Finite Group Databasebeta

← #508626

Subgroups — #508626

Order 512.

484 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33257C211yesmaximal, characteristic, complemented
2#21275C211yesmaximal, characteristic
3#13516C211yesmaximal, characteristic, complemented
4#13516C211yesmaximal, characteristic, complemented
5#13515C211yesmaximal, characteristic, complemented
6#13514C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic
8#13476C211yesmaximal, characteristic
9#13474C211yesmaximal, characteristic
10#13474C211yesmaximal, characteristic
11#13178C211yesmaximal, characteristic, complemented
12#13178C211yesmaximal, characteristic, complemented
13#12898C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic
15#12813C211yesmaximal, characteristic
16C2×C4×D8C2211yescharacteristic
17#2864—21no—
18#2293—21no—
19C2×#367C2211yescharacteristic
20C2×#367C2211yescharacteristic
21C2×#366C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33#1970C2211yescharacteristic
34#1970C2211yescharacteristic
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1941—21no—
42#1941—21no—
43#1934—21no—
44#1934—21no—
45#1846C2211yescharacteristic
46#1846C2211yescharacteristic
47#1835—21no—
48C4×#107C2211yescharacteristic
49C4×#102—21no—
50C42×C8C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C2×C4×D4—21no—
61C2×#117—41no—
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C4×D8—22no—
67C4×D8Q814yes—
68C2×#107C2311yescharacteristic
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#106—21no—
76C2×#106—21no—
77C2×#102—21no—
78C2×#102—21no—
79C2×#102—21no—
80C2×C4×C8D411yescharacteristic
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86#390—21no—
87#390—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#388—41no—
95#388—41no—
96#387—41no—
97#387—41no—
98#387—41no—
99#387—41no—
100#382—21no—
101#382—21no—
102C4×C4⋊C4—21no—
103C4×C22⋊C4—41no—
104C43D411yescharacteristic
105#367—41no—
106#367—41no—
107#366—41no—
108#366—41no—
109C4⋊C16—22no—
110C4⋊C16—22no—
111C4⋊C16—41no—
112C4⋊C16—41no—
113C4⋊C16—41no—
114C4⋊C16—41no—
115#357—41no—
116#357—41no—
117#357—41no—
118#357—41no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126C22×D4—21no—
127C23×C4—41no—
128C2×D8—21no—
129C2×D8—21no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8C2×Q812yes—
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2411yescharacteristic
138C4×D4—24no—
139C4×D4—42no—
140#117—84no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C42—41no—
160C2×C42D811yescharacteristic
161C2×C42D811yescharacteristic
162C2×C42D811yescharacteristic
163C2×C42D811yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C16—21no—
168C2×C16—21no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179#107—21no—
180#107—21no—
181#107—21no—
182#107—21no—
183#107—21no—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107C2×Q812yes—
193#106—42no—
194#106—42no—
195#102—24no—
196#102—42no—
197#102—42no—
198C4×C8—21no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8—22no—
203C4×C8—41no—
204C4×C8—41no—
205C4×C8C2×Q812yes—
206C4×C8#4112yes—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218C24—41no—
219C2×D4—22no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C22×C4—41no—
226C22×C4—44no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C22×D411yescharacteristic
242D8—22no—
243D8—22no—
244D8—41no—
245D8—41no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8C22⋊Q812yes—
255C2×C8C22×Q811yescharacteristic
256C2×C8C2×#4111yescharacteristic
257C2×C8H(5,𝔽2)11yescharacteristic
258C2×C8H(5,𝔽2)11yescharacteristic
259C2×C82−1+411yescharacteristic
260C2×C82−1+411yescharacteristic
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C22⋊C4—44no—
282C22⋊C4—82no—
283C22⋊C4—82no—
284C22⋊C4—82no—
285C22⋊C4—82no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—22no—
291C42—22no—
292C42—41no—
293C42—41no—
294C42—82no—
295C42C22⋊Q812yes—
296C42#13712yes—
297C16—41no—
298C16—41no—
299C16—41no—
300C16—41no—
301C23—42no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23C22×D811yescharacteristic
309D4—42no—
310D4—42no—
311D4—81no—
312D4—81no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—88no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4#47412yes—
350C2×C4#47412yes—
351C2×C4#50812yes—
352C2×C4#50912yes—
353C2×C4C2×C22⋊Q811yescharacteristic
354C2×C4#53711yescharacteristic
355C2×C4#53711yescharacteristic
356C2×C4C2×#13711yescharacteristic
357C2×C4#57511yescharacteristic
358C2×C4#57511yescharacteristic
359C8—22no—
360C8—41no—
361C8—41no—
362C8#54011yescharacteristic
363C8#54111yescharacteristic
364C8#54411yescharacteristic
365C8#54411yescharacteristic
366C22—82no—
367C22—82no—
368C22—82no—
369C22—161no—
370C22—161no—
371C22—161no—
372C22—161no—
373C22C2×#47411yescharacteristic
374C22#324511yescharacteristic
375C22#324511yescharacteristic
376C22C2×#50811yescharacteristic
377C22C2×#50911yescharacteristic
378C22#341411yescharacteristic
379C22#341411yescharacteristic
380C4—22no—
381C4—22no—
382C4—22no—
383C4—22no—
384C4—41no—
385C4—41no—
386C4—82no—
387C4—161no—
388C4—161no—
389C4—161no—
390C4—161no—
391C4—161no—
392C4—161no—
393C4#262911yescharacteristic
394C4#308011yescharacteristic
395C4#325211yescharacteristic
396C4#325211yescharacteristic
397C2—162no—
398C2#2999411yescharacteristic
399C2#3362611yescharacteristic
400C2#3362711yescharacteristic
401C2#3381311yescharacteristic
402C2#3381611yescharacteristic
403C2#3381711yescharacteristic
404C2#3382211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33257C2semidirect
3. #13516C2semidirect
4. #13516C2semidirect
5. #13515C2semidirect
6. #13514C2semidirect
11. #13178C2semidirect
12. #13178C2semidirect
13. #12898C2semidirect

Maximal subgroups

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

subgroupindex
#332572
#212752
#135162
#135152
#135142
#134762
#134742
#131782
#128982
#128132