Finite Group Databasebeta

← #120227

Subgroups — #120227

Order 512.

639 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 134 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#21948C211yesmaximal, characteristic
2#21932C211yesmaximal, characteristic
3#8107C212yesmaximal
4#8019C212yesmaximal
5C2×#1289C211yesmaximal, characteristic
6C2×#407C2212yes—
7C2×#407C2211yescharacteristic
8C22×C22⋊C8C2211yescharacteristic
9C2×#380C2212yes—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1855—24no—
12#1851C412yescomplemented
13#1851C412yescomplemented
14#1850—22no—
15#1810—22no—
16#1797—21no—
17#1797—21no—
18#1305—24no—
19#1294—22no—
20#1294—22no—
21#1289—21no—
22#1289—21no—
23C24×C4C2311yescharacteristic
24C22×M16—22no—
25C22×M16C2×C412yes—
26C23×C8D412yes—
27C22×C4⋊C4—22no—
28C22×C22⋊C4—21no—
29C22×C22⋊C4—22no—
30C22×C42—21no—
31C2×#108—24no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—22no—
34#407—22no—
35#407—22no—
36#407—22no—
37#407—22no—
38#407—22no—
39#407D414yes—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8C2×C412yes—
47C2×C22⋊C8D411yescharacteristic
48C2×C22⋊C8D411yescharacteristic
49#380—22no—
50#380—22no—
51#380—22no—
52#380Q814yes—
53C4×C22⋊C4—21no—
54C4×C22⋊C4—21no—
55C4×C22⋊C4C2×C411yescharacteristic
56C4×C22⋊C4C2×C411yescharacteristic
57C4×C22⋊C4Q811yescharacteristic
58C4×C22⋊C4Q811yescharacteristic
59C2×#95—22no—
60C2×#95D411yescharacteristic
61C2×#95D411yescharacteristic
62#341—24no—
63#336—22no—
64#336—22no—
65C25#4111yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—42no—
69C23×C4—82no—
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D412yes—
74C23×C4C2×D411yescharacteristic
75C23×C4C2×Q811yescharacteristic
76C23×C4C2×Q812yes—
77C23×C4#4112yes—
78C23×C4#4112yes—
79C23×C4#4111yescharacteristic
80C2×M16—24no—
81C2×M16—24no—
82C2×M16—42no—
83C2×M16—82no—
84C2×M16—84no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—42no—
89C22×C8—82no—
90C22×C8C22⋊C412yes—
91C22×C8C22⋊C412yes—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—84no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—44no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—82no—
103C2×C22⋊C4—82no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—82no—
109#108—84no—
110C4⋊C8—82no—
111C4⋊C8—82no—
112C22⋊C8—24no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115C22⋊C8—24no—
116C22⋊C8—24no—
117C22⋊C8—41no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—41no—
121C22⋊C8—42no—
122#95—21no—
123#95—21no—
124#95—24no—
125#95—42no—
126#95C4⋊C411yescharacteristic
127#95C4⋊C411yescharacteristic
128#95C4⋊C411yescharacteristic
129#95C4⋊C411yescharacteristic
130C24—22no—
131C24—21no—
132C24—21no—
133C24—42no—
134C24—81no—
135C24#11711yescharacteristic
136C24#12111yescharacteristic
137C24C22⋊Q811yescharacteristic
138C24C22⋊Q812yes—
139C24#12612yes—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—44no—
155C22×C4—81no—
156C22×C4—81no—
157C22×C4—82no—
158C22×C4—81no—
159C22×C4—82no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4—84no—
163C22×C4—84no—
164C22×C4—84no—
165C22×C4C2×C22⋊C411yescharacteristic
166C22×C4C2×C4⋊C411yescharacteristic
167C22×C4C4×D412yes—
168C22×C4C4×Q812yes—
169C22×C4#12011yescharacteristic
170C22×C4#12112yes—
171C22×C4#12112yes—
172C22×C4#12111yescharacteristic
173C22×C4C22⋊Q812yes—
174C22×C4C22⋊Q812yes—
175C22×C4C22⋊Q811yescharacteristic
176C22×C4#12311yescharacteristic
177C22×C4#12312yes—
178C22×C4#12411yescharacteristic
179C22×C4#12512yes—
180C22×C4#12512yes—
181C22×C4C4⋊Q812yes—
182C22×C4C4⋊Q811yescharacteristic
183M16—48no—
184M16—84no—
185M16—84no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—44no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—44no—
195C2×C8—82no—
196C2×C8—84no—
197C4⋊C4—84no—
198C4⋊C4—84no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—82no—
204C22⋊C4—82no—
205C22⋊C4—84no—
206C42—42no—
207C42—42no—
208C42—81no—
209C42—81no—
210C23—22no—
211C23—22no—
212C23—21no—
213C23—21no—
214C23—22no—
215C23—24no—
216C23—22no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—41no—
221C23—41no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—42no—
227C23—42no—
228C23—84no—
229C23—82no—
230C23—82no—
231C23#38011yescharacteristic
232C23#38712yes—
233C23#39511yescharacteristic
234C23#39612yes—
235C23#40011yescharacteristic
236C23#47112yes—
237C23#47211yescharacteristic
238C23#47311yescharacteristic
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
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—84no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4—84no—
275C2×C4—168no—
276C2×C4#38611yescharacteristic
277C2×C4#38912yes—
278C2×C4#39111yescharacteristic
279C2×C4#39411yescharacteristic
280C2×C4#39812yes—
281C2×C4#39811yescharacteristic
282C8—42no—
283C8—42no—
284C8—84no—
285C8—82no—
286C22—21no—
287C22—22no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—22no—
293C22—21no—
294C22—22no—
295C22—21no—
296C22—22no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—44no—
305C22—81no—
306C22—81no—
307C22—82no—
308C22—82no—
309C22#144111yescharacteristic
310C22#181511yescharacteristic
311C22#181611yescharacteristic
312C22#206312yes—
313C22#206412yes—
314C22#210212yes—
315C4—21no—
316C4—22no—
317C4—21no—
318C4—22no—
319C4—21no—
320C4—42no—
321C4—42no—
322C4—81no—
323C4—81no—
324C4—164no—
325C4#191312yes—
326C2—21no—
327C2—21no—
328C2—22no—
329C2—22no—
330C2—22no—
331C2—42no—
332C2—41no—
333C2#1120511yescharacteristic
334C2#1124211yescharacteristic
335C2#1124411yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1851C4semidirect
13. #1851C4semidirect

Maximal subgroups

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

subgroupindex
#219482
#219322
#81072
#80192
C2×#12892