic
7#72230779C411yescharacteristic
8#72230778C811yescharacteristic
9#72230775C811yescharacteristic
10#72230770C811yescharacteristic
11#72230767C811yescharacteristic
12#72230761C211yesmaximal, characteristic, complemented
13#72230759C211yesmaximal, characteristic, complemented
14#72230758—21no—
15#72230755—41no—
16#72230752—41no—
17#72230749—21no—
18#72230747C2211yescharacteristic
19#72230742C2×C411yescharacteristic
20#72230741—21no—
21#72230739C2×C411yescharacteristic
22#72230735C2211yescharacteristic
23#72230730C211yesmaximal, characteristic, complemented
24#72230729C2211yescharacteristic
25#72230728—21no—
26#72230725C2211yescharacteristic
27#72230722C2×C411yescharacteristic
28#72230722C2×C411yescharacteristic
29#72230720C211yesmaximal, characteristic, complemented
30#72230718—21no—
31#72230715—21no—
32#72230713—41no—
33#72230710—41no—
34#72230706C2211yescharacteristic
35#72230705C2×C411yescharacteristic
36#72230702—21no—
37#72230701C2×C411yescharacteristic
38#72230700—21no—
39#72230696C2211yescharacteristic
40#72230695—21no—
41#72230692C2211yescharacteristic
42#72230689D411yescharacteristic
43#72230688C2311yescharacteristic
44#72227187—21no—
45#72225369—21no—
46#72225140D411yescharacteristic
47#153706C1611yescharacteristic
48#153596C2×C811yescharacteristic
49#153594C2×C811yescharacteristic
50#153587—81no—
51#151101—81no—
52#151082—21no—
53#151080—21no—
54#151073—41no—
55#122946—41no—
56#94843C1611yescharacteristic
57#94836C1611yescharacteristic
58#94835C1611yescharacteristic
59#94815—21no—
60#94815—41no—
61#94815C2×C811yescharacteristic
62#94815C2×C811yescharacteristic
63#94753C2×C811yescharacteristic
64#94752C2×C811yescharacteristic
65#94746—41no—
66#94612—21no—
67#94609—21no—
68#94606—81no—
69#94508—21no—
70#94505C22×C411yescharacteristic
71#94505#4111yescharacteristic
72#93885—21no—
73#93884—21no—
74#93876—41no—
75#93755—81no—
76#93752—41no—
77#93749—41no—
78#93667—21no—
79#93666C2×D411yescharacteristic
80#13741C2×C1611yescharacteristic
81C2×D64C2×C1611yescharacteristic
82#13737—161no—
83#13655—21no—
84C4×D32—21no—
85#13647—81no—
86D4×C32—161no—
87#12046—41no—
88C8×D16—41no—
89C16×D8—81no—
90SD256C3212yescomplemented
91D128C3212yescomplemented
92M256—21no—
93M256—41no—
94M256—81no—
95M256C2×C1611yescharacteristic
96M256C2×C1611yescharacteristic
97#7547C2×C1611yescharacteristic
98#7546C2×C1611yescharacteristic
99#7544—81no—
100#7539—21no—
101#7537—21no—
102#7514—161no—
103#7510—21no—
104#7510—41no—
105#7510C22×C811yescharacteristic
106#7510#13111yescharacteristic
107#7462—21no—
108#7461—21no—
109#7459—41no—
110#7455—81no—
111#7387—41no—
112#7383—41no—
113#7381—81no—
114D4⋊C32—161no—
115#7331—41no—
116#7329C4×D411yescharacteristic
117#7329AGL(1,ℤ/8)11yescharacteristic
118#7101—41no—
119#7068—81no—
120#7052AGL(1,ℤ/8)11yescharacteristic
121#2262—21no—
122C2×D32—21no—
123#2258—161no—
124C22×C32—321no—
125#2178—41no—
126C4×D16—41no—
127#2170—81no—
128D4×C16—161no—
129C8×D8—81no—
130Q128C2×C3211yescharacteristic, complemented
131SD128—21no—
132D64—21no—
133D64C2×C3212yes—
134D64C2×C3211yescharacteristic, complemented
135M128—161no—
136M128—321no—
137C2×C64—21no—
138C2×C64—41no—
139C2×C64—81no—
140C2×C64—321no—
141C2×C64C22×C1611yescharacteristic
142C2×C64#50411yescharacteristic
143#1425—21no—
144#1423—21no—
145#1422—81no—
146C4⋊C32—161no—
147#1417—41no—
148#1415—41no—
149#1401—161no—
150C22⋊C32—321no—
151C4×C32—21no—
152C4×C32—81no—
153C4×C32C8×D411yescharacteristic
154C4×C32#44411yescharacteristic
