Finite Group Databasebeta

← #100613

Subgroups — #100613

Order 512.

416 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 106 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#29799C211yesmaximal, characteristic
2#22723C211yesmaximal, characteristic
3#9905C211yesmaximal, characteristic
4#9896C211yesmaximal, characteristic
5#7577C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic
7#7022C211yesmaximal, characteristic
8C2×#462—21no—
9C2×#446—21no—
10#2868C2211yescharacteristic
11#2295C2211yescharacteristic
12#2043—21no—
13#2043C412yes—
14#2016—21no—
15#2016C412yes—
16#1981—21no—
17#1940—21no—
18#1940—21no—
19#1939—21no—
20#1843—21no—
21#1842C2211yescharacteristic
22#1831C2211yescharacteristic
23#1831C2211yescharacteristic
24#1768—21no—
25#1767C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27#1296—21no—
28#1296—21no—
29#1296—21no—
30#1296—21no—
31#1272—21no—
32#1271—21no—
33C22×Q16—22no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8C2×C412yes—
37C2×C4×Q8D411yescharacteristic
38#462—42no—
39#462—42no—
40#446—42no—
41#446—42no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8D411yescharacteristic
52C2×C4⋊C8D411yescharacteristic
53C2×C4⋊C8D411yescharacteristic
54C2×C4⋊C8D411yescharacteristic
55C2×#103—22no—
56C2×#103—22no—
57C2×#103—22no—
58C2×#97—21no—
59C2×#97—21no—
60C2×C4×C8—21no—
61C2×C4×C8—21no—
62#391C2×C411yescharacteristic
63#391D411yescharacteristic
64#389C2×C411yescharacteristic
65#389D411yescharacteristic
66#387—21no—
67#387—21no—
68#384—21no—
69#384—21no—
70#384—21no—
71#384—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4D411yescharacteristic
74#376—21no—
75C43C2311yescharacteristic
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336C2×C412yes—
84C22×Q8—22no—
85C22×Q8C22⋊C411yescharacteristic
86C2×Q16—44no—
87C2×Q16—82no—
88C2×Q16—82no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—41no—
95C22×C8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—82no—
101C4⋊Q8—82no—
102C4×Q8—21no—
103C4×Q8—21no—
104C4×Q8D812yes—
105C4×Q8SD1612yes—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4C22⋊C411yescharacteristic
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C42—21no—
119C2×C42—21no—
120C2×C42C22×C411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42#4111yescharacteristic
127C4⋊C8—22no—
128C4⋊C8—21no—
129C4⋊C8—21no—
130C4⋊C8—21no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—42no—
135C4⋊C8—42no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—81no—
145C4⋊C8—81no—
146C4⋊C8Q1612yescomplemented
147#103—44no—
148#103—82no—
149#103—82no—
150#97—42no—
151#97—41no—
152#97—41no—
153C4×C8—42no—
154C4×C8—81no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—21no—
162C2×Q8—21no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8#10212yes—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—42no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4#12011yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12311yescharacteristic
184Q16—88no—
185C2×C8—22no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C4⋊C4—21no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—81no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4#10212yes—
223C42—22no—
224C42—22no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—41no—
230C42—41no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42C2×D811yescharacteristic
235C42C2×SD1611yescharacteristic
236C42C2×Q1611yescharacteristic
237C42#13511yescharacteristic
238C42AGL(1,ℤ/8)11yescharacteristic
239C42#13711yescharacteristic
240C42#13711yescharacteristic
241C42#13711yescharacteristic
242C23#38611yescharacteristic
243Q8—42no—
244Q8—81no—
245Q8—82no—
246Q8—82no—
247Q8—82no—
248Q8—82no—
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—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4C2×#10211yescharacteristic
278C2×C4#41711yescharacteristic
279C2×C4C4×Q1611yescharacteristic
280C2×C4#44111yescharacteristic
281C2×C4#44711yescharacteristic
282C2×C4#44811yescharacteristic
283C2×C4#45111yescharacteristic
284C2×C4#45211yescharacteristic
285C2×C4#45511yescharacteristic
286C2×C4#45611yescharacteristic
287C2×C4#46211yescharacteristic
288C2×C4#46411yescharacteristic
289C2×C4#47111yescharacteristic
290C2×C4#47311yescharacteristic
291C2×C4#48011yescharacteristic
292C2×C4#48111yescharacteristic
293C8—42no—
294C8—82no—
295C8—82no—
296C8—81no—
297C8—81no—
298C8—81no—
299C8—161no—
300C22#187411yescharacteristic
301C22#187811yescharacteristic
302C22#188211yescharacteristic
303C22#190211yescharacteristic
304C22#190811yescharacteristic
305C22#191011yescharacteristic
306C22#191411yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—42no—
319C4—42no—
320C4—81no—
321C4—81no—
322C4—162no—
323C4#164111yescharacteristic
324C4#164511yescharacteristic
325C4#164911yescharacteristic
326C4#165311yescharacteristic
327C2#898711yescharacteristic
328C2#917111yescharacteristic
329C2#919711yescharacteristic
330C2#926811yescharacteristic
331C2#929011yescharacteristic
332C2#933711yescharacteristic
333C2#937611yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
146. C4⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#297992
#227232
#99052
#98962
#75772
#70402
#70222