Finite Group Databasebeta

← #6434605

Subgroups — #6434605

Order 512.

1830 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C212yesmaximal, complemented
2#30384C214yesmaximal, complemented
3#30342C214yesmaximal, complemented
4#30293C211yesmaximal, characteristic, complemented
5#30239C211yesmaximal, characteristic
6#30185C212yesmaximal, complemented
7#29772C214yesmaximal, complemented
8#23781C214yesmaximal, complemented
9#23682C214yesmaximal, complemented
10#21526C214yesmaximal, complemented
11C2×#1767C211yesmaximal, characteristic, complemented
12C2×#529—22no—
13C2×#466C2214yes—
14C2×#442C2214yes—
15C22×C4⋊C8C2212yes—
16C22×#97C2212yes—
17#2861—22no—
18#2860—22no—
19#2860C2214yes—
20#2859C2212yes—
21#2858C2212yes—
22#2857C2212yes—
23#2856C2212yes—
24#2854—22no—
25#2854C2214yes—
26#2850—22no—
27C4×C4⋊Q8C2212yes—
28C4×#127C2212yes—
29C4×#124—22no—
30C4×#121—22no—
31C4×#121C2214yes—
32#2293—22no—
33#2293C2214yes—
34C2×C43C2211yescharacteristic
35#2099C2214yes—
36#2093C2214yes—
37#2087C2218yes—
38#2084C2214yes—
39#2071C2214yes—
40#2065C2218yes—
41#2065C2218yes—
42#2058C2214yes—
43#2054C2218yes—
44#2041C2214yes—
45#2023C2214yes—
46#2011C2214yes—
47#1972C2214yes—
48#1943C2214yes—
49#1936C2218yes—
50#1893C2218yes—
51#1807C2214yes—
52#1776C2214yes—
53#1767C2214yes—
54#1767C2214yes—
55#1762C2218yes—
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C22×D8C2314yes—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×C4⋊Q8C2312yes—
62C2×#127C2312yes—
63#529—416no—
64C2×#126—24no—
65C2×#125—22no—
66C2×#124—22no—
67C2×#124—22no—
68C2×#123—24no—
69C2×C22⋊Q8—22no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121C2314yes—
73C2×C4×Q8—22no—
74C2×C4×D4—24no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4C2314yes—
78C2×#117—24no—
79C22×C42D412yes—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#466—216no—
85#442—48no—
86C2×#107C2314yes—
87C2×#106C2314yes—
88C2×#106C2314yes—
89C2×C4⋊C8—28no—
90C2×C4⋊C8C2318yes—
91C2×#102C2318yes—
92C2×#102C2318yes—
93C2×#102C2318yes—
94C2×#97—44no—
95C2×#97C2314yes—
96C2×#97C2314yes—
97#391—22no—
98#391—22no—
99#390—22no—
100#390—22no—
101#390—22no—
102#390—22no—
103#390—22no—
104#390—22no—
105#390C2314yes—
106#390C2314yes—
107#390C2314yes—
108#390C2314yes—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389C2314yes—
114#389C2314yes—
115#389C2314yes—
116#389C2314yes—
117#388—24no—
118#388—24no—
119#388—24no—
120#388—24no—
121#388—24no—
122#388—24no—
123#388—24no—
124#388—24no—
125#386—24no—
126#386—24no—
127#384C2312yes—
128#384C2312yes—
129#382—22no—
130#380—22no—
131#380C2314yes—
132C4×C4⋊C4—22no—
133C4×C4⋊C4C2314yes—
134C4×C22⋊C4—24no—
135C4×C22⋊C4—24no—
136C43D412yes—
137C43D412yes—
138C43C2312yes—
139C43C2312yes—
140#340C2318yes—
141#340C2318yes—
142#340C2318yes—
143#336C2314yes—
144#336C2314yes—
145C22×Q8—22no—
146C22×D4—24no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4C2414yes—
150C23×C4—24no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2411yescharacteristic
156C2×D8—28no—
157C2×D8—44no—
158C2×D8—44no—
159C22×C8—24no—
160C22×C8—42no—
161C22×C8—42no—
162C22×C8—42no—
163C22×C8—42no—
164C22×C8#4118yes—
165C22×C8C2414yes—
166C22×C8C2414yes—
167C4⋊Q8—44no—
168#127—44no—
169#126—416no—
170#125—48no—
171#124—48no—
172#124—48no—
173#123—416no—
174C22⋊Q8—48no—
175#121—216no—
176#121—48no—
177#121—48no—
178C4×Q8—48no—
179C4×D4—28no—
180C4×D4—28no—
181C4×D4—48no—
182C4×D4—48no—
183C4×D4—416no—
184#117—416no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2414yes—
