Finite Group Databasebeta

← #98538

Subgroups — #98538

Order 512.

808 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2550C211yesmaximal, characteristic
2#8560C211yesmaximal, characteristic
3#8560C211yesmaximal, characteristic
4#8340C212yesmaximal
5#7641C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7C22×#125C411yescharacteristic
8#2550—24no—
9#2550—24no—
10C2×#398—21no—
11C2×#395—21no—
12C2×#391—21no—
13C2×#389—21no—
14C2×#389C411yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#382C411yescharacteristic
17C2×C4×C4⋊C4C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1797C2212yes—
20#1797C2212yes—
21C2×#332—24no—
22C2×#324—21no—
23C2×#324—21no—
24#1280—21no—
25#1280—21no—
26#1271—24no—
27C23×Q8—21no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—44no—
30C2×C4⋊Q8D414yes—
31C2×#125—22no—
32C2×#125—22no—
33C2×#125—44no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53#398—28no—
54#395—28no—
55#391—28no—
56#389—28no—
57#389—22no—
58#389—22no—
59#389—22no—
60#389D414yes—
61#382—22no—
62#382—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74#336—22no—
75#336—22no—
76#332—48no—
77#324—42no—
78#324—42no—
79C22×Q8—24no—
80C22×Q8—44no—
81C22×Q8—42no—
82C22×Q8—41no—
83C22×Q8—41no—
84C22×Q8—82no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C22×C8—42no—
97C22×C8—42no—
98C4⋊Q8—28no—
99C4⋊Q8—44no—
100C4⋊Q8—48no—
101#125—48no—
102#125—44no—
103#125—44no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—82no—
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4C22⋊C412yes—
134C2×C4⋊C4C22⋊C412yes—
135C2×C4⋊C4C22⋊C412yes—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—24no—
143C2×C42—42no—
144C2×C42—42no—
145C2×C42—82no—
146C2×C42C2×D412yes—
147C2×C42C2×D412yes—
148C4⋊C8—44no—
149C4⋊C8—44no—
150C22⋊C8—42no—
151C22⋊C8—42no—
152C22⋊C8—42no—
153C22⋊C8—42no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—22no—
160#95—22no—
161C24C2×C22⋊C411yescharacteristic
162C2×Q8—48no—
163C2×Q8—44no—
164C2×Q8—44no—
165C2×Q8—84no—
166C2×Q8—88no—
167C2×Q8—84no—
168C2×Q8—84no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
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#10112yes—
218C22×C4#10112yes—
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4#12012yes—
222C22×C4#12012yes—
223C22×C4#13512yes—
224C22×C4#13512yes—
225C22×C4AGL(1,ℤ/8)11yescharacteristic
226C22×C4AGL(1,ℤ/8)11yescharacteristic
227C22×C4#13711yescharacteristic
228C22×C4#13711yescharacteristic
229C2×C8—82no—
230C2×C8—82no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—82no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—24no—
240C4⋊C4—48no—
241C4⋊C4—42no—
242C4⋊C4—48no—
243C4⋊C4—48no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—84no—
253C4⋊C4#9914yes—
254C4⋊C4#10114yes—
255C42—24no—
256C42—24no—
257C42—24no—
258C42—42no—
259C42—42no—
260C42—42no—
261C42—84no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23#37911yescharacteristic
267C23C2×#10111yescharacteristic
268C23C2×#10111yescharacteristic
269C23#41611yescharacteristic
270C23#41611yescharacteristic
271C23#41711yescharacteristic
272C23#41711yescharacteristic
273Q8—88no—
274Q8—88no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4C2×#9912yes—
334C2×C4C2×#10112yes—
335C2×C4#44912yes—
336C2×C4#44912yes—
337C2×C4#45212yes—
338C2×C4#45212yes—
339C8—162no—
340C8—162no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#154411yescharacteristic
354C22#155211yescharacteristic
355C22#155211yescharacteristic
356C22#156111yescharacteristic
357C22#178111yescharacteristic
358C22#178411yescharacteristic
359C22#178411yescharacteristic
360C22#178811yescharacteristic
361C22#178811yescharacteristic
362C22#179511yescharacteristic
363C22#179611yescharacteristic
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—22no—
369C4—22no—
370C4—42no—
371C4—42no—
372C4—44no—
373C4—44no—
374C4—81no—
375C4—82no—
376C4—81no—
377C4—82no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2#834511yescharacteristic
383C2#837711yescharacteristic
384C2#838111yescharacteristic
385C2#839011yescharacteristic
386C2#839811yescharacteristic
387C2#852511yescharacteristic
388C2#855111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25502
#85602
#83402
#76412