Finite Group Databasebeta

← #140658

Subgroups — #140658

Order 512.

438 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 96 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#21050C211yesmaximal, characteristic
2#12786C211yesmaximal, characteristic, complemented
3#12602C211yesmaximal, characteristic, complemented
4C2×#1328C212yesmaximal, complemented
5C2×#1320C212yesmaximal
6C2×#424C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C22×C22⋊C8C411yescharacteristic
9C2×#405C2211yescharacteristic
10#2280C411yescharacteristic
11#2150—21no—
12C2×#349C2212yes—
13C2×#349C2212yes—
14C4×M32—21no—
15#1924—21no—
16#1924—21no—
17#1923—21no—
18#1923—21no—
19#1729C411yescharacteristic
20#1729C411yescharacteristic
21#1377—21no—
22#1377—21no—
23#1328—24no—
24#1320—24no—
25C24×C4C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C22×M16C2311yescharacteristic
28C23×C8C2×C411yescharacteristic
29C23×C8C2×C411yescharacteristic
30C2×#117C2×C411yescharacteristic
31C2×#117C2×C411yescharacteristic
32C2×M32—22no—
33C2×M32—22no—
34#424—22no—
35#424—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38#407—22no—
39#407—22no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—41no—
45C2×C22⋊C8—41no—
46#405—41no—
47#405—41no—
48C2×#97D411yescharacteristic
49C2×#97D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52#381C811yescharacteristic
53#381C811yescharacteristic
54#380—21no—
55#380—21no—
56#379C811yescharacteristic
57#379C811yescharacteristic
58C4×C22⋊C4—21no—
59C4×C22⋊C4—21no—
60C4⋊C16—41no—
61C4⋊C16—41no—
62#349—22no—
63#349—22no—
64#349—22no—
65#349D414yes—
66#346—41no—
67C4×C16—41no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72C25C2×C811yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—41no—
78C23×C4—41no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4C2×C811yescharacteristic
82C23×C4C22×C411yescharacteristic
83C2×M16—22no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16C2×D412yes—
89C2×M16#4112yes—
90C22×C8—22no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—22no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8C22⋊C411yescharacteristic
100C22×C8C22⋊C411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103#117—22no—
104#117—22no—
105#117—41no—
106#117—41no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4M1611yescharacteristic
109C2×C4⋊C4M1611yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4C2×C811yescharacteristic
115C2×C22⋊C4C2×C811yescharacteristic
116C2×C22⋊C4C2×C811yescharacteristic
117C2×C22⋊C4C2×C811yescharacteristic
118C2×C42—21no—
119C2×C42C22⋊C411yescharacteristic
120C2×C42C22⋊C411yescharacteristic
121M32—24no—
122M32—42no—
123M32—42no—
124M32—42no—
125C2×C16—42no—
126C2×C16—42no—
127C4⋊C8—42no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C22⋊C8—42no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—42no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136#97—41no—
137#97—41no—
138C4×C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—21no—
146C24—21no—
147C24—22no—
148C24—41no—
149C24—42no—
150C24—41no—
151C24—81no—
152C24C22×C811yescharacteristic
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4C22⋊C811yescharacteristic
177C22×C4C22⋊C811yescharacteristic
178C22×C4C22⋊C811yescharacteristic
179C22×C4C22⋊C811yescharacteristic
180C22×C4C2×C22⋊C411yescharacteristic
181C22×C4C4×D412yes—
182C22×C4C2×M1611yescharacteristic
183C22×C4#13112yes—
184M16—42no—
185M16—42no—
186M16—82no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—82no—
205C2×C8#9911yescharacteristic
206C2×C8#9911yescharacteristic
207C2×C8#12112yes—
208C2×C8#12411yescharacteristic
209C2×C8#12711yescharacteristic
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C22⋊C4—22no—
214C22⋊C4—22no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—81no—
220C22⋊C4—81no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—41no—
226C42—41no—
227C16—44no—
228C16—82no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—42no—
238C23—42no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—82no—
243C23—84no—
244C23—84no—
245C23—82no—
246C23C2×C22⋊C811yescharacteristic
247C23C8×D412yes—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
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—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—84no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4#39011yescharacteristic
278C2×C4#40811yescharacteristic
279C2×C4C2×#9911yescharacteristic
280C2×C4#41011yescharacteristic
281C2×C4#41311yescharacteristic
282C2×C4#41311yescharacteristic
283C2×C4#43612yes—
284C8—22no—
285C8—21no—
286C8—21no—
287C8—21no—
288C8—21no—
289C8—41no—
290C8—81no—
291C8—81no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—81no—
303C22—81no—
304C22—84no—
305C22—84no—
306C22—81no—
307C22—81no—
308C22#145811yescharacteristic
309C22#145911yescharacteristic
310C22#195411yescharacteristic
311C4—22no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4#146811yescharacteristic
320C4#147311yescharacteristic
321C4#196411yescharacteristic
322C4#196711yescharacteristic
323C2—21no—
324C2—21no—
325C2—42no—
326C2—82no—
327C2#965111yescharacteristic
328C2#1148311yescharacteristic
329C2#1148411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C22×C8
Frattini, cosocleC22×M16C23
socleC22#1459
prime radicalC25C2×C8
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12786C2semidirect
3. #12602C2semidirect
4. C2×#1328C2semidirect

Maximal subgroups

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

subgroupindex
#210502
#127862
#126022
C2×#13282
C2×#13202