Finite Group Databasebeta

← #99997

Subgroups — #99997

Order 512.

446 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 96 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#23326C211yesmaximal, characteristic
2#22081C211yesmaximal, characteristic
3#9879C211yesmaximal, characteristic, complemented
4#7593C211yesmaximal, characteristic, complemented
5#7582C211yesmaximal, characteristic
6#7033C211yesmaximal, characteristic, complemented
7#7021C211yesmaximal, characteristic, complemented
8C2×#480C411yescharacteristic
9C4×#121C2211yescharacteristic
10C2×#389C2211yescharacteristic
11#1954—21no—
12#1943—21no—
13#1936—21no—
14#1923—21no—
15#1921—21no—
16#1916—21no—
17#1874C411yescharacteristic
18#1869—21no—
19#1869—21no—
20#1864—21no—
21#1853C411yescharacteristic
22#1850—21no—
23#1823C411yescharacteristic
24#1817C2211yescharacteristic
25#1817C2211yescharacteristic
26#1797C2211yescharacteristic
27#1749C2211yescharacteristic
28C4×C22⋊C8C2211yescharacteristic
29#1280—21no—
30#1280—21no—
31#1271—21no—
32#1271—21no—
33#1271—21no—
34#1270—21no—
35C23×C8—21no—
36C2×#121C2×C411yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4D411yescharacteristic
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C42C2311yescharacteristic
43#480—22no—
44#480C814yescomplemented
45C2×#107—21no—
46C2×#106—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×#102—21no—
50C2×#102—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8C2×C411yescharacteristic
58C2×C22⋊C8C2×C411yescharacteristic
59C2×C22⋊C8D411yescharacteristic
60C2×C22⋊C8D411yescharacteristic
61C2×#97—21no—
62C2×C4×C8—21no—
63C2×C4×C8—21no—
64#390D411yescharacteristic
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389D411yescharacteristic
70#388—21no—
71#386—21no—
72#380C2×C411yescharacteristic
73C4×C4⋊C4D411yescharacteristic
74C4×C22⋊C4—21no—
75C2×#95—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
90C22×D4—21no—
91C22×D4C22⋊C411yescharacteristic
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4#4111yescharacteristic
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—81no—
107C22×C8—81no—
108C22×C8—81no—
109#121—21no—
110#121—21no—
111#121C2×C812yes—
112#121M1612yes—
113C4×D4—81no—
114C4×D4—81no—
115C4×D4—81no—
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—21no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—81no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C2×C812yescomplemented
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C42—21no—
138C2×C42—81no—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143#107—24no—
144#106—81no—
145C4⋊C8—81no—
146C4⋊C8—81no—
147#102—24no—
148#102—81no—
149C22⋊C8—21no—
150C22⋊C8—22no—
151C22⋊C8—21no—
152C22⋊C8—21no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8—81no—
157C22⋊C8—81no—
158C22⋊C8—81no—
159C22⋊C8—81no—
160C22⋊C8—81no—
161C22⋊C8C2×C812yescomplemented
162#97—81no—
163C4×C8—81no—
164C4×C8—81no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—41no—
169#95—41no—
170C24—21no—
171C24C4×D411yescharacteristic
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—81no—
178C2×D4—81no—
179C2×D4C22⋊C812yes—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4#12011yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12311yescharacteristic
213C22×C4C22×C811yescharacteristic
214C22×C4C2×M1611yescharacteristic
215C22×C4#13111yescharacteristic
216C22×C4#13111yescharacteristic
217C2×C8—22no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—42no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—81no—
237C2×C8—81no—
238C2×C8—84no—
239C2×C8—81no—
240C2×C8—81no—
241C2×C8—81no—
242C4⋊C4—21no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4C22⋊C812yes—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—81no—
259C22⋊C4—81no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C42—41no—
263C42—41no—
264C42—41no—
265C42—41no—
266C42—41no—
267C42—41no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23#38611yescharacteristic
282C23C8×D411yescharacteristic
283C23#43511yescharacteristic
284D4—42no—
285D4—81no—
286D4—82no—
287D4—82no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—162no—
332C2×C4C2×C22⋊C811yescharacteristic
333C2×C4#40811yescharacteristic
334C2×C4C8×D411yescharacteristic
335C2×C4#43511yescharacteristic
336C2×C4#45511yescharacteristic
337C2×C4#45611yescharacteristic
338C8—42no—
339C8—81no—
340C8—81no—
341C8—81no—
342C8—81no—
343C8—81no—
344C8—82no—
345C8—161no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22#186911yescharacteristic
364C22#188111yescharacteristic
365C22#188211yescharacteristic
366C22#190311yescharacteristic
367C22#190411yescharacteristic
368C22#191311yescharacteristic
369C22#191511yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—42no—
380C4—81no—
381C4—81no—
382C4—82no—
383C4—161no—
384C2—41no—
385C2—41no—
386C2—82no—
387C2#895311yescharacteristic
388C2#908911yescharacteristic
389C2#910811yescharacteristic
390C2#933211yescharacteristic
391C2#933511yescharacteristic
392C2#933811yescharacteristic
393C2#934111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC2×#121C2×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
3. #9879C2semidirect
4. #7593C2semidirect
6. #7033C2semidirect
7. #7021C2semidirect
44. #480C8semidirect
131. C2×C4⋊C4C2×C8semidirect
161. C22⋊C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#233262
#220812
#98792
#75932
#75822
#70332
#70212