Finite Group Databasebeta

← #140613

Subgroups — #140613

Order 512.

642 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 120 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#14155C211yesmaximal, characteristic
2C2×#1319C214yesmaximal
3#8558C211yesmaximal, characteristic
4C4×#324C211yesmaximal, characteristic
5C22×C4⋊Q8C411yescharacteristic
6C2×#391C412yes—
7C2×#391C2212yes—
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1797—21no—
11C4×C22⋊C8—21no—
12C2×#324C2212yes—
13C2×#324C2212yes—
14#1445—21no—
15#1445—21no—
16C4⋊#95—21no—
17C4⋊#95—21no—
18#1319—28no—
19#1280—21no—
20#1280—21no—
21C23×Q8C2×C412yes—
22C2×C4⋊Q8—22no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—41no—
25C2×C4⋊Q8—42no—
26C2×C4⋊Q8C814yescomplemented
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2311yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C412yes—
34C22×C42D411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C4⋊C8—41no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C4×C8—41no—
40#391—24no—
41#391—24no—
42#391—44no—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—42no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—21no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95D411yescharacteristic
52C2×#95D411yescharacteristic
53#336—41no—
54#336—41no—
55#324—22no—
56#324—24no—
57#324D414yes—
58C22×Q8—22no—
59C22×Q8—24no—
60C22×Q8—24no—
61C22×Q8—42no—
62C22×Q8—42no—
63C22×Q8—42no—
64C22×Q8—42no—
65C22×Q8C2×C814yes—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C22×C8—42no—
76C22×C8—42no—
77C4⋊Q8—44no—
78C4⋊Q8—44no—
79C4⋊Q8—44no—
80C4⋊Q8—44no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4C2×D412yes—
97C2×C4⋊C4#4112yes—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—22no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—24no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42—42no—
110C2×C42C2×C812yes—
111C2×C42M1612yes—
112C4⋊C8—82no—
113C22⋊C8—24no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117C4×C8—82no—
118#95—41no—
119#95—41no—
120#95—42no—
121#95—44no—
122#95—44no—
123#95—44no—
124#95—42no—
125#95—42no—
126C24C2×C22⋊C411yescharacteristic
127C2×Q8—28no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—44no—
131C2×Q8—44no—
132C2×Q8—44no—
133C2×Q8—84no—
134C2×Q8—84no—
135C2×Q8—84no—
136C2×Q8—84no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4C22⋊C812yes—
179C22×C4C22⋊C812yes—
180C22×C4#9911yescharacteristic
181C22×C4#9911yescharacteristic
182C22×C4#10012yes—
183C22×C4#10012yes—
184C22×C4C4×D412yes—
185C22×C4#12112yes—
186C22×C4#12411yescharacteristic
187C22×C4#12711yescharacteristic
188C22×C4C22×C811yescharacteristic
189C22×C4C2×M1611yescharacteristic
190C22×C4#13112yes—
191C2×C8—44no—
192C2×C8—82no—
193C2×C8—82no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8—82no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—44no—
200C4⋊C4—48no—
201C4⋊C4—48no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C42—22no—
205C42—22no—
206C42—22no—
207C42—22no—
208C42—42no—
209C42—42no—
210C42—81no—
211C42—81no—
212C42—84no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23#39011yescharacteristic
218C23C2×C22⋊C811yescharacteristic
219C23#40811yescharacteristic
220C23C2×#9911yescharacteristic
221C23#41011yescharacteristic
222C23C2×#10011yescharacteristic
223C23C2×#10011yescharacteristic
224Q8—88no—
225Q8—88no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4#32312yes—
280C2×C4#32312yes—
281C2×C4#35412yes—
282C2×C4#35612yes—
283C2×C4C8×D412yes—
284C2×C4#43612yes—
285C8—84no—
286C8—162no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22C2×#32311yescharacteristic
300C22#146011yescharacteristic
301C22#146911yescharacteristic
302C22#147111yescharacteristic
303C22#195411yescharacteristic
304C22#196411yescharacteristic
305C22#196511yescharacteristic
306C22C2×#35411yescharacteristic
307C22#212911yescharacteristic
308C22C2×#35611yescharacteristic
309C22#213511yescharacteristic
310C4—22no—
311C4—22no—
312C4—22no—
313C4—42no—
314C4—42no—
315C4—41no—
316C4—41no—
317C4—42no—
318C4—84no—
319C4—81no—
320C4—84no—
321C4#131914yescomplemented
322C2—21no—
323C2—21no—
324C2—21no—
325C2—21no—
326C2#964911yescharacteristic
327C2C2×#131911yescharacteristic
328C2#1143411yescharacteristic
329C2#1156411yescharacteristic
330C2#1156811yescharacteristic
331C2#1322411yescharacteristic
332C2#1322711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. C2×C4⋊Q8C8semidirect
321. C4#1319semidirect

Maximal subgroups

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

subgroupindex
#141552
C2×#13192
#85582
C4×#3242