196C2×C4⋊C4C2414yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C42—28no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—44no—
214C2×C42—42no—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2412yes—
228C2×C42C2411yescharacteristic
229C2×C42C2412yes—
230C2×C42C2411yescharacteristic
231#107—28no—
232#106—44no—
233#106—44no—
234C4⋊C8—216no—
235#102—216no—
236#102—48no—
237#102—48no—
238#97—28no—
239#97—44no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246C24—24no—
247C24C22×D411yescharacteristic
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×D4—28no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—28no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—48no—
260C2×D4—48no—
261C2×D4—48no—
262C2×D4—44no—
263C2×D4—44no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4AGL(1,ℤ/8)18yes—
283C22×C4C22×D412yes—
284C22×C4C22×D412yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D412yes—
287C22×C4C22×D412yes—
288C22×C4C22×D412yes—
289C22×C4C22×D412yes—
290C22×C4C2×#4114yes—
291C22×C4C2511yescharacteristic
292D8—416no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—48no—
299C2×C8—48no—
300C2×C8C2×#4118yes—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—28no—
305C4⋊C4—48no—
306C4⋊C4—44no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C4⋊C4—44no—
310C4⋊C4—48no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C22⋊C4—48no—
315C22⋊C4—48no—
316C22⋊C4—48no—
317C22⋊C4—48no—
318C22⋊C4—48no—
319C22⋊C4—48no—
320C22⋊C4—48no—
321C22⋊C4—48no—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C22⋊C4—48no—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—21no—
335C42—21no—
336C42—24no—
337C42—41no—
338C42—41no—
339C42—42no—
340C42—42no—
341C42—48no—
342C42H(5,𝔽2)12yes—
343C422−1+412yes—
344C23—21no—
345C23—21no—
346C23—22no—
347C23—44no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23#53212yes—
355C23C2×AGL(1,ℤ/8)14yes—
356C23C23×D411yescharacteristic
357Q8—88no—
358D4—48no—
359D4—48no—
360D4—88no—
361D4—88no—
362D4—816no—
363C2×C4—28no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—816no—
395C2×C4#53212yes—
396C2×C4#53714yes—
397C2×C4C2×AGL(1,ℤ/8)14yes—
398C2×C4C2×AGL(1,ℤ/8)14yes—
399C2×C4#57514yes—
400C2×C4C22×#4112yes—
401C2×C4C2×H(5,𝔽2)11yescharacteristic
402C2×C4C2×2−1+411yescharacteristic
403C2×C4#58512yes—
404C8—28no—
405C8—44no—
406C22—22no—
407C22—24no—
408C22—22no—
409C22—24no—
410C22—84no—
411C22—84no—
412C22—84no—
413C22—84no—
414C22—84no—
415C22—84no—
416C22—84no—
417C22#318414yes—
418C22C2×#53211yescharacteristic
419C22C2×#53711yescharacteristic
420C22#344911yescharacteristic
421C22C22×AGL(1,ℤ/8)12yes—
422C22C2×#57512yes—
423C4—24no—
424C4—24no—
425C4—24no—
426C4—22no—
427C4—24no—
428C4—42no—
429C4—42no—
430C4—84no—
431C4—84no—
432C4—84no—
433C4#345912yes—
434C4#346112yes—
435C2—24no—
436C2—164no—
437C2#3459411yescharacteristic
438C2#6068811yescharacteristic
439C2C2×#318411yescharacteristic
440C2#6091612yes—
441C2#6095812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#30239C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36631C2semidirect
2. #30384C2semidirect
3. #30342C2semidirect
4. #30293C2semidirect
6. #30185C2semidirect
7. #29772C2semidirect
8. #23781C2semidirect
9. #23682C2semidirect
10. #21526C2semidirect
11. C2×#1767C2semidirect

Maximal subgroups

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

subgroupindex
#366312
#303842
#303422
#302932
#302392
#301852
#297722
#237812
#236822
#215262
C2×#17672