Finite Group Databasebeta

← #116912

Subgroups — #116912

Order 512.

669 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 669 conjugacy classes of subgroups — too many to draw. Only its 126 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#21960C211yesmaximal, characteristic, complemented
2#21851C211yesmaximal, characteristic
3#20748C212yesmaximal
4#7988C211yesmaximal, characteristic, complemented
5#7977C212yesmaximal, complemented
6C2×C4×#41C2211yescharacteristic
7#2920C2212yes—
8C2×#432C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10#2283C2212yes—
11#1985C412yescomplemented
12#1955C412yescomplemented
13#1949—24no—
14#1869—21no—
15#1869—21no—
16#1826—22no—
17#1808—21no—
18#1807—21no—
19#1806—24no—
20#1797—22no—
21#1789—22no—
22#1759—24no—
23#1290—22no—
24#1288—24no—
25C22×#41D411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C4×#41—22no—
29C4×#41—22no—
30C4×#41—22no—
31C4×#41—22no—
32C2×C4×Q8C2×C411yescharacteristic
33C2×C4×Q8Q811yescharacteristic
34C2×C4×D4—21no—
35C2×C4×D4—21no—
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4Q811yescharacteristic
38C2×#117—21no—
39C2×#117D411yescharacteristic
40C22×C42—21no—
41C22×C42C2311yescharacteristic
42#432—22no—
43#432—22no—
44#432—22no—
45#432—22no—
46#431—22no—
47#431—22no—
48C2×#107—22no—
49C2×#106—22no—
50#423—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8C2×C412yes—
55C2×C4⋊C8D412yes—
56C2×#104—28no—
57#407—22no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C4×M16D414yes—
62C2×#97C2×C412yes—
63#383—22no—
64#382—22no—
65#381—22no—
66#380—22no—
67C4×C4⋊C4D414yes—
68C4×C22⋊C4—22no—
69#376—22no—
70#340—22no—
71#340—22no—
72#339—28no—
73#336—22no—
74#336—22no—
75C2×#41—22no—
76C2×#41—22no—
77C2×#41—44no—
78C22×Q8C4⋊C411yescharacteristic
79C22×D4—21no—
80C22×D4C4⋊C411yescharacteristic
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4#4111yescharacteristic
87C2×M16—24no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—81no—
94C22×C8—81no—
95C22×C8C22⋊C412yes—
96C22×C8C22⋊C412yes—
97C4×Q8—22no—
98C4×Q8—22no—
99C4×Q8—22no—
100C4×Q8—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—41no—
106C4×D4—42no—
107C4×D4—41no—
108C4×D4—41no—
109C4×D4—42no—
110C4×D4—41no—
111#117—22no—
112#117—22no—
113#117—42no—
114#117—42no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4C4⋊C411yescharacteristic
119C2×C4⋊C4C4⋊C411yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—42no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—24no—
128C2×C42—42no—
129C2×C42—42no—
130C2×C42C22×C411yescharacteristic
131C2×C42C2×D412yes—
132C2×C42C2×D412yes—
133C2×C42C2×D411yescharacteristic
134C2×C42C2×Q811yescharacteristic
135C2×C42#4112yes—
136C2×C42#4112yes—
137C2×C42#4111yescharacteristic
138#107—82no—
139#106—82no—
140C4⋊C8—24no—
141C4⋊C8—24no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145#104—416no—
146C22⋊C8—42no—
147C22⋊C8—42no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—82no—
151C22⋊C8—82no—
152#97—24no—
153#97—24no—
154C4×C8—24no—
155#95—24no—
156#95—42no—
157#95—42no—
158#95—42no—
159C24—21no—
160C24#12111yescharacteristic
161#41—28no—
162#41—44no—
163#41—44no—
164#41—44no—
165C2×Q8—22no—
166C2×Q8—42no—
167C2×D4—24no—
168C2×D4—22no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—41no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4C2×C4⋊C411yescharacteristic
192C22×C4#11711yescharacteristic
193C22×C4#12012yes—
194C22×C4#12011yescharacteristic
195C22×C4#12112yes—
196C22×C4#12111yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4C22⋊Q811yescharacteristic
199C22×C4#12311yescharacteristic
200C22×C4#12312yes—
201C22×C4#12411yescharacteristic
202C22×C4#12612yes—
203C22×C4#12711yescharacteristic
204M16—48no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—44no—
215C2×C8—81no—
216C2×C8—81no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—42no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—84no—
230C42—48no—
231C42—41no—
232C42—41no—
233C42—42no—
234C42—42no—
235C42C4×D412yes—
236C42C4×D412yes—
237C42#12112yes—
238C42#12112yes—
239C42C22⋊Q812yes—
240C42C22⋊Q812yes—
241C42#12312yes—
242C42#12412yes—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—41no—
251C23#38011yescharacteristic
252C23#39211yescharacteristic
253C23#39711yescharacteristic
254Q8—24no—
255Q8—42no—
256D4—24no—
257D4—42no—
258D4—44no—
259D4—44no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—48no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—82no—
286C2×C4—84no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—84no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4#38512yes—
293C2×C4#38611yescharacteristic
294C2×C4#38812yes—
295C2×C4#39011yescharacteristic
296C2×C4#39311yescharacteristic
297C2×C4#39412yes—
298C2×C4#39612yes—
299C2×C4#39911yescharacteristic
300C8—42no—
301C8—42no—
302C8—82no—
303C8—84no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—44no—
312C22—44no—
313C22—42no—
314C22—82no—
315C22—81no—
316C22—81no—
317C22#143911yescharacteristic
318C22#181412yes—
319C22#201012yes—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—22no—
324C4—22no—
325C4—22no—
326C4—22no—
327C4—42no—
328C4—81no—
329C4—81no—
330C4—164no—
331C4#191111yescharacteristic
332C4#191211yescharacteristic
333C2—22no—
334C2—22no—
335C2—41no—
336C2—82no—
337C2#1107211yescharacteristic
338C2#1110011yescharacteristic
339C2#1110411yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21960C2semidirect
4. #7988C2semidirect
5. #7977C2semidirect
11. #1985C4semidirect
12. #1955C4semidirect

Maximal subgroups

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

subgroupindex
#219602
#218512
#207482
#79882
#79772