Finite Group Databasebeta

← #101271

Subgroups — #101271

Order 512.

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

normal subgroups only

This group has 521 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#23734C211yesmaximal, characteristic
2#21936C211yesmaximal, characteristic
3#8725C212yesmaximal, complemented
4#7597C211yesmaximal, characteristic
5#7045C212yesmaximal, complemented
6C2×#428—21no—
7C2×#417—21no—
8C2×#408—21no—
9#2297C2211yescharacteristic
10C2×#384C2211yescharacteristic
11#2096C412yes—
12#2095—21no—
13#2094—21no—
14#2093C412yes—
15#1851—22no—
16#1850—22no—
17#1798—21no—
18#1797C2212yes—
19#1797C2212yes—
20#1728C2211yescharacteristic
21#1296—22no—
22#1296—22no—
23#1272—21no—
24#1270—21no—
25C22×#41D411yescharacteristic
26C22×M16—21no—
27C22×M16—21no—
28C23×C8—21no—
29C2×#117D411yescharacteristic
30C22×C4⋊C4—21no—
31C22×C4⋊C4—22no—
32C22×C42C2311yescharacteristic
33#428—42no—
34#428—42no—
35C2×#107—22no—
36C2×#106—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8D412yes—
41C2×C4⋊C8D412yes—
42#417—42no—
43#417—42no—
44C2×#103—22no—
45C2×#102—22no—
46#408—42no—
47#408—42no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51#391C2×C411yescharacteristic
52#391D411yescharacteristic
53#390C2×C411yescharacteristic
54#390D411yescharacteristic
55#386—22no—
56#384—22no—
57#384—21no—
58#384C2×C412yes—
59C4×C22⋊C4—22no—
60#336—22no—
61#336—22no—
62#336—21no—
63#336—22no—
64#336C2×C412yes—
65C2×#41—21no—
66C2×#41—21no—
67C2×#41—41no—
68C2×#41—41no—
69C2×#41—41no—
70C2×#41—41no—
71C2×#41SD1612yes—
72C2×#41Q1612yes—
73C22×Q8C22⋊C411yescharacteristic
74C22×D4—21no—
75C22×D4C22⋊C411yescharacteristic
76C23×C4—21no—
77C23×C4—41no—
78C23×C4C2×D411yescharacteristic
79C23×C4#4112yes—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—42no—
87C2×M16—81no—
88C2×M16—81no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8—81no—
95C22×C8—81no—
96#117—41no—
97#117—41no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—84no—
107C2×C4⋊C4C22⋊C411yescharacteristic
108C2×C4⋊C4C22⋊C411yescharacteristic
109C2×C22⋊C4—22no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—22no—
112C2×C42—21no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D412yes—
117C2×C42C2×D411yescharacteristic
118#107—44no—
119#106—44no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123C4⋊C8—42no—
124C4⋊C8—81no—
125C4⋊C8—81no—
126#103—44no—
127#102—44no—
128C22⋊C8—44no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C22⋊C8—82no—
132#95—22no—
133#95—22no—
134C24—21no—
135C24#12311yescharacteristic
136#41—42no—
137#41—42no—
138#41—42no—
139#41—42no—
140#41—81no—
141#41—81no—
142#41—81no—
143#41—81no—
144C2×Q8—21no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8#10312yes—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—21no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—81no—
154C2×D4—81no—
155C2×D4#10312yes—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—84no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C4×D412yes—
179C22×C4#12011yescharacteristic
180C22×C4#12112yes—
181C22×C4C2×SD1611yescharacteristic
182C22×C4C2×Q1611yescharacteristic
183C22×C4AGL(1,ℤ/8)11yescharacteristic
184C22×C4#13711yescharacteristic
185M16—42no—
186M16—42no—
187M16—42no—
188M16—42no—
189M16—84no—
190C2×C8—24no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—84no—
200C2×C8—82no—
201C2×C8—82no—
202C2×C8—82no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—84no—
208C4⋊C4—84no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—82no—
214C22⋊C4—81no—
215C42—22no—
216C42—22no—
217C42—41no—
218C42—41no—
219C42—41no—
220C42—41no—
221C23—21no—
222C23—21no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—41no—
227C23—41no—
228C23—42no—
229C23—42no—
230C23#38611yescharacteristic
231C23#48311yescharacteristic
232C23#48411yescharacteristic
233Q8—42no—
234Q8—81no—
235D4—42no—
236D4—81no—
237D4—82no—
238D4—82no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
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—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—162no—
270C2×C4—162no—
271C2×C4C2×#10311yescharacteristic
272C2×C4#41711yescharacteristic
273C2×C4#45011yescharacteristic
274C2×C4#45111yescharacteristic
275C2×C4#45311yescharacteristic
276C2×C4#45311yescharacteristic
277C2×C4#45811yescharacteristic
278C2×C4#45811yescharacteristic
279C2×C4#47112yes—
280C2×C4#47312yes—
281C8—44no—
282C8—82no—
283C8—82no—
284C8—162no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—42no—
293C22—44no—
294C22—82no—
295C22—81no—
296C22—81no—
297C22#187211yescharacteristic
298C22#188711yescharacteristic
299C22#189011yescharacteristic
300C22#190312yes—
301C22#190812yes—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—21no—
306C4—42no—
307C4—42no—
308C4—81no—
309C4—81no—
310C4—162no—
311C4#161311yescharacteristic
312C4#161311yescharacteristic
313C4#161411yescharacteristic
314C4#161411yescharacteristic
315C2—41no—
316C2—41no—
317C2—82no—
318C2#902111yescharacteristic
319C2#916912yes—
320C2#942311yescharacteristic
321C2#943511yescharacteristic
322C2#952112yes—
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#41D4
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
3. #8725C2semidirect
5. #7045C2semidirect

Maximal subgroups

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

subgroupindex
#237342
#219362
#87252
#75972
#70452