Finite Group Databasebeta

← #99696

Subgroups — #99696

Order 512.

397 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 98 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#23679C211yesmaximal, characteristic
2#22824C211yesmaximal, characteristic
3#9874C211yesmaximal, characteristic
4#9869C211yesmaximal, characteristic
5#7562C211yesmaximal, characteristic
6#7031C211yesmaximal, characteristic
7#7027C211yesmaximal, characteristic
8C2×C4×Q16—21no—
9C4×C4⋊Q8C2211yescharacteristic
10#2295C2211yescharacteristic
11#2083C411yescharacteristic
12#2072C411yescharacteristic
13#2068—21no—
14#2056—21no—
15#2043C411yescharacteristic
16#2016C411yescharacteristic
17#1985—21no—
18#1917—21no—
19#1916—21no—
20#1842C2211yescharacteristic
21#1832—21no—
22#1771—21no—
23#1768—21no—
24#1767C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×#103—21no—
28#1744C2211yescharacteristic
29#1279—21no—
30#1279—21no—
31#1279—21no—
32#1279—21no—
33#1272—21no—
34#1271—21no—
35C22×Q16—21no—
36C2×C4⋊Q8C2×C411yescharacteristic
37C2×C4×Q8—21no—
38C2×C4×Q8D411yescharacteristic
39C4×Q16—42no—
40C4×Q16—42no—
41C2×#107—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×#103—21no—
51C2×#103—21no—
52C2×#103—21no—
53C2×#103—21no—
54C2×#97—21no—
55C2×#97—21no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60#391—21no—
61#391C2×C411yescharacteristic
62#391D411yescharacteristic
63#389—21no—
64#389—21no—
65#389C2×C411yescharacteristic
66#389D411yescharacteristic
67#387—21no—
68#387—21no—
69#384C2×C411yescharacteristic
70C4×C4⋊C4—21no—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4D411yescharacteristic
74#376—21no—
75C43C2311yescharacteristic
76#340—21no—
77#340—21no—
78#336—21no—
79#336—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#336C2×C411yescharacteristic
89#336C2×C411yescharacteristic
90C22×Q8—21no—
91C22×Q8C22⋊C411yescharacteristic
92C2×Q16—42no—
93C2×Q16—42no—
94C2×Q16—42no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—41no—
102C22×C8—41no—
103C4⋊Q8—42no—
104C4×Q8—21no—
105C4×Q8—21no—
106C4×Q8—21no—
107C4×Q8—21no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×Q8—41no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
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—41no—
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C4⋊C4C22⋊C411yescharacteristic
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—41no—
127C2×C42C22×C411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4111yescharacteristic
133C2×C42#4111yescharacteristic
134#107—42no—
135C4⋊C8—22no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—42no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8Q1612yescomplemented
148#103—42no—
149#103—42no—
150#103—42no—
151#103—42no—
152#97—41no—
153#97—41no—
154#97—81no—
155C4×C8—22no—
156C4×C8—41no—
157C4×C8—41no—
158C4×C8—41no—
159C4×C8—41no—
160C4×C8—41no—
161C4×C8—81no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—41no—
170#95—41no—
171C2×Q8—22no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4C4×D411yescharacteristic
189C22×C4C4×D411yescharacteristic
190C22×C4#12011yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12311yescharacteristic
194Q16—48no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—81no—
216C2×C8—81no—
217C2×C8—81no—
218C2×C8—81no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C42—81no—
251C42—81no—
252C42C2×Q1611yescharacteristic
253C42#13511yescharacteristic
254C42#13511yescharacteristic
255C42#13511yescharacteristic
256C42AGL(1,ℤ/8)11yescharacteristic
257C42AGL(1,ℤ/8)11yescharacteristic
258C42#13711yescharacteristic
259C42#13711yescharacteristic
260C23#38611yescharacteristic
261Q8—42no—
262Q8—42no—
263Q8—42no—
264Q8—81no—
265Q8—81no—
266Q8—81no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4#41611yescharacteristic
299C2×C4#41811yescharacteristic
300C2×C4C4×Q1611yescharacteristic
301C2×C4#44111yescharacteristic
302C2×C4#44311yescharacteristic
303C2×C4#44411yescharacteristic
304C2×C4#44911yescharacteristic
305C2×C4#44911yescharacteristic
306C2×C4#45111yescharacteristic
307C2×C4#45211yescharacteristic
308C2×C4#45511yescharacteristic
309C2×C4#45611yescharacteristic
310C2×C4#46211yescharacteristic
311C2×C4#46411yescharacteristic
312C2×C4#48211yescharacteristic
313C2×C4#48211yescharacteristic
314C8—42no—
315C8—42no—
316C8—81no—
317C8—82no—
318C8—82no—
319C8—81no—
320C8—81no—
321C8—81no—
322C22#187511yescharacteristic
323C22#187611yescharacteristic
324C22#187811yescharacteristic
325C22#188111yescharacteristic
326C22#188211yescharacteristic
327C22#188411yescharacteristic
328C22#189111yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—41no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—82no—
344C4—81no—
345C4—161no—
346C4#164011yescharacteristic
347C4#164411yescharacteristic
348C4#164811yescharacteristic
349C4#165211yescharacteristic
350C2#898611yescharacteristic
351C2#899611yescharacteristic
352C2#899811yescharacteristic
353C2#903611yescharacteristic
354C2#904811yescharacteristic
355C2#905111yescharacteristic
356C2#905611yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socle, prime radicalC23#386
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
147. C4⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#236792
#228242
#98742
#98692
#75622
#70312
#70272