Finite Group Databasebeta

← #511661

Subgroups — #511661

Order 512.

432 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33216C211yesmaximal, characteristic
2#33215C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13539C211yesmaximal, characteristic, complemented
5#13538C211yesmaximal, characteristic, complemented
6#13537C211yesmaximal, characteristic, complemented
7#13494C211yesmaximal, characteristic
8#13494C211yesmaximal, characteristic
9#13493C211yesmaximal, characteristic
10#13492C211yesmaximal, characteristic
11#13453C211yesmaximal, characteristic, complemented
12#13453C211yesmaximal, characteristic, complemented
13#13452C211yesmaximal, characteristic, complemented
14#13451C211yesmaximal, characteristic, complemented
15#12845C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#405C2211yescharacteristic
19#2629—21no—
20#2628—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C22⋊C16C2211yescharacteristic
26C2×C22⋊C16C2211yescharacteristic
27#1993—21no—
28#1992—21no—
29#1944—21no—
30#1943—21no—
31#1942C2211yescharacteristic
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1846C2211yescharacteristic
36#1845C2211yescharacteristic
37#1809—21no—
38#1809—21no—
39#1808C2211yescharacteristic
40#1808C2211yescharacteristic
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43#1380C2211yescharacteristic
44#1380C2211yescharacteristic
45#1380C2211yescharacteristic
46#1380C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×M16D411yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×#125—21no—
63C2×#117D411yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68#500—41no—
69#500—41no—
70#499—41no—
71#499—41no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#106—21no—
83C2×#106—21no—
84#405—41no—
85#405—41no—
86C2×#97D411yescharacteristic
87C2×#97D411yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#400—41no—
91#399—41no—
92#398—41no—
93#398—41no—
94#396—41no—
95#396—41no—
96#395—41no—
97#393—41no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#384—21no—
103#382—21no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108#367—41no—
109#367—41no—
110#366—41no—
111#366—41no—
112C22⋊C16—41no—
113C22⋊C16—41no—
114#340—21no—
115#340—21no—
116#340—21no—
117#340—21no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122C22×Q8—41no—
123C23×C4C2×D411yescharacteristic
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—41no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22×C8C2411yescharacteristic
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139#125—42no—
140#125—42no—
141#117—41no—
142#117—41no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4D811yescharacteristic
161C2×C4⋊C4D811yescharacteristic
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4D811yescharacteristic
167C2×C22⋊C4D811yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
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—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—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#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—42no—
199#107—42no—
200#106—42no—
201#106—42no—
202#97—41no—
203#97—41no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—21no—
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—
218C24C2×D811yescharacteristic
219C2×Q8—82no—
220C2×Q8—161no—
221C2×Q8—161no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C22×D411yescharacteristic
242M16—44no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8#13712yes—
255C2×C8#13712yes—
256C2×C8H(5,𝔽2)11yescharacteristic
257C2×C8H(5,𝔽2)11yescharacteristic
258C2×C82−1+411yescharacteristic
259C2×C82−1+411yescharacteristic
260C2×C82−1+411yescharacteristic
261C2×C82−1+411yescharacteristic
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—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—161no—
284C4⋊C4—161no—
285C4⋊C4—161no—
286C4⋊C4—161no—
287C4⋊C4—161no—
288C4⋊C4—161no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—161no—
292C22⋊C4—161no—
293C22⋊C4—161no—
294C22⋊C4—161no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C16—41no—
300C16—41no—
301C16—41no—
302C16—41no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23C22×D811yescharacteristic
311Q8—164no—
312C2×C4—21no—
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—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
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—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4#53411yescharacteristic
356C2×C4#53611yescharacteristic
357C2×C4#53611yescharacteristic
358C2×C4C2×#13711yescharacteristic
359C2×C4C2×#13711yescharacteristic
360C2×C4#57511yescharacteristic
361C8—22no—
362C8—22no—
363C8—41no—
364C8#56311yescharacteristic
365C8#56311yescharacteristic
366C8#56311yescharacteristic
367C8#56311yescharacteristic
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22#318611yescharacteristic
376C22#322611yescharacteristic
377C22#322611yescharacteristic
378C22#341511yescharacteristic
379C22#341611yescharacteristic
380C22#341611yescharacteristic
381C22#341711yescharacteristic
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—161no—
388C4—161no—
389C4—161no—
390C4—161no—
391C4—161no—
392C4—161no—
393C4—161no—
394C4—161no—
395C4#281711yescharacteristic
396C4#319511yescharacteristic
397C4#323711yescharacteristic
398C4#323711yescharacteristic
399C2—81no—
400C2#3278611yescharacteristic
401C2#3376411yescharacteristic
402C2#3376711yescharacteristic
403C2#3380711yescharacteristic
404C2#3380811yescharacteristic
405C2#3380911yescharacteristic
406C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13539C2semidirect
4. #13539C2semidirect
5. #13538C2semidirect
6. #13537C2semidirect
11. #13453C2semidirect
12. #13453C2semidirect
13. #13452C2semidirect
14. #13451C2semidirect

Maximal subgroups

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

subgroupindex
#332162
#332152
#135392
#135382
#135372
#134942
#134932
#134922
#134532
#134522
#134512
#128452