Finite Group Databasebeta

← #102313

Subgroups — #102313

Order 512.

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

normal subgroups only

This group has 514 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#33172C211yesmaximal, characteristic
2#10797C211yesmaximal, characteristic
3#9909C212yesmaximal
4#8596C211yesmaximal, characteristic
5#7668C212yesmaximal
6C2×#494C411yescharacteristic, complemented
7C2×#467—21no—
8C2×#467—21no—
9C22×C4×C8C2211yescharacteristic
10#2857C2212yes—
11#1969—21no—
12#1969—21no—
13#1956C412yescomplemented
14#1845C411yescharacteristic, complemented
15#1797C2212yes—
16C2×#336—21no—
17C2×#336—21no—
18C2×#336C2211yescharacteristic
19C2×#335—21no—
20C4×#95C2211yescharacteristic
21#1294—24no—
22#1277—21no—
23#1271—24no—
24C22×Q16—21no—
25C22×Q16—21no—
26C23×C8—21no—
27C23×C8D411yescharacteristic
28C23×C8D411yescharacteristic
29C2×C4⋊Q8C2×C412yes—
30C2×C22⋊Q8—22no—
31C22×C42—21no—
32C22×C42C2311yescharacteristic
33#494—42no—
34#467—42no—
35#467—42no—
36#467—42no—
37#467—42no—
38C2×#107—21no—
39C2×#107—21no—
40C2×C4⋊C8—22no—
41C2×#103—22no—
42C2×#103—22no—
43C2×C22⋊C8—22no—
44C2×C4×C8—42no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49#391—22no—
50#389—22no—
51#389—22no—
52#384C2×C412yes—
53#380—22no—
54#380—22no—
55C2×#95—21no—
56C2×#95D412yes—
57#336—22no—
58#336—21no—
59#336—42no—
60#336—42no—
61#336D412yes—
62#335—42no—
63C22×Q8—22no—
64C23×C4—22no—
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C23×C4#4111yescharacteristic
68C2×Q16—42no—
69C2×Q16—42no—
70C2×Q16—42no—
71C2×Q16—42no—
72C2×Q16—42no—
73C2×Q16—42no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—42no—
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8D812yes—
93C22×C8SD1612yes—
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C22⋊Q8—42no—
97C22⋊Q8—42no—
98C22⋊Q8—42no—
99C22⋊Q8—42no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C42—41no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42—41no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42#4111yescharacteristic
119#107—42no—
120#107—42no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123#103—44no—
124#103—44no—
125C22⋊C8—44no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—41no—
129C4×C8—41no—
130#95—22no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95Q1614yescomplemented
135C24#12111yescharacteristic
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4C2×C22⋊C411yescharacteristic
161C22×C4C4×D411yescharacteristic
162C22×C4C4×D411yescharacteristic
163C22×C4#12111yescharacteristic
164C22×C4#12411yescharacteristic
165C22×C4#12711yescharacteristic
166C22×C4C2×D811yescharacteristic
167C22×C4C2×SD1611yescharacteristic
168C22×C4C2×Q1612yes—
169C22×C4#13512yes—
170C22×C4AGL(1,ℤ/8)11yescharacteristic
171C22×C4AGL(1,ℤ/8)11yescharacteristic
172Q16—84no—
173Q16—84no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—81no—
191C2×C8—81no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8#9912yes—
198C2×C8#10212yes—
199C2×C8#10212yes—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—41no—
215C42—41no—
216C42—81no—
217C42—81no—
218C42—81no—
219C42—81no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23#39011yescharacteristic
225C23#45911yescharacteristic
226C23#46011yescharacteristic
227C23#46712yes—
228C23#47312yes—
229Q8—82no—
230Q8—82no—
231Q8—82no—
232Q8—82no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4C2×#9911yescharacteristic
269C2×C4#41011yescharacteristic
270C2×C4C2×#10211yescharacteristic
271C2×C4#41811yescharacteristic
272C2×C4#42111yescharacteristic
273C2×C4#42111yescharacteristic
274C2×C4C4×Q1612yes—
275C2×C4#44412yes—
276C2×C4#45911yescharacteristic
277C2×C4#46011yescharacteristic
278C2×C4#49411yescharacteristic
279C2×C4#49511yescharacteristic
280C8—22no—
281C8—22no—
282C8—22no—
283C8—41no—
284C8—82no—
285C8—162no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—21no—
293C22—21no—
294C22#166912yes—
295C22#167012yes—
296C22#196311yescharacteristic
297C22#196411yescharacteristic
298C22#196911yescharacteristic
299C22#197511yescharacteristic
300C22#197611yescharacteristic
301C22#197911yescharacteristic
302C22#198211yescharacteristic
303C4—21no—
304C4—22no—
305C4—21no—
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—84no—
311C4—162no—
312C4—162no—
313C4#151011yescharacteristic
314C4#151411yescharacteristic
315C4#151711yescharacteristic
316C4#151911yescharacteristic
317C2—22no—
318C2—22no—
319C2#971511yescharacteristic
320C2#973311yescharacteristic
321C2#975311yescharacteristic
322C2#976511yescharacteristic
323C2#978211yescharacteristic
324C2#981711yescharacteristic
325C2#984811yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
6. C2×#494C4semidirect
13. #1956C4semidirect
14. #1845C4semidirect
134. #95Q16semidirect

Maximal subgroups

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

subgroupindex
#331722
#107972
#99092
#85962
#76682