Finite Group Databasebeta

← #99719

Subgroups — #99719

Order 512.

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

normal subgroups only

This group has 491 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#23627C211yesmaximal, characteristic
2#21936C211yesmaximal, characteristic
3#8709C212yesmaximal, complemented
4#7563C211yesmaximal, characteristic
5#7029C212yesmaximal, complemented
6C2×#416—21no—
7C2×#408—21no—
8#2297C2211yescharacteristic
9#2285C2211yescharacteristic
10#2074—21no—
11#2073—21no—
12#2072C412yes—
13#2071C412yes—
14#1851—22no—
15#1850—22no—
16#1798—21no—
17#1797C2212yes—
18C4×C22⋊C8C2212yes—
19#1735—21no—
20C2×#336C2211yescharacteristic
21#1279—22no—
22#1279—22no—
23#1272—21no—
24#1270—21no—
25C22×#41D411yescharacteristic
26C22×M16—21no—
27C23×C8—21no—
28C23×C8—21no—
29C2×#117—21no—
30C2×#117D411yescharacteristic
31C22×C42C2311yescharacteristic
32C2×C4⋊C8—21no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8D412yes—
35C2×C4⋊C8D412yes—
36#416—42no—
37#416—42no—
38C2×#103—22no—
39C2×#102—22no—
40#408—42no—
41#408—42no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C4×C8—22no—
46#391C2×C411yescharacteristic
47#391D411yescharacteristic
48#390C2×C411yescharacteristic
49#390D411yescharacteristic
50#386—22no—
51#384C2×C412yes—
52#382—21no—
53#380—22no—
54C4×C4⋊C4—22no—
55C4×C22⋊C4—22no—
56#340—22no—
57#340—22no—
58#336—22no—
59#336—22no—
60#336—21no—
61#336—22no—
62#336C2×C412yes—
63C2×#41—21no—
64C2×#41—21no—
65C2×#41—41no—
66C2×#41—41no—
67C2×#41—41no—
68C2×#41—41no—
69C2×#41D812yes—
70C2×#41SD1612yes—
71C22×Q8C22⋊C411yescharacteristic
72C22×D4—21no—
73C22×D4C22⋊C411yescharacteristic
74C23×C4—21no—
75C23×C4C2×D411yescharacteristic
76C23×C4#4112yes—
77C2×M16—42no—
78C2×M16—81no—
79C2×M16—81no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—81no—
92C22×C8—81no—
93#117—41no—
94#117—41no—
95#117—41no—
96#117—41no—
97#117—41no—
98#117—41no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4C22⋊C411yescharacteristic
103C2×C4⋊C4C22⋊C411yescharacteristic
104C2×C22⋊C4—22no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—41no—
108C2×C42—21no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42C22×C411yescharacteristic
113C2×C42C2×D412yes—
114C2×C42C2×D411yescharacteristic
115C4⋊C8—42no—
116C4⋊C8—42no—
117C4⋊C8—81no—
118C4⋊C8—81no—
119#103—44no—
120#102—44no—
121C22⋊C8—24no—
122C22⋊C8—42no—
123C22⋊C8—44no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C4×C8—42no—
127C4×C8—42no—
128#95—22no—
129#95—22no—
130#95—42no—
131C24—21no—
132C24#12311yescharacteristic
133#41—42no—
134#41—42no—
135#41—42no—
136#41—42no—
137#41—81no—
138#41—81no—
139#41—81no—
140#41—81no—
141C2×Q8—21no—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8#10212yes—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—21no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—81no—
151C2×D4—81no—
152C2×D4#10212yes—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C4×D412yes—
173C22×C4#12011yescharacteristic
174C22×C4#12112yes—
175C22×C4C2×D811yescharacteristic
176C22×C4C2×SD1611yescharacteristic
177C22×C4AGL(1,ℤ/8)11yescharacteristic
178C22×C4#13711yescharacteristic
179M16—84no—
180C2×C8—44no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—82no—
191C2×C8—82no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—84no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—44no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—82no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C42—22no—
211C42—22no—
212C42—41no—
213C42—41no—
214C42—41no—
215C42—41no—
216C42—81no—
217C42—81no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—22no—
223C23—42no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23#38611yescharacteristic
228C23#48011yescharacteristic
229C23#48111yescharacteristic
230Q8—42no—
231Q8—81no—
232D4—42no—
233D4—81no—
234D4—82no—
235D4—82no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
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—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4C2×#10211yescharacteristic
268C2×C4#41711yescharacteristic
269C2×C4#44312yes—
270C2×C4#44412yes—
271C2×C4#44711yescharacteristic
272C2×C4#44811yescharacteristic
273C2×C4#45511yescharacteristic
274C2×C4#45511yescharacteristic
275C2×C4#45811yescharacteristic
276C2×C4#45811yescharacteristic
277C8—82no—
278C8—84no—
279C8—82no—
280C8—82no—
281C22—22no—
282C22—22no—
283C22—21no—
284C22—21no—
285C22—41no—
286C22—42no—
287C22—41no—
288C22—44no—
289C22—42no—
290C22—81no—
291C22—82no—
292C22—81no—
293C22#187411yescharacteristic
294C22#187612yes—
295C22#188112yes—
296C22#188511yescharacteristic
297C22#189011yescharacteristic
298C4—21no—
299C4—21no—
300C4—22no—
301C4—22no—
302C4—42no—
303C4—42no—
304C4—81no—
305C4—81no—
306C4—162no—
307C4#160711yescharacteristic
308C4#160711yescharacteristic
309C4#160811yescharacteristic
310C4#160811yescharacteristic
311C2—41no—
312C2—41no—
313C2—82no—
314C2#899512yes—
315C2#900511yescharacteristic
316C2#902511yescharacteristic
317C2#903812yes—
318C2#906711yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8709C2semidirect
5. #7029C2semidirect

Maximal subgroups

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

subgroupindex
#236272
#219362
#87092
#75632
#70292