Finite Group Databasebeta

← #141513

Subgroups — #141513

Order 512.

335 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 78 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#28677C211yesmaximal, characteristic
2#13302C211yesmaximal, characteristic
3#13227C211yesmaximal, characteristic
4#11908C211yesmaximal, characteristic
5#11883C211yesmaximal, characteristic
6#8617C211yesmaximal, characteristic
7#8165C211yesmaximal, characteristic
8C2×#498C411yescharacteristic
9C2×C8×Q8C2211yescharacteristic
10#2567C2211yescharacteristic
11#2546C2211yescharacteristic
12C2×#356—22no—
13#2088—21no—
14#2088—21no—
15#1988C411yescharacteristic
16#1987C411yescharacteristic
17#1941—21no—
18#1941—21no—
19#1801C2211yescharacteristic
20#1801C2211yescharacteristic
21C4×#106C411yescharacteristic
22C4×#101C2211yescharacteristic
23#1567—21no—
24#1461C2211yescharacteristic
25#1451—21no—
26#1394—22no—
27#1354—22no—
28#1346—22no—
29#1292—21no—
30#1283—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8C2×C411yescharacteristic
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C4×Q8C2311yescharacteristic
35#498—41no—
36#498—41no—
37C8×Q8—41no—
38C8×Q8—41no—
39#431—21no—
40C2×#106—21no—
41C2×#106—21no—
42C2×#106—21no—
43#423—21no—
44#423—21no—
45C2×C4⋊C8C2×C411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×#101—21no—
49C2×#101—21no—
50C2×#101D412yes—
51C4×M16—21no—
52C2×C4×C8C2×C411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C2×C4×C8D411yescharacteristic
55#398—21no—
56#398—21no—
57#395—21no—
58#395—21no—
59#391—21no—
60#391—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#389—21no—
65#384—21no—
66C4×C4⋊C4C2×C411yescharacteristic
67C4×C4⋊C4C2×C411yescharacteristic
68#356—44no—
69#343—21no—
70#343—21no—
71#343—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#335—41no—
78#335—41no—
79#324—21no—
80#324D412yes—
81#322—41no—
82#322—41no—
83C22×Q8—21no—
84C22×Q8—21no—
85C22×Q8C2×D411yescharacteristic
86C2×M16—22no—
87C2×M16—41no—
88C2×M16—41no—
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—42no—
98C4⋊Q8—42no—
99C4×Q8—41no—
100C4×Q8—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4C2×D411yescharacteristic
112C2×C4⋊C4#4111yescharacteristic
113C2×C4⋊C4#4111yescharacteristic
114C2×C42—21no—
115C2×C42—21no—
116C2×C42C22×C411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119#106—24no—
120#106—42no—
121#106—41no—
122#106—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131#101—22no—
132#101—22no—
133#101—42no—
134#101—42no—
135C22⋊C8—22no—
136#97—41no—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8—21no—
143C4×C8—41no—
144C4×C8—41no—
145#95—21no—
146#95—21no—
147#95—41no—
148#95—41no—
149#95—41no—
150#95—41no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—42no—
154C2×Q8—81no—
155C2×Q8—81no—
156C2×Q8—81no—
157C2×Q8—161no—
158C2×Q8AGL(1,ℤ/8)12yes—
159C2×Q8#13712yes—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4C4×D411yescharacteristic
166C22×C4C4×D411yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4#12111yescharacteristic
169C22×C4#12411yescharacteristic
170C22×C4#12711yescharacteristic
171M16—42no—
172M16—42no—
173M16—81no—
174M16—81no—
175M16—81no—
176M16—81no—
177C2×C8—22no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—81no—
187C2×C8—81no—
188C2×C8#9911yescharacteristic
189C2×C8#9911yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—82no—
204C4⋊C4—81no—
205C4⋊C4—81no—
206C4⋊C4—161no—
207C4⋊C4—161no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—42no—
215C42—81no—
216C23#39011yescharacteristic
217Q8—42no—
218Q8—42no—
219Q8—162no—
220Q8—162no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4C2×#9911yescharacteristic
248C2×C4#41011yescharacteristic
249C2×C4#44012yes—
250C2×C4#46812yes—
251C2×C4#49611yescharacteristic
252C2×C4#49711yescharacteristic
253C8—21no—
254C8—41no—
255C8—41no—
256C8—41no—
257C8—82no—
258C8—82no—
259C8—161no—
260C8—161no—
261C8#35612yescomplemented
262C22—21no—
263C22—21no—
264C22#196411yescharacteristic
265C22#197311yescharacteristic
266C22#197611yescharacteristic
267C4—22no—
268C4—21no—
269C4—22no—
270C4—21no—
271C4—21no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—82no—
276C4—82no—
277C4—161no—
278C4—161no—
279C4C2×#35611yescharacteristic
280C4#213511yescharacteristic
281C2—21no—
282C2—21no—
283C2#978611yescharacteristic
284C2#1322711yescharacteristic
285C2#1323211yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC23#390
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
261. C8#356semidirect

Maximal subgroups

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

subgroupindex
#286772
#133022
#132272
#119082
#118832
#86172
#81652