Finite Group Databasebeta

← #99535

Subgroups — #99535

Order 512.

438 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 104 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#23346C211yesmaximal, characteristic
2C4×#489C211yesmaximal, characteristic
3#9873C211yesmaximal, characteristic, complemented
4#9641C211yesmaximal, characteristic, complemented
5#7567C211yesmaximal, characteristic
6#7030C211yesmaximal, characteristic, complemented
7#7030C211yesmaximal, characteristic, complemented
8C2×#489C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×#124C2211yescharacteristic
11#1993C411yescharacteristic
12#1984—21no—
13#1959C411yescharacteristic
14#1936—21no—
15#1935—21no—
16#1923—21no—
17#1923—21no—
18#1869—21no—
19#1850—21no—
20#1850—21no—
21#1836C411yescharacteristic
22#1831C2211yescharacteristic
23#1831C2211yescharacteristic
24C4×#103—21no—
25C4×#102—21no—
26#1744C2211yescharacteristic
27#1744C2211yescharacteristic
28C4×#97C2211yescharacteristic
29C2×Q8⋊C8—21no—
30C2×Q8⋊C8—21no—
31#1279—21no—
32#1279—21no—
33#1272—21no—
34#1270—21no—
35C23×C8—21no—
36C2×C4⋊Q8C2×C411yescharacteristic
37C2×#124C2×C411yescharacteristic
38C2×C4×Q8—21no—
39C2×C4×Q8D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41#489—22no—
42#489C814yescomplemented
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×C4⋊C8D411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×#103—21no—
49C2×#103—21no—
50C2×#102—21no—
51C2×#102—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×#97—21no—
56C2×#97C2×C411yescharacteristic
57C2×#97C2×C411yescharacteristic
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61C2×C4×C8—21no—
62#391—21no—
63#391D411yescharacteristic
64#390D411yescharacteristic
65#389—21no—
66#389—21no—
67#388—21no—
68#388—21no—
69#384C2×C411yescharacteristic
70#382C2×C411yescharacteristic
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C4×C22⋊C4—21no—
74C4×C22⋊C4—21no—
75C43C2311yescharacteristic
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#336—21no—
89#336—21no—
90Q8⋊C8—42no—
91Q8⋊C8—42no—
92C22×Q8—21no—
93C22×Q8C22⋊C411yescharacteristic
94C22×D4C22⋊C411yescharacteristic
95C23×C4—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—81no—
105C22×C8—81no—
106C22×C8—81no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8C2×C812yescomplemented
111#124—21no—
112#124—21no—
113#124C2×C812yes—
114#124M1612yes—
115C4×Q8—41no—
116C4×Q8—41no—
117C4×Q8—41no—
118C4×Q8—41no—
119C4×Q8—41no—
120C4×Q8—41no—
121C4×D4—41no—
122C4×D4—41no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
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—41no—
139C2×C42C22×C411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42#4111yescharacteristic
145C2×C42#4111yescharacteristic
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—81no—
151#103—24no—
152#103—42no—
153#102—24no—
154#102—42no—
155C22⋊C8—81no—
156C22⋊C8—81no—
157C22⋊C8—81no—
158#97—21no—
159#97—21no—
160#97—21no—
161#97—42no—
162#97C2×C812yescomplemented
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—41no—
166C4×C8—41no—
167C4×C8—41no—
168C4×C8—41no—
169C4×C8—81no—
170#95—21no—
171#95—21no—
172#95—21no—
173#95—21no—
174#95—41no—
175#95—41no—
176C24—21no—
177C2×Q8—21no—
178C2×Q8—22no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8C22⋊C812yes—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4C22⋊C812yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
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×C4C2×C22⋊C411yescharacteristic
203C22×C4C4×D411yescharacteristic
204C22×C4C4×D411yescharacteristic
205C22×C4#12011yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12311yescharacteristic
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—42no—
216C2×C8—42no—
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—41no—
229C2×C8—81no—
230C2×C8—81no—
231C2×C8—81no—
232C2×C8—81no—
233C2×C8—84no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—81no—
253C22⋊C4—81no—
254C22⋊C4—81no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—41no—
262C42—41no—
263C42—41no—
264C42—41no—
265C42—41no—
266C42—81no—
267C42—81no—
268C42C22×C811yescharacteristic
269C42C2×M1611yescharacteristic
270C42#13111yescharacteristic
271C42#13111yescharacteristic
272C23—22no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23#38611yescharacteristic
280Q8—42no—
281Q8—42no—
282Q8—42no—
283Q8—81no—
284Q8—81no—
285Q8—81no—
286D4—42no—
287D4—81no—
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—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4C2×C22⋊C811yescharacteristic
323C2×C4#40811yescharacteristic
324C2×C4C8×D411yescharacteristic
325C2×C4C8×D411yescharacteristic
326C2×C4#43511yescharacteristic
327C2×C4#43511yescharacteristic
328C2×C4#45311yescharacteristic
329C2×C4#45411yescharacteristic
330C2×C4#45511yescharacteristic
331C2×C4#45511yescharacteristic
332C2×C4#45511yescharacteristic
333C2×C4#45611yescharacteristic
334C8—42no—
335C8—42no—
336C8—82no—
337C8—82no—
338C8—81no—
339C8—81no—
340C8—81no—
341C8—81no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22#186911yescharacteristic
350C22#188111yescharacteristic
351C22#188111yescharacteristic
352C22#188211yescharacteristic
353C22#188311yescharacteristic
354C22#188411yescharacteristic
355C22#188511yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—82no—
369C4—161no—
370C4#165911yescharacteristic
371C4#166111yescharacteristic
372C4#166211yescharacteristic
373C4#166411yescharacteristic
374C2—82no—
375C2#895311yescharacteristic
376C2#895411yescharacteristic
377C2#895711yescharacteristic
378C2#899711yescharacteristic
379C2#899811yescharacteristic
380C2#899911yescharacteristic
381C2#900511yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C22×C8
Frattini, cosocleC43C23
socleC23#386
prime radicalC22×D4C22⋊C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9873C2semidirect
4. #9641C2semidirect
6. #7030C2semidirect
7. #7030C2semidirect
42. #489C8semidirect
110. C4⋊Q8C2×C8semidirect
162. #97C2×C8semidirect

Maximal subgroups

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

subgroupindex
#233462
C4×#4892
#98732
#96412
#75672
#70302