Finite Group Databasebeta

← #99475

Subgroups — #99475

Order 512.

444 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 94 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#22170C211yesmaximal, characteristic
2#11269C211yesmaximal, characteristic, complemented
3#8944C211yesmaximal, characteristic
4#8832C211yesmaximal, characteristic, complemented
5#7565C211yesmaximal, characteristic
6#7029C211yesmaximal, characteristic, complemented
7#7026C211yesmaximal, characteristic, complemented
8C2×#435—21no—
9C2×#423C2211yescharacteristic
10C4×#121C2211yescharacteristic
11#1924—21no—
12#1917—21no—
13#1870—21no—
14#1869—21no—
15#1850—21no—
16#1797C2211yescharacteristic
17C4×C22⋊C8C2211yescharacteristic
18C4×C22⋊C8C2211yescharacteristic
19C2×#341—21no—
20C2×#341—21no—
21#1728C2211yescharacteristic
22C2×#336C2211yescharacteristic
23C2×#335—21no—
24C2×D4⋊C8—21no—
25#1279—21no—
26#1277—21no—
27#1271C411yescharacteristic
28#1271C411yescharacteristic
29#1270C411yescharacteristic
30#1270C411yescharacteristic
31C22×M16—21no—
32C22×M16D411yescharacteristic
33C22×M16D411yescharacteristic
34C23×C8—21no—
35C23×C8—21no—
36C2×#121Q811yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4D411yescharacteristic
39C22×C42C2311yescharacteristic
40#435—42no—
41#435—42no—
42#423—42no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×#97—21no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57#390C2×C411yescharacteristic
58#389C2×C411yescharacteristic
59#388—21no—
60#386—21no—
61#380Q811yescharacteristic
62C4×C4⋊C4D411yescharacteristic
63C4×C22⋊C4—21no—
64#341—42no—
65#341—42no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336C2×C411yescharacteristic
74#336C2×C411yescharacteristic
75#336C2×C411yescharacteristic
76#336C2×C411yescharacteristic
77#335—42no—
78D4⋊C8—42no—
79C22×D4—21no—
80C22×D4C4⋊C411yescharacteristic
81C23×C4—21no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×Q811yescharacteristic
84C23×C4#4111yescharacteristic
85C2×M16—21no—
86C2×M16—22no—
87C2×M16—21no—
88C2×M16—21no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16D812yes—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—42no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—81no—
117C22×C8—81no—
118#121—22no—
119#121—41no—
120C4×D4—41no—
121C4×D4—41no—
122C4×D4—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4C4⋊C411yescharacteristic
128C2×C4⋊C4C4⋊C411yescharacteristic
129C2×C4⋊C4C4⋊C411yescharacteristic
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C42—21no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42C22×C411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42#4111yescharacteristic
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—42no—
149C22⋊C8—22no—
150C22⋊C8—42no—
151C22⋊C8—42no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—81no—
156C22⋊C8—81no—
157#97—42no—
158C4×C8—41no—
159C4×C8—41no—
160C4×C8—41no—
161C4×C8—41no—
162#95—21no—
163#95—21no—
164#95—21no—
165C24—21no—
166C24C22⋊Q811yescharacteristic
167C2×D4—21no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4—81no—
174C2×D4#10812yes—
175C22×C4—21no—
176C22×C4—21no—
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—21no—
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×C4C2×C4⋊C411yescharacteristic
197C22×C4C4×D411yescharacteristic
198C22×C4C4×D411yescharacteristic
199C22×C4#12011yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201C22×C4#12311yescharacteristic
202C22×C4C2×D811yescharacteristic
203C22×C4#13511yescharacteristic
204C22×C4AGL(1,ℤ/8)11yescharacteristic
205C22×C4#13711yescharacteristic
206M16—42no—
207M16—42no—
208M16—42no—
209M16—42no—
210M16—81no—
211M16—81no—
212M16—81no—
213M16—81no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—42no—
222C2×C8—41no—
223C2×C8—41no—
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—41no—
237C2×C8—41no—
238C2×C8—42no—
239C2×C8—41no—
240C2×C8—84no—
241C2×C8—81no—
242C2×C8—81no—
243C2×C8—81no—
244C2×C8—81no—
245C2×C8—81no—
246C2×C8—81no—
247C2×C8—81no—
248C2×C8—81no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C22⋊C4—42no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—81no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—41no—
268C42—41no—
269C42—41no—
270C42—41no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23#38511yescharacteristic
285C23#47411yescharacteristic
286C23#47811yescharacteristic
287D4—42no—
288D4—81no—
289D4—82no—
290D4—82no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
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—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4C2×#10811yescharacteristic
329C2×C4#43011yescharacteristic
330C2×C4C4×D811yescharacteristic
331C2×C4#44011yescharacteristic
332C2×C4#44311yescharacteristic
333C2×C4#44411yescharacteristic
334C2×C4#44711yescharacteristic
335C2×C4#45211yescharacteristic
336C2×C4#45411yescharacteristic
337C2×C4#45511yescharacteristic
338C2×C4B(4,𝔽2)11yescharacteristic
339C2×C4#45811yescharacteristic
340C8—42no—
341C8—42no—
342C8—81no—
343C8—81no—
344C8—81no—
345C8—82no—
346C8—81no—
347C8—82no—
348C8—81no—
349C8—81no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22#185511yescharacteristic
368C22#185811yescharacteristic
369C22#186011yescharacteristic
370C22#186111yescharacteristic
371C22#186211yescharacteristic
372C22#186611yescharacteristic
373C22#186711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4#159811yescharacteristic
386C4#160611yescharacteristic
387C4#160711yescharacteristic
388C4#161511yescharacteristic
389C2—41no—
390C2—41no—
391C2—82no—
392C2#885911yescharacteristic
393C2#886911yescharacteristic
394C2#887511yescharacteristic
395C2#890011yescharacteristic
396C2#890611yescharacteristic
397C2#891711yescharacteristic
398C2#893011yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#385
prime radicalC2×#121Q8
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11269C2semidirect
4. #8832C2semidirect
6. #7029C2semidirect
7. #7026C2semidirect

Maximal subgroups

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

subgroupindex
#221702
#112692
#89442
#88322
#75652
#70292
#70262