Finite Group Databasebeta

← #99820

Subgroups — #99820

Order 512.

558 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 124 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#21951C211yesmaximal, characteristic
2#14320C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7575C212yesmaximal
5#7024C212yesmaximal
6C2×#431—21no—
7C2×#395C411yescharacteristic
8C2×#391C2211yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1939—24no—
11#1797—21no—
12#1797C2212yes—
13C4×C22⋊C8C2212yes—
14C2×#324—21no—
15#1446C411yescharacteristic
16#1445—21no—
17#1442—21no—
18#1442C411yescharacteristic
19#1441C411yescharacteristic
20#1436—21no—
21#1434—21no—
22#1434C2211yescharacteristic
23#1280—21no—
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C42D411yescharacteristic
31C22×C42D411yescharacteristic
32C22×C42C2311yescharacteristic
33#431—42no—
34#431—42no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C4×C8—22no—
41C2×C4×C8—22no—
42#395—22no—
43#395C814yescomplemented
44#391—22no—
45#391—22no—
46C4×C4⋊C4—22no—
47C4×C4⋊C4—22no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95D411yescharacteristic
63#336—22no—
64#336—22no—
65#336D414yes—
66#324—24no—
67C22×Q8—21no—
68C22×Q8—22no—
69C22×Q8—22no—
70C22×Q8—81no—
71C22×Q8—81no—
72C22×Q8C2×C812yes—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—24no—
85C22×C8—42no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4—81no—
95C2×C4⋊C4—81no—
96C2×C4⋊C4C2×C812yes—
97C2×C4⋊C4M1612yes—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—81no—
105C2×C42—81no—
106C2×C42SD1612yes—
107C2×C42Q1612yes—
108C2×C42C2×D412yes—
109C2×C42#4112yes—
110C4⋊C8—44no—
111C4⋊C8—82no—
112C4⋊C8—82no—
113C22⋊C8—24no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117C4×C8—44no—
118C4×C8—82no—
119#95—24no—
120#95—22no—
121#95—22no—
122#95—21no—
123#95—22no—
124#95—42no—
125#95—42no—
126#95—41no—
127#95—42no—
128#95—41no—
129#95—41no—
130#95—41no—
131#95—81no—
132#95—81no—
133#95—81no—
134#95—81no—
135#95—81no—
136#95C2×C812yes—
137C24C2×C22⋊C411yescharacteristic
138C2×Q8—24no—
139C2×Q8—42no—
140C2×Q8—82no—
141C2×Q8—82no—
142C2×Q8—82no—
143C2×Q8—82no—
144C2×Q8—84no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4C22⋊C812yes—
199C22×C4C22⋊C812yes—
200C22×C4#10312yes—
201C22×C4#10312yes—
202C22×C4#10412yes—
203C22×C4C4×D412yes—
204C22×C4#12011yescharacteristic
205C22×C4#12112yes—
206C22×C4#12311yescharacteristic
207C22×C4C22×C811yescharacteristic
208C22×C4C2×M1611yescharacteristic
209C22×C4#13112yes—
210C22×C4C2×SD1611yescharacteristic
211C22×C4C2×Q1611yescharacteristic
212C22×C4AGL(1,ℤ/8)11yescharacteristic
213C22×C4#13711yescharacteristic
214C2×C8—44no—
215C2×C8—44no—
216C2×C8—44no—
217C2×C8—44no—
218C2×C8—82no—
219C2×C8—82no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—44no—
223C4⋊C4—82no—
224C4⋊C4—84no—
225C42—22no—
226C42—22no—
227C42—22no—
228C42—22no—
229C42—42no—
230C42—42no—
231C42—42no—
232C42—81no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23#38611yescharacteristic
238C23C2×C22⋊C811yescharacteristic
239C23#40811yescharacteristic
240C23C2×#10311yescharacteristic
241C23#41711yescharacteristic
242C23C2×#10411yescharacteristic
243C23#42111yescharacteristic
244Q8—84no—
245Q8—84no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4Q8⋊C814yes—
314C2×C4C8×D412yes—
315C2×C4#43512yes—
316C2×C4#45011yescharacteristic
317C2×C4#45111yescharacteristic
318C2×C4#48311yescharacteristic
319C2×C4#48411yescharacteristic
320C8—84no—
321C8—84no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22C2×Q8⋊C811yescharacteristic
335C22#147711yescharacteristic
336C22#148111yescharacteristic
337C22#148311yescharacteristic
338C22#186911yescharacteristic
339C22#187111yescharacteristic
340C22#187211yescharacteristic
341C22#190911yescharacteristic
342C22#191211yescharacteristic
343C22#191311yescharacteristic
344C22#191411yescharacteristic
345C4—22no—
346C4—22no—
347C4—24no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—82no—
356C4—82no—
357C4—81no—
358C4—161no—
359C4—161no—
360C2—21no—
361C2—21no—
362C2—21no—
363C2—21no—
364C2#893711yescharacteristic
365C2#910011yescharacteristic
366C2#910711yescharacteristic
367C2#918411yescharacteristic
368C2#918811yescharacteristic
369C2#919111yescharacteristic
370C2#920011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
43. #395C8semidirect

Maximal subgroups

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

subgroupindex
#219512
#143202
C4×#3242
#75752
#70242