Finite Group Databasebeta

← #97018

Subgroups — #97018

Order 512.

894 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2319C211yesmaximal, characteristic
2#8699C211yesmaximal, characteristic, complemented
3#8699C211yesmaximal, characteristic, complemented
4#7593C212yesmaximal, complemented
5#7592C212yesmaximal
6C22×C4×D4C411yescharacteristic
7#2319—24no—
8#2319—24no—
9C2×#390C411yescharacteristic
10C2×#390C2211yescharacteristic
11C2×#389C411yescharacteristic
12C2×#389C2211yescharacteristic
13C2×#388—21no—
14C2×#388—21no—
15C2×#381—21no—
16C2×#380—21no—
17C2×C4×C4⋊C4C411yescharacteristic
18C2×#376C2211yescharacteristic
19#1817C2212yes—
20#1817C2212yes—
21#1280—21no—
22#1280—21no—
23#1280—21no—
24#1280—21no—
25#1271—24no—
26#1270—24no—
27C23×D4C2×C411yescharacteristic
28C24×C4—21no—
29C2×C4×D4—22no—
30C2×C4×D4—22no—
31C2×C4×D4—22no—
32C2×C4×D4—22no—
33C2×C4×D4—22no—
34C2×C4×D4C814yescomplemented
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C22⋊C4—21no—
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51#390—22no—
52#390—22no—
53#390—22no—
54#390D414yes—
55#389—22no—
56#389—22no—
57#389—22no—
58#389D414yes—
59#388—28no—
60#388—28no—
61#381—28no—
62#380—28no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65#376—22no—
66#376—22no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95D411yescharacteristic
71C2×#95D411yescharacteristic
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74#336—22no—
75#336—22no—
76#336—22no—
77#336—22no—
78C25—21no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—42no—
83C22×D4—42no—
84C22×D4C22⋊C412yes—
85C22×D4C2×C812yes—
86C22×D4M1612yes—
87C23×C4—21no—
88C23×C4—24no—
89C23×C4—24no—
90C23×C4—24no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—82no—
94C23×C4—82no—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C22×C8—42no—
103C22×C8—42no—
104C4×D4—28no—
105C4×D4—28no—
106C4×D4—28no—
107C4×D4—28no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—82no—
118C2×C4⋊C4—82no—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4C22⋊C412yes—
121C2×C4⋊C4C22⋊C412yes—
122C2×C4⋊C4C2×C812yes—
123C2×C4⋊C4M1612yes—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—82no—
132C2×C22⋊C4—82no—
133C2×C22⋊C4—82no—
134C2×C22⋊C4—82no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—24no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42—82no—
145C2×C42C2×C812yes—
146C2×C42M1612yes—
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C22⋊C8—44no—
150C22⋊C8—44no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—22no—
155#95—22no—
156#95—22no—
157#95—22no—
158C24—22no—
159C24—22no—
160C24—22no—
161C24—42no—
162C24—42no—
163C24—42no—
164C24—42no—
165C24—42no—
166C24—42no—
167C24C2×C22⋊C411yescharacteristic
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—44no—
174C2×D4—44no—
175C2×D4—44no—
176C2×D4—44no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4C22⋊C814yes—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—216no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—88no—
219C22×C4—88no—
220C22×C4—84no—
221C22×C4—84no—
222C22×C4C22⋊C812yes—
223C22×C4C22⋊C812yes—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4#12012yes—
227C22×C4#12012yes—
228C22×C4C22×C811yescharacteristic
229C22×C4C2×M1611yescharacteristic
230C22×C4C2×M1611yescharacteristic
231C22×C4C2×M1611yescharacteristic
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—82no—
235C2×C8—82no—
236C2×C8—82no—
237C2×C8—82no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—84no—
246C4⋊C4—84no—
247C4⋊C4C22⋊C814yes—
248C22⋊C4—216no—
249C22⋊C4—84no—
250C22⋊C4—84no—
251C22⋊C4—84no—
252C22⋊C4—84no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—42no—
258C42—42no—
259C42—42no—
260C42—42no—
261C42—84no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—24no—
267C23—44no—
268C23—44no—
269C23—44no—
270C23—44no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—44no—
276C23—44no—
277C23—42no—
278C23—42no—
279C23—84no—
280C23—84no—
281C23—84no—
282C23—84no—
283C23#37911yescharacteristic
284C23C2×C22⋊C811yescharacteristic
285C23#40711yescharacteristic
286C23#41012yes—
287C23#41312yes—
288D4—44no—
289D4—44no—
290D4—44no—
291D4—44no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—816no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4C2×C22⋊C811yescharacteristic
341C2×C4C2×C22⋊C811yescharacteristic
342C2×C4#40711yescharacteristic
343C2×C4#40711yescharacteristic
344C8—162no—
345C8—162no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—44no—
355C22—44no—
356C22—44no—
357C22—88no—
358C22—88no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22—82no—
363C22#145812yes—
364C22#146412yes—
365C22#177911yescharacteristic
366C22#178211yescharacteristic
367C22#178511yescharacteristic
368C22#179511yescharacteristic
369C22#179511yescharacteristic
370C22#179511yescharacteristic
371C22#179611yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—42no—
377C4—42no—
378C4—44no—
379C4—42no—
380C4—42no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4—82no—
385C2—22no—
386C2—22no—
387C2—84no—
388C2#830011yescharacteristic
389C2#832211yescharacteristic
390C2#832411yescharacteristic
391C2#851811yescharacteristic
392C2#851911yescharacteristic
393C2#852311yescharacteristic
394C2#852611yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×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
2. #8699C2semidirect
3. #8699C2semidirect
4. #7593C2semidirect
34. C2×C4×D4C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23192
#86992
#75932
#75922