Finite Group Databasebeta

← #102160

Subgroups — #102160

Order 512.

632 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 118 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#33171C211yesmaximal, characteristic
2#10820C211yesmaximal, characteristic, complemented
3#9680C212yesmaximal, complemented
4#9680C212yesmaximal, complemented
5#8145C211yesmaximal, characteristic, complemented
6C2×#493C411yescharacteristic, complemented
7C2×#466—21no—
8C2×#466—21no—
9C22×C4×C8C2211yescharacteristic
10#2856C2212yes—
11#1970—21no—
12#1970—21no—
13#1957C412yescomplemented
14#1845C411yescharacteristic, complemented
15C4×C22⋊C8C2211yescharacteristic
16C4×C22⋊C8C2211yescharacteristic
17C2×#340—21no—
18C2×#340—21no—
19C2×#327—24no—
20C2×#327—24no—
21C2×#322—21no—
22C4⋊#95C2212yes—
23#1273—21no—
24C22×D8—21no—
25C22×D8—21no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#127C2×C412yes—
29C2×#121—22no—
30C2×#121—22no—
31C2×#121—22no—
32C22×C4⋊C4—22no—
33C22×C42C2311yescharacteristic
34#493—42no—
35#466—44no—
36#466—44no—
37C2×#107—21no—
38C2×#107—21no—
39C2×#102—22no—
40C2×#102—22no—
41C2×C22⋊C8D412yes—
42C2×C22⋊C8D412yes—
43C2×C4×C8—21no—
44C2×C4×C8—21no—
45C2×C4×C8—42no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#390—22no—
51#390—22no—
52#390—22no—
53#384C2×C412yes—
54C2×#95—22no—
55C2×#95—22no—
56#340—24no—
57#340—42no—
58#336—21no—
59#336—21no—
60#327—48no—
61#327—48no—
62#322—42no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—42no—
67C23×C4—42no—
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C2×D8—42no—
72C2×D8—42no—
73C2×D8—42no—
74C2×D8—42no—
75C2×D8—42no—
76C2×D8—42no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8C22⋊C411yescharacteristic
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89#127—42no—
90#127—42no—
91#121—44no—
92#121—44no—
93#121—44no—
94#121—44no—
95#121—42no—
96#121—42no—
97#121—42no—
98#121—42no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C42—41no—
112C2×C42—41no—
113C2×C42C22×C411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42#4111yescharacteristic
116C2×C42#4111yescharacteristic
117#107—42no—
118#107—42no—
119#102—44no—
120#102—44no—
121C22⋊C8—22no—
122C22⋊C8—24no—
123C22⋊C8D814yescomplemented
124C4×C8—22no—
125C4×C8—22no—
126C4×C8—41no—
127C4×C8—41no—
128C4×C8—41no—
129C4×C8—41no—
130C4×C8—41no—
131C4×C8—41no—
132#95—44no—
133#95—44no—
134C24—42no—
135C24#12711yescharacteristic
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—44no—
143C2×D4—44no—
144C2×D4—82no—
145C2×D4—82no—
146C2×D4—82no—
147C2×D4—82no—
148C2×D4—82no—
149C2×D4—82no—
150C2×D4—82no—
151C2×D4—82no—
152C2×D4—82no—
153C2×D4—82no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4C2×C22⋊C411yescharacteristic
171C22×C4C4×D411yescharacteristic
172C22×C4C4×D411yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12411yescharacteristic
176C22×C4C2×D812yes—
177C22×C4#13512yes—
178C22×C4AGL(1,ℤ/8)12yes—
179C22×C4AGL(1,ℤ/8)12yes—
180C22×C4AGL(1,ℤ/8)14yes—
181D8—84no—
182D8—84no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—24no—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8#9912yes—
202C2×C8#10412yes—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C22⋊C4—84no—
212C22⋊C4—84no—
213C22⋊C4—82no—
214C22⋊C4—82no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—41no—
220C42—41no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23—82no—
229C23—82no—
230C23—82no—
231C23—82no—
232C23#39011yescharacteristic
233C23#49311yescharacteristic
234C23#49511yescharacteristic
235C23#49612yes—
236C23#49612yes—
237D4—82no—
238D4—82no—
239D4—82no—
240D4—82no—
241D4—164no—
242D4—164no—
243D4—162no—
244D4—162no—
245D4—162no—
246D4—162no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—84no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—162no—
271C2×C4—162no—
272C2×C4—162no—
273C2×C4—162no—
274C2×C4C2×#9911yescharacteristic
275C2×C4#41011yescharacteristic
276C2×C4#41811yescharacteristic
277C2×C4#41811yescharacteristic
278C2×C4C2×#10411yescharacteristic
279C2×C4#42111yescharacteristic
280C2×C4C4×D812yes—
281C2×C4#44212yes—
282C2×C4#46612yes—
283C2×C4#46912yes—
284C2×C4#49012yes—
285C8—22no—
286C8—22no—
287C8—44no—
288C8—41no—
289C8—41no—
290C8—82no—
291C22—22no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—22no—
296C22—21no—
297C22—21no—
298C22—22no—
299C22—162no—
300C22—162no—
301C22—162no—
302C22—162no—
303C22—162no—
304C22—162no—
305C22—162no—
306C22#171712yes—
307C22#172112yes—
308C22#196111yescharacteristic
309C22#196411yescharacteristic
310C22#197011yescharacteristic
311C22#197211yescharacteristic
312C22#197611yescharacteristic
313C22#197611yescharacteristic
314C22#198011yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—164no—
321C4—162no—
322C4#150511yescharacteristic
323C4#150711yescharacteristic
324C4#151011yescharacteristic
325C4#151411yescharacteristic
326C2—22no—
327C2—22no—
328C2—322no—
329C2#968411yescharacteristic
330C2#971911yescharacteristic
331C2#972111yescharacteristic
332C2#976011yescharacteristic
333C2#978311yescharacteristic
334C2#978511yescharacteristic
335C2#983211yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radical#33171C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #10820C2semidirect
3. #9680C2semidirect
4. #9680C2semidirect
5. #8145C2semidirect
6. C2×#493C4semidirect
13. #1957C4semidirect
14. #1845C4semidirect
123. C22⋊C8D8semidirect

Maximal subgroups

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

subgroupindex
#331712
#108202
#96802
#81452