Finite Group Databasebeta

← #97620

Subgroups — #97620

Order 512.

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

normal subgroups only

This group has 736 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
1C2×#2497C211yesmaximal, characteristic
2#8707C212yesmaximal, complemented
3#8557C211yesmaximal, characteristic
4C4×#323C211yesmaximal, characteristic
5#7576C212yesmaximal, complemented
6#2497—24no—
7#2497—24no—
8C2×#400—21no—
9C2×#396—21no—
10C2×#389C411yescharacteristic
11C2×#389C2211yescharacteristic
12C2×#388—21no—
13C2×#387—21no—
14C2×#384C411yescharacteristic
15C2×#380C411yescharacteristic
16C2×#380C411yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×#323—21no—
22C2×#323—21no—
23#1280—21no—
24#1280—21no—
25#1279—24no—
26#1271—24no—
27C24×C4C2×C411yescharacteristic
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4D411yescharacteristic
39C22×C22⋊C4D411yescharacteristic
40C22×C42—21no—
41C22×C42C2×C411yescharacteristic
42C22×C42C2311yescharacteristic
43C2×C4⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C4×C8—22no—
47#400—28no—
48#396—28no—
49#389—22no—
50#389—22no—
51#389—22no—
52#389D414yes—
53#388—28no—
54#387—28no—
55#384—22no—
56#384—22no—
57#380—22no—
58#380—22no—
59#380—22no—
60#380—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4D414yes—
63C4×C22⋊C4—22no—
64C4×C22⋊C4—22no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95D411yescharacteristic
71C2×#95D411yescharacteristic
72#336—22no—
73#336—22no—
74#323—24no—
75#323—42no—
76C25C22⋊C411yescharacteristic
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—42no—
94C22×C8—42no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—82no—
112C2×C4⋊C4—82no—
113C2×C4⋊C4—82no—
114C2×C4⋊C4C22⋊C412yes—
115C2×C4⋊C4C22⋊C412yes—
116C2×C4⋊C4C22⋊C412yes—
117C2×C4⋊C4C22⋊C412yes—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—82no—
126C2×C22⋊C4—82no—
127C2×C22⋊C4SD1612yes—
128C2×C22⋊C4Q1612yes—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—24no—
132C2×C42—42no—
133C2×C42—42no—
134C2×C42—82no—
135C2×C42C2×D412yes—
136C2×C42C2×D412yes—
137C4⋊C8—44no—
138C22⋊C8—24no—
139C22⋊C8—42no—
140C22⋊C8—42no—
141C22⋊C8—42no—
142C4×C8—44no—
143#95—24no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—22no—
149#95—22no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24#10312yes—
161C24C2×C22⋊C411yescharacteristic
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—84no—
210C22×C4—84no—
211C22×C4#10312yes—
212C22×C4#10412yes—
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4#12012yes—
216C22×C4#12012yes—
217C22×C4#13112yes—
218C22×C4#13112yes—
219C22×C4C2×SD1611yescharacteristic
220C22×C4C2×Q1611yescharacteristic
221C22×C4#13712yes—
222C2×C8—44no—
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—48no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—84no—
238C4⋊C4—84no—
239C4⋊C4—84no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—44no—
243C22⋊C4—84no—
244C22⋊C4—84no—
245C42—24no—
246C42—42no—
247C42—42no—
248C42—42no—
249C42—84no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23#37911yescharacteristic
278C23#40811yescharacteristic
279C23#40811yescharacteristic
280C23C2×#10311yescharacteristic
281C23#41911yescharacteristic
282C23C2×#10411yescharacteristic
283C23#42111yescharacteristic
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—84no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4#40812yes—
336C2×C4#40812yes—
337C2×C4#41012yes—
338C2×C4#41312yes—
339C2×C4#44812yes—
340C2×C4#45112yes—
341C8—84no—
342C8—162no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22#148111yescharacteristic
367C22#148511yescharacteristic
368C22#149311yescharacteristic
369C22#149711yescharacteristic
370C22#178011yescharacteristic
371C22#178211yescharacteristic
372C22#178511yescharacteristic
373C22#178711yescharacteristic
374C22#178911yescharacteristic
375C22#179511yescharacteristic
376C22#179511yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—42no—
381C4—42no—
382C4—42no—
383C4—81no—
384C4—82no—
385C4—82no—
386C4—82no—
387C4—82no—
388C4#147314yes—
389C2—21no—
390C2—21no—
391C2—21no—
392C2—21no—
393C2—82no—
394C2#830911yescharacteristic
395C2#831411yescharacteristic
396C2#832611yescharacteristic
397C2#836011yescharacteristic
398C2#837311yescharacteristic
399C2#838011yescharacteristic
400C2#839511yescharacteristic
subgroupG/H
center, socleC23#379
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
2. #8707C2semidirect
5. #7576C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24972
#87072
#85572
C4×#3232
#75762