Finite Group Databasebeta

← #99939

Subgroups — #99939

Order 512.

503 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 112 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#22260C211yesmaximal, characteristic
2#22220C211yesmaximal, characteristic
3#9641C211yesmaximal, characteristic, complemented
4#7588C211yesmaximal, characteristic, complemented
5#7587C211yesmaximal, characteristic
6#7018C211yesmaximal, characteristic, complemented
7#7014C211yesmaximal, characteristic, complemented
8C2×#486C411yescharacteristic
9C2×C8×D4—21no—
10C4×C4⋊Q8C2211yescharacteristic
11C4×#127C2211yescharacteristic
12#1990C411yescharacteristic
13#1957C411yescharacteristic
14#1954—21no—
15#1925—22no—
16#1869—22no—
17#1869—22no—
18#1862—22no—
19#1845C411yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1744C2211yescharacteristic
23#1744C2211yescharacteristic
24#1744C2211yescharacteristic
25C2×Q8⋊C8—21no—
26C2×Q8⋊C8—21no—
27#1271—22no—
28#1270—22no—
29C23×C8—22no—
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×#127C2×C411yescharacteristic
32C2×C4×Q8—21no—
33C2×C4×D4—21no—
34C2×C4×D4—21no—
35C2×C4×D4D412yes—
36#486—22no—
37#486C814yescomplemented
38C8×D4—42no—
39C8×D4—42no—
40C2×#107—22no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×#102—22no—
44C2×#102—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C4×C8—21no—
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53#391—21no—
54#390—21no—
55#390—21no—
56#390D412yes—
57#389—21no—
58#389—21no—
59#384C2×C411yescharacteristic
60#384C2×C411yescharacteristic
61C4×C4⋊C4—21no—
62C4×C4⋊C4—21no—
63C43C2311yescharacteristic
64#340—22no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336D412yes—
75Q8⋊C8—42no—
76Q8⋊C8—42no—
77C22×Q8—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4C22⋊C412yes—
81C23×C4—22no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—21no—
87C22×C8—44no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—82no—
91C22×C8—82no—
92C4⋊Q8—21no—
93C4⋊Q8—21no—
94C4⋊Q8—21no—
95C4⋊Q8C2×C812yescomplemented
96#127—21no—
97#127—21no—
98#127C2×C812yes—
99#127M1612yes—
100C4×Q8—41no—
101C4×Q8—41no—
102C4×Q8—41no—
103C4×Q8—41no—
104C4×D4—41no—
105C4×D4—41no—
106C4×D4—41no—
107C4×D4—41no—
108C4×D4—82no—
109C4×D4—82no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4C22⋊C412yes—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C42—41no—
123C2×C42C22×C411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42#4111yescharacteristic
130#107—82no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135#102—28no—
136#102—82no—
137C22⋊C8—44no—
138C22⋊C8—82no—
139C22⋊C8—82no—
140C4×C8—22no—
141C4×C8—21no—
142C4×C8—21no—
143C4×C8—21no—
144C4×C8—21no—
145C4×C8—21no—
146C4×C8—21no—
147C4×C8—41no—
148C4×C8C2×C812yescomplemented
149C4×C8D812yes—
150#95—41no—
151#95—41no—
152C24—22no—
153C2×Q8—22no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4C22⋊C814yes—
173C22×C4—22no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4C2×C22⋊C411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4#12011yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12311yescharacteristic
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—42no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—88no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8—81no—
209C2×C8—81no—
210C2×C8—81no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C42—21no—
230C42—21no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—81no—
236C42—81no—
237C42C22×C811yescharacteristic
238C42C2×M1611yescharacteristic
239C42#13111yescharacteristic
240C42#13111yescharacteristic
241C42C2×D811yescharacteristic
242C42#13511yescharacteristic
243C42AGL(1,ℤ/8)11yescharacteristic
244C42AGL(1,ℤ/8)11yescharacteristic
245C23—24no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23#38611yescharacteristic
253Q8—42no—
254Q8—42no—
255Q8—81no—
256Q8—81no—
257D4—44no—
258D4—42no—
259D4—42no—
260D4—42no—
261D4—42no—
262D4—82no—
263D4—82no—
264D4—82no—
265C2×C4—21no—
266C2×C4—21no—
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—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4C2×C22⋊C811yescharacteristic
293C2×C4#40811yescharacteristic
294C2×C4C8×D411yescharacteristic
295C2×C4C8×D411yescharacteristic
296C2×C4#43511yescharacteristic
297C2×C4#43511yescharacteristic
298C2×C4C4×D811yescharacteristic
299C2×C4#44211yescharacteristic
300C2×C4#44311yescharacteristic
301C2×C4#44411yescharacteristic
302C2×C4#44711yescharacteristic
303C2×C4#44911yescharacteristic
304C2×C4#45911yescharacteristic
305C2×C4#46311yescharacteristic
306C8—42no—
307C8—42no—
308C8—81no—
309C8—81no—
310C8—82no—
311C8—82no—
312C8—161no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—82no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22#186911yescharacteristic
321C22#187611yescharacteristic
322C22#187911yescharacteristic
323C22#190211yescharacteristic
324C22#190411yescharacteristic
325C22#190911yescharacteristic
326C22#191111yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—82no—
338C4—82no—
339C4—161no—
340C4C8×D811yescharacteristic
341C4#158011yescharacteristic
342C4#158111yescharacteristic
343C4#158611yescharacteristic
344C2—84no—
345C2#894711yescharacteristic
346C2#908711yescharacteristic
347C2#909911yescharacteristic
348C2#927911yescharacteristic
349C2#928511yescharacteristic
350C2#929111yescharacteristic
351C2#929911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9641C2semidirect
4. #7588C2semidirect
6. #7018C2semidirect
7. #7014C2semidirect
37. #486C8semidirect
95. C4⋊Q8C2×C8semidirect
148. C4×C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#222602
#222202
#96412
#75882
#75872
#70182
#70142