Finite Group Databasebeta

← #97595

Subgroups — #97595

Order 512.

752 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 116 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×#2513C211yesmaximal, characteristic
2#8707C212yesmaximal, complemented
3#8558C211yesmaximal, characteristic, complemented
4C4×#324C211yesmaximal, characteristic, complemented
5#7572C212yesmaximal
6#2513—24no—
7#2513—24no—
8C2×#397—21no—
9C2×#397—21no—
10C2×#390C411yescharacteristic
11C2×#390C411yescharacteristic
12C2×#390C2211yescharacteristic
13C2×#389C411yescharacteristic
14C2×#388—21no—
15C2×#388—21no—
16C2×#382C411yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×#324—21no—
22C2×#324—21no—
23#1280—21no—
24#1280—21no—
25#1279—24no—
26#1270—24no—
27C23×D4C2×C411yescharacteristic
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C22⋊C4—21no—
40C22×C42—21no—
41C22×C42C2×C411yescharacteristic
42C22×C42C2×C411yescharacteristic
43C22×C42C2311yescharacteristic
44C2×C4⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C4×C8—22no—
48#397—28no—
49#397—28no—
50#390—22no—
51#390—22no—
52#390—22no—
53#390—22no—
54#390—22no—
55#390D414yes—
56#389—22no—
57#389—22no—
58#388—28no—
59#388—28no—
60#382—22no—
61#382—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4D414yes—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95D411yescharacteristic
71C2×#95D411yescharacteristic
72#336—22no—
73#336—22no—
74#324—24no—
75#324—42no—
76C25—21no—
77C22×D4—21no—
78C22×D4—22no—
79C22×D4—22no—
80C22×D4—42no—
81C22×D4—42no—
82C22×D4C22⋊C412yes—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4C22⋊C411yescharacteristic
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—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—82no—
111C2×C4⋊C4C22⋊C412yes—
112C2×C4⋊C4C22⋊C412yes—
113C2×C4⋊C4C22⋊C412yes—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—82no—
126C2×C22⋊C4—82no—
127C2×C22⋊C4—82no—
128C2×C22⋊C4—82no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—24no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42—82no—
139C2×C42C2×D412yes—
140C2×C42C2×D412yes—
141C4⋊C8—44no—
142C22⋊C8—24no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145C22⋊C8—42no—
146C4×C8—44no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—22no—
152#95—22no—
153C24—22no—
154C24—42no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24C2×C22⋊C411yescharacteristic
165C2×D4—24no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—84no—
172C2×D4—84no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4#10412yes—
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4#12012yes—
221C22×C4#12012yes—
222C22×C4#13112yes—
223C22×C4#13112yes—
224C22×C4#13512yes—
225C22×C4AGL(1,ℤ/8)11yescharacteristic
226C22×C4#13711yescharacteristic
227C2×C8—44no—
228C2×C8—82no—
229C2×C8—82no—
230C2×C8—82no—
231C2×C8—82no—
232C2×C8—82no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—84no—
243C22⋊C4—48no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—84no—
247C22⋊C4—84no—
248C22⋊C4—84no—
249C22⋊C4—84no—
250C42—24no—
251C42—42no—
252C42—42no—
253C42—42no—
254C42—42no—
255C42—42no—
256C42—84no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—82no—
269C23—82no—
270C23—82no—
271C23—82no—
272C23—82no—
273C23—82no—
274C23—82no—
275C23—82no—
276C23—82no—
277C23—82no—
278C23—82no—
279C23—82no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23#37911yescharacteristic
285C23#40811yescharacteristic
286C23#40811yescharacteristic
287C23#41611yescharacteristic
288C23#41711yescharacteristic
289C23C2×#10411yescharacteristic
290C23#42111yescharacteristic
291D4—84no—
292D4—84no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4#40812yes—
345C2×C4#40812yes—
346C2×C4#41012yes—
347C2×C4#41312yes—
348C2×C4#44912yes—
349C2×C4#45212yes—
350C8—84no—
351C8—162no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—162no—
372C22—162no—
373C22—162no—
374C22—162no—
375C22#148211yescharacteristic
376C22#148311yescharacteristic
377C22#149411yescharacteristic
378C22#149511yescharacteristic
379C22#178011yescharacteristic
380C22#178211yescharacteristic
381C22#178511yescharacteristic
382C22#178811yescharacteristic
383C22#178911yescharacteristic
384C22#179511yescharacteristic
385C22#179511yescharacteristic
386C4—22no—
387C4—22no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—81no—
392C4—82no—
393C4—81no—
394C4—82no—
395C4—81no—
396C4—82no—
397C4#147314yes—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2—162no—
403C2#830911yescharacteristic
404C2#831511yescharacteristic
405C2#832611yescharacteristic
406C2#836011yescharacteristic
407C2#837011yescharacteristic
408C2#837811yescharacteristic
409C2#838611yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×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
3. #8558C2semidirect
4. C4×#324C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25132
#87072
#85582
C4×#3242
#75722