155#1373—41no—
156#1371—41no—
157#1370—81no—
158#1335—81no—
159#1331—81no—
160D4⋊C16—161no—
161C8×C16—21no—
162C8×C16#44211yescharacteristic
163C128—21no—
164C128—41no—
165C128—81no—
166C128—161no—
167C128C2×C3212yescomplemented
168#508—41no—
169C2×D16—41no—
170#504—161no—
171C22×C16—321no—
172#443—81no—
173C4×D8—81no—
174C8×D4—161no—
175Q64—21no—
176SD64—41no—
177D32—22no—
178D32—21no—
179D32—41no—
180M64—161no—
181M64—321no—
182C2×C32—21no—
183C2×C32—41no—
184C2×C32—161no—
185C2×C32—321no—
186C2×C32—321no—
187C2×C32—642no—
188C2×C32D4×C1611yescharacteristic
189C2×C32#217111yescharacteristic
190#368—41no—
191#366—41no—
192#364—81no—
193C4⋊C16—161no—
194#359—81no—
195#357—81no—
196#350—161no—
197C22⋊C16—321no—
198C4×C16—41no—
199C4×C16#158011yescharacteristic
200C4×C16AGL(1,ℤ/16)11yescharacteristic
201#335—81no—
202D4⋊C8—161no—
203C82AGL(1,ℤ/16)11yescharacteristic
204C64—21no—
205C64—41no—
206C64—81no—
207C64—161no—
208C64—321no—
209C64C22×C3211yescharacteristic
210C64#225811yescharacteristic
211#135—81no—
212C2×D8—81no—
213#131—161no—
214C22×C8—321no—
215C4×D4—161no—
216Q32—41no—
217SD32—81no—
218D16—42no—
219D16—41no—
220D16—81no—
221M32—161no—
222M32—321no—
223C2×C16—21no—
224C2×C16—81no—
225C2×C16—321no—
226C2×C16—321no—
227C2×C16—641no—
228C2×C16—641no—
229C2×C16#1206211yescharacteristic
230C2×C16#1206811yescharacteristic
231#108—81no—
232#107—81no—
233C4⋊C8—161no—
234#104—161no—
235#102—161no—
236C22⋊C8—321no—
237C4×C8—21no—
238C4×C8#1206711yescharacteristic
239C32—21no—
240C32—41no—
241C32—81no—
242C32—321no—
243C32—321no—
244C32—641no—
245C32D4×C3211yescharacteristic
246C32#1364811yescharacteristic
247#41—161no—
248C2×D4—161no—
249C22×C4—321no—
250Q16—81no—
251SD16—161no—
252D8—82no—
253D8—81no—
254D8—161no—
255M16—161no—
256M16—321no—
257C2×C8—41no—
258C2×C8—321no—
259C2×C8—321no—
260C2×C8—641no—
261C2×C8—641no—
262C2×C8#12296311yescharacteristic
263C2×C8AGL(1,ℤ/32)11yescharacteristic
264C4⋊C4—161no—
265C22⋊C4—321no—
266C42AGL(1,ℤ/32)11yescharacteristic
267C16—21no—
268C16—41no—
269C16—161no—
270C16—321no—
271C16—641no—
272C16#15110111yescharacteristic
273C16#15110311yescharacteristic
274C23—321no—
275Q8—161no—
276D4—162no—
277D4—161no—
278D4—321no—
279D4—321no—
280C2×C4—21no—
281C2×C4—321no—
282C2×C4—321no—
283C2×C4—641no—
284C2×C4—641no—
285C2×C4#7222463911yescharacteristic
286C8—21no—
287C8—81no—
288C8—321no—
289C8—641no—
290C8#7222464711yescharacteristic
291C8#7223071011yescharacteristic
292C22—321no—
293C22—322no—
294C22—641no—
295C22—641no—
296C22#7222466211yescharacteristic
297C4—41no—
298C4—321no—
299C4—641no—
300C4#7222466211yescharacteristic
301C4#7223071511yescharacteristic
302C2—21no—
303C2—641no—
304C2—642no—
305C2#7223072011yescharacteristic
subgroupG/H
center, socleC2#72230720
commutatorC64C22×C32
Frattini, cosocle#72230688C23
prime radical#153706C16
LCSG ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C8×C16 ◁ #93666 ◁ #72230692 ◁ G
LECSG ▷ #72230688 ▷ #7329 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72230812C2semidirect
2. #72230810C2semidirect
12. #72230761C2semidirect
13. #72230759C2semidirect
23. #72230730C2semidirect
29. #72230720C2semidirect
90. SD256C32semidirect
91. D128C32semidirect
130. Q128C2×C32semidirect
134. D64C2×C32semidirect
167. C128C2×C32semidirect

Maximal subgroups

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

subgroupindex
#722308122
#722308102
#722308052
#722307612
#722307592
#722307302
#722307202