Finite Group Databasebeta

← #100612

Subgroups — #100612

Order 512.

392 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 98 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#22824C211yesmaximal, characteristic
3#9905C211yesmaximal, characteristic
4#9896C211yesmaximal, characteristic
5#7577C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic
7#7022C211yesmaximal, characteristic
8C2×#462—21no—
9#2868C2211yescharacteristic
10#2295C2211yescharacteristic
11#2043—21no—
12#2043C412yes—
13#2016—21no—
14#2016C412yes—
15#1985—21no—
16#1981—21no—
17#1917—21no—
18#1916—21no—
19#1842C2211yescharacteristic
20#1832—21no—
21#1831C2211yescharacteristic
22#1831C2211yescharacteristic
23#1771—21no—
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—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8D411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×C4⋊C8D411yescharacteristic
49C2×#103—22no—
50C2×#103—22no—
51C2×#103—22no—
52C2×#97—21no—
53C2×#97—21no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56#391C2×C411yescharacteristic
57#391D411yescharacteristic
58#389C2×C411yescharacteristic
59#389D411yescharacteristic
60#387—21no—
61#387—21no—
62#384—21no—
63#384—21no—
64#384—21no—
65#384—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4D411yescharacteristic
68#376—21no—
69C43C2311yescharacteristic
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336C2×C412yes—
82C22×Q8—22no—
83C22×Q8C22⋊C411yescharacteristic
84C2×Q16—44no—
85C2×Q16—82no—
86C2×Q16—82no—
87C22×C8—21no—
88C22×C8—22no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—41no—
93C22×C8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—82no—
99C4⋊Q8—82no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C22⋊C411yescharacteristic
114C2×C4⋊C4C22⋊C411yescharacteristic
115C2×C4⋊C4C22⋊C411yescharacteristic
116C2×C42—21no—
117C2×C42—21no—
118C2×C42C22×C411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125C4⋊C8—22no—
126C4⋊C8—21no—
127C4⋊C8—21no—
128C4⋊C8—21no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8Q1612yescomplemented
141#103—44no—
142#103—82no—
143#103—82no—
144#97—41no—
145#97—41no—
146#97—81no—
147C4×C8—81no—
148C4×C8—81no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C4×D411yescharacteristic
173C22×C4C4×D411yescharacteristic
174C22×C4#12011yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12311yescharacteristic
178Q16—88no—
179C2×C8—22no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—81no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—41no—
222C42—41no—
223C42—41no—
224C42—41no—
225C42—41no—
226C42—41no—
227C42—41no—
228C42C2×Q1611yescharacteristic
229C42#13511yescharacteristic
230C42#13511yescharacteristic
231C42#13511yescharacteristic
232C42AGL(1,ℤ/8)11yescharacteristic
233C42AGL(1,ℤ/8)11yescharacteristic
234C42#13711yescharacteristic
235C42#13711yescharacteristic
236C23#38611yescharacteristic
237Q8—42no—
238Q8—81no—
239Q8—82no—
240Q8—82no—
241Q8—82no—
242Q8—82no—
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—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4#41611yescharacteristic
272C2×C4#41811yescharacteristic
273C2×C4C4×Q1611yescharacteristic
274C2×C4#44111yescharacteristic
275C2×C4#44911yescharacteristic
276C2×C4#44911yescharacteristic
277C2×C4#45111yescharacteristic
278C2×C4#45211yescharacteristic
279C2×C4#45511yescharacteristic
280C2×C4#45611yescharacteristic
281C2×C4#46211yescharacteristic
282C2×C4#46411yescharacteristic
283C2×C4#47111yescharacteristic
284C2×C4#47311yescharacteristic
285C2×C4#48211yescharacteristic
286C2×C4#48211yescharacteristic
287C8—42no—
288C8—82no—
289C8—82no—
290C8—81no—
291C8—81no—
292C8—81no—
293C8—161no—
294C22#187511yescharacteristic
295C22#187811yescharacteristic
296C22#188211yescharacteristic
297C22#190211yescharacteristic
298C22#190811yescharacteristic
299C22#191011yescharacteristic
300C22#191411yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—41no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—162no—
317C4#164011yescharacteristic
318C4#164411yescharacteristic
319C4#164811yescharacteristic
320C4#165211yescharacteristic
321C2#898611yescharacteristic
322C2#917211yescharacteristic
323C2#919811yescharacteristic
324C2#926811yescharacteristic
325C2#929011yescharacteristic
326C2#933711yescharacteristic
327C2#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
140. C4⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#297992
#228242
#99052
#98962
#75772
#70402
#70222