Finite Group Databasebeta

← #99835

Subgroups — #99835

Order 512.

602 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 100 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#21906C211yesmaximal, characteristic
2#14319C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7575C212yesmaximal, complemented
5#7024C212yesmaximal, complemented
6C2×#393C411yescharacteristic
7C2×#391C2211yescharacteristic
8C2×C4×C22⋊C4C2211yescharacteristic
9#1924—24no—
10#1797C2212yes—
11#1779—21no—
12#1749—21no—
13C4×C22⋊C8C2212yes—
14C2×#323—21no—
15#1446C411yescharacteristic
16#1444C411yescharacteristic
17#1443—21no—
18#1439—21no—
19#1439C411yescharacteristic
20#1436—21no—
21#1434C2211yescharacteristic
22C22⋊#95—21no—
23#1280—21no—
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4D411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42C2311yescharacteristic
35C2×C4⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×#97—22no—
41C2×C4×C8—22no—
42#393—22no—
43#393—22no—
44#391—22no—
45#391—22no—
46C4×C22⋊C4—22no—
47C4×C22⋊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×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95D411yescharacteristic
61#336—22no—
62#336—22no—
63#336D414yes—
64#323—24no—
65C25C22⋊C411yescharacteristic
66C22×Q8—21no—
67C22×Q8—22no—
68C22×Q8—22no—
69C22×Q8—81no—
70C22×Q8—81no—
71C22×Q8M1612yes—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—81no—
81C23×C4—81no—
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C22×C8—24no—
89C22×C8—42no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4—81no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—81no—
105C2×C22⋊C4—81no—
106C2×C22⋊C4—81no—
107C2×C22⋊C4—81no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—81no—
111C2×C42—81no—
112C2×C42C2×D412yes—
113C2×C42#4112yes—
114C4⋊C8—82no—
115C22⋊C8—24no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C22⋊C8—82no—
120C22⋊C8—82no—
121#97—82no—
122C4×C8—82no—
123#95—24no—
124#95—22no—
125#95—22no—
126#95—21no—
127#95—22no—
128#95—41no—
129#95—42no—
130#95—41no—
131#95—42no—
132#95—81no—
133#95—81no—
134#95—81no—
135#95—81no—
136#95—81no—
137#95M1612yes—
138C24—21no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—41no—
145C24—41no—
146C24—41no—
147C24—41no—
148C24#10412yes—
149C24C2×C22⋊C411yescharacteristic
150C2×Q8—24no—
151C2×Q8—42no—
152C2×Q8—82no—
153C2×Q8—82no—
154C2×Q8—82no—
155C2×Q8—82no—
156C2×Q8—84no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—42no—
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—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—84no—
215C22×C4—84no—
216C22×C4C4×D412yes—
217C22×C4#12011yescharacteristic
218C22×C4#12112yes—
219C22×C4#12311yescharacteristic
220C22×C4C2×M1611yescharacteristic
221C22×C4C2×M1611yescharacteristic
222C22×C4#13112yes—
223C22×C4#13512yes—
224C22×C4#13712yes—
225C2×C8—44no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8—82no—
230C2×C8—82no—
231C4⋊C4—82no—
232C4⋊C4—84no—
233C22⋊C4—44no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—84no—
237C22⋊C4—84no—
238C42—42no—
239C42—42no—
240C42—42no—
241C42—81no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23#38611yescharacteristic
270C23#40711yescharacteristic
271C23#40811yescharacteristic
272C23#41611yescharacteristic
273C23#41911yescharacteristic
274C23C2×#10411yescharacteristic
275C23#42111yescharacteristic
276Q8—84no—
277Q8—84no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4#43512yes—
340C2×C4#43612yes—
341C2×C4#45212yes—
342C2×C4#48512yes—
343C8—84no—
344C8—84no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—44no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—82no—
362C22—82no—
363C22—82no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22#148811yescharacteristic
369C22#149111yescharacteristic
370C22#149411yescharacteristic
371C22#149711yescharacteristic
372C22#187011yescharacteristic
373C22#187111yescharacteristic
374C22#187311yescharacteristic
375C22#190911yescharacteristic
376C22#191111yescharacteristic
377C22#191311yescharacteristic
378C22#191411yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—42no—
384C4—82no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4—81no—
389C4—161no—
390C4—161no—
391C2—21no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2—82no—
396C2#894111yescharacteristic
397C2#910311yescharacteristic
398C2#910911yescharacteristic
399C2#918311yescharacteristic
400C2#918711yescharacteristic
401C2#919211yescharacteristic
402C2#920111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×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
4. #7575C2semidirect
5. #7024C2semidirect

Maximal subgroups

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

subgroupindex
#219062
#143192
C4×#3232
#75752
#70242