Finite Group Databasebeta

← #101800

Subgroups — #101800

Order 512.

408 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 84 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#27877C211yesmaximal, characteristic
2#22602C211yesmaximal, characteristic
3#9683C211yesmaximal, characteristic
4#8716C211yesmaximal, characteristic
5#7728C211yesmaximal, characteristic
6#7024C211yesmaximal, characteristic
7#7021C211yesmaximal, characteristic
8C2×#433—21no—
9#2857C2211yescharacteristic
10#2285C2211yescharacteristic
11#2075—21no—
12#2075—21no—
13#1990C411yescharacteristic, complemented
14#1990C411yescharacteristic, complemented
15#1940—21no—
16#1918—21no—
17#1918—21no—
18#1917—21no—
19#1916—21no—
20#1845C411yescharacteristic, complemented
21#1834C411yescharacteristic, complemented
22#1818—21no—
23#1817C2211yescharacteristic
24#1797C2211yescharacteristic
25#1771—21no—
26#1749C2211yescharacteristic
27C4×C22⋊C8C2211yescharacteristic
28C2×#328—21no—
29C2×#328—21no—
30#1434C2211yescharacteristic
31#1294—21no—
32#1294—21no—
33#1271—21no—
34#1271—21no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C22⋊Q8—21no—
37C2×#117—21no—
38C22×C42C2311yescharacteristic
39#433—42no—
40#433—42no—
41C2×#107—21no—
42C2×#107—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8D411yescharacteristic
50C2×C22⋊C8D411yescharacteristic
51C2×#97—21no—
52C2×#97—21no—
53C2×C4×C8—21no—
54C2×C4×C8C2×C411yescharacteristic
55C2×C4×C8C2×C411yescharacteristic
56#391—21no—
57#389—21no—
58#389—21no—
59#384C2×C411yescharacteristic
60#384C2×C411yescharacteristic
61#384C2×C411yescharacteristic
62#382—21no—
63#380—21no—
64#380—21no—
65#380—21no—
66#380D411yescharacteristic
67#380D411yescharacteristic
68C4×C4⋊C4—21no—
69C4×C4⋊C4D411yescharacteristic
70C4×C4⋊C4D411yescharacteristic
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#340—21no—
78#340—21no—
79#340—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336D411yescharacteristic
89#336D411yescharacteristic
90#328—42no—
91#328—42no—
92C22×Q8—21no—
93C23×C4—41no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4#4111yescharacteristic
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—41no—
104C4⋊Q8—41no—
105C4⋊Q8—41no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110#117—41no—
111#117—41no—
112#117—41no—
113#117—41no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—41no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42C22×C411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42#4111yescharacteristic
139#107—42no—
140#107—42no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145C22⋊C8—21no—
146C22⋊C8—21no—
147C22⋊C8—21no—
148C22⋊C8—42no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8D812yes—
154#97—41no—
155#97—41no—
156#97—42no—
157C4×C8—41no—
158C4×C8—41no—
159C4×C8—42no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—22no—
165#95—41no—
166#95—41no—
167#95—41no—
168#95—41no—
169#95—41no—
170#95—41no—
171#95—41no—
172C24#12111yescharacteristic
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—81no—
176C2×Q8—81no—
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—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4C2×C22⋊C411yescharacteristic
201C22×C4C4×D411yescharacteristic
202C22×C4C4×D411yescharacteristic
203C22×C4#12011yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12311yescharacteristic
206C22×C4C2×D811yescharacteristic
207C22×C4#13511yescharacteristic
208C22×C4AGL(1,ℤ/8)11yescharacteristic
209C22×C4AGL(1,ℤ/8)11yescharacteristic
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—81no—
229C2×C8—81no—
230C2×C8—81no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—81no—
258C22⋊C4—81no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—41no—
264C42—41no—
265C42—41no—
266C42—41no—
267C42—41no—
268C42—41no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23#38611yescharacteristic
276C23#45911yescharacteristic
277C23#46311yescharacteristic
278Q8—161no—
279Q8—161no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
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—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—161no—
324C2×C4—161no—
325C2×C4—161no—
326C2×C4—161no—
327C2×C4#41011yescharacteristic
328C2×C4#41011yescharacteristic
329C2×C4C4×D811yescharacteristic
330C2×C4#44211yescharacteristic
331C2×C4#44311yescharacteristic
332C2×C4#44411yescharacteristic
333C2×C4#44711yescharacteristic
334C2×C4#44911yescharacteristic
335C8—42no—
336C8—42no—
337C8—81no—
338C8—81no—
339C8—81no—
340C8—81no—
341C8—161no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22#187611yescharacteristic
353C22#187911yescharacteristic
354C22#188411yescharacteristic
355C22#189911yescharacteristic
356C22#190211yescharacteristic
357C22#190511yescharacteristic
358C22#190911yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—162no—
374C4—161no—
375C4—161no—
376C2—41no—
377C2—41no—
378C2#905311yescharacteristic
379C2#924011yescharacteristic
380C2#926011yescharacteristic
381C2#928011yescharacteristic
382C2#929111yescharacteristic
383C2#941611yescharacteristic
384C2#944411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ 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
13. #1990C4semidirect
14. #1990C4semidirect
20. #1845C4semidirect
21. #1834C4semidirect

Maximal subgroups

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

subgroupindex
#278772
#226022
#96832
#87162
#77282
#70242
#70212