Finite Group Databasebeta

← #496875

Subgroups — #496875

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2233C211yesmaximal, characteristic
2#21524C211yesmaximal, characteristic
3#21158C211yesmaximal, characteristic, complemented
4#13167C211yesmaximal, characteristic, complemented
5#13166C211yesmaximal, characteristic, complemented
6#13149C211yesmaximal, characteristic, complemented
7#13147C211yesmaximal, characteristic, complemented
8#13111C211yesmaximal, characteristic, complemented
9#13110C211yesmaximal, characteristic, complemented
10#12992C211yesmaximal, characteristic
11#12937C211yesmaximal, characteristic, complemented
12#12935C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic, complemented
14#12866C211yesmaximal, characteristic, complemented
15#12600C211yesmaximal, characteristic, complemented
16C2×#466C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C2×#405C2211yescharacteristic
21#2324—21no—
22#2316—21no—
23#2233C418yescomplemented
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×#346C2211yescharacteristic
35C2×C4×C16C2211yescharacteristic
36#1970C2211yescharacteristic
37#1943—21no—
38#1943—21no—
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1936—21no—
45#1895—21no—
46#1808C2211yescharacteristic
47C4×#107C2211yescharacteristic
48#1762—21no—
49#1736—21no—
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×D8C2311yescharacteristic
61C22×M16D411yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#121—21no—
64C2×C4×D4—21no—
65C2×#117—21no—
66C2×#117D411yescharacteristic
67C22×C4⋊C4—21no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#466—22no—
73#466C2×C414yes—
74C4×D8—41no—
75C4×D8—41no—
76#427—41no—
77#427—41no—
78C2×#107—21no—
79C2×#107—22no—
80C2×#107C2×C414yescomplemented
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#106—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92#405—41no—
93#405—41no—
94C2×#97D411yescharacteristic
95C2×#97D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#390—21no—
99#390—41no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—41no—
106#389—41no—
107#388—41no—
108#386—41no—
109#385—41no—
110#380—21no—
111#380—41no—
112#380—41no—
113C4×C4⋊C4—21no—
114#367—41no—
115#367C2×C414yescomplemented
116#367C2×C414yescomplemented
117#366—41no—
118#357—41no—
119#357—41no—
120#357C2×C414yes—
121#357C2×C414yes—
122C22⋊C16—22no—
123C22⋊C16C2×C414yescomplemented
124#346—41no—
125C4×C16—41no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133C22×D4—21no—
134C22×D4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4C2×D411yescharacteristic
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8C22×C412yes—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—41no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8C22×C412yes—
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8#4112yes—
156C22×C8C2411yescharacteristic
157#121—24no—
158#121—42no—
159C4×D4—42no—
160C4×D4—42no—
161#117—41no—
162#117—41no—
163#117—42no—
164#117—42no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4D811yescharacteristic
181C2×C4⋊C4D811yescharacteristic
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4D811yescharacteristic
189C2×C22⋊C4D811yescharacteristic
190C2×C42—41no—
191C2×C42—41no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16C22×C412yes—
205C2×C16C22×C412yes—
206#107—24no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107C22×C412yes—
219#107C22×C412yes—
220#107C22×C412yes—
221#106—42no—
222#102—24no—
223#102—42no—
224#102—42no—
225#97—41no—
226#97—41no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238C24—41no—
239C24C2×D811yescharacteristic
240C2×D4—22no—
241C2×D4—44no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—161no—
248C2×D4—161no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—44no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—82no—
262C22×C4—161no—
263C22×C4—161no—
264C22×C4—161no—
265C22×C4—161no—
266C22×C4C4×D412yes—
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4#13512yes—
273C22×C4C22×D411yescharacteristic
274D8—22no—
275D8—22no—
276D8—41no—
277D8—41no—
278M16—44no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8C4×D412yes—
291C2×C8#13712yes—
292C2×C8C23×C411yescharacteristic
293C2×C8C2×#4111yescharacteristic
294C2×C8H(5,𝔽2)11yescharacteristic
295C2×C8H(5,𝔽2)11yescharacteristic
296C2×C82−1+411yescharacteristic
297C2×C82−1+411yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—44no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—161no—
316C4⋊C4—161no—
317C4⋊C4—161no—
318C4⋊C4—161no—
319C4⋊C4—161no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—44no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—161no—
326C22⋊C4—161no—
327C22⋊C4—161no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—82no—
333C42—161no—
334C16—22no—
335C16—22no—
336C16—41no—
337C16—41no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—42no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23C4×D812yes—
352C23C22×D811yescharacteristic
353D4—42no—
354D4—42no—
355D4—81no—
356D4—81no—
357D4—164no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—164no—
398C2×C4—164no—
399C2×C4C4×D812yes—
400C2×C4C2×C4×D411yescharacteristic
401C2×C4#53411yescharacteristic
402C2×C4#53611yescharacteristic
403C2×C4C2×#13511yescharacteristic
404C2×C4C2×#13711yescharacteristic
405C2×C4#57511yescharacteristic
406C8—22no—
407C8—22no—
408C8—41no—
409C8#52011yescharacteristic
410C8#52011yescharacteristic
411C8#53911yescharacteristic
412C8#53911yescharacteristic
413C22—22no—
414C22—22no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—161no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22C2×C4×D811yescharacteristic
428C22#318611yescharacteristic
429C22#322611yescharacteristic
430C22#341511yescharacteristic
431C22#341611yescharacteristic
432C22#341611yescharacteristic
433C22#341611yescharacteristic
434C4—22no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—82no—
440C4—82no—
441C4—82no—
442C4—161no—
443C4—161no—
444C4—161no—
445C4—161no—
446C4#232211yescharacteristic
447C4C4×#13711yescharacteristic
448C4#319011yescharacteristic
449C4#323211yescharacteristic
450C2—42no—
451C2—162no—
452C2#2207511yescharacteristic
453C2#3345411yescharacteristic
454C2#3345711yescharacteristic
455C2#3376411yescharacteristic
456C2#3377011yescharacteristic
457C2#3380711yescharacteristic
458C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C2×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. #21158C2semidirect
4. #13167C2semidirect
5. #13166C2semidirect
6. #13149C2semidirect
7. #13147C2semidirect
8. #13111C2semidirect
9. #13110C2semidirect
11. #12937C2semidirect
12. #12935C2semidirect
13. #12921C2semidirect
14. #12866C2semidirect
15. #12600C2semidirect
23. #2233C4semidirect
80. C2×#107C2×C4semidirect
115. #367C2×C4semidirect
116. #367C2×C4semidirect
123. C22⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22332
#215242
#211582
#131672
#131662
#131492
#131472
#131112
#131102
#129922
#129372
#129352
#129212
#128662
#126002