Finite Group Databasebeta

← #7500254

Subgroups — #7500254

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59822C211yesmaximal, characteristic, complemented
2#39506C211yesmaximal, characteristic, complemented
3#32623C214yesmaximal, complemented
4#32178C214yesmaximal, complemented
5#32096C214yesmaximal, complemented
6#30076C212yesmaximal, complemented
7#29364C212yesmaximal, complemented
8#27504C214yesmaximal, complemented
9#27374C214yesmaximal, complemented
10#26334C214yesmaximal, complemented
11#23366C211yesmaximal, characteristic, complemented
12C2×#555—21no—
13#2859C2211yescharacteristic
14#2783—24no—
15#2733—22no—
16#2621—21no—
17#2548C2214yes—
18#2541C2214yes—
19#2538C2212yes—
20#2537—21no—
21#2513C2214yes—
22#2508C2214yes—
23#2489—22no—
24#2414—24no—
25#2355C2212yes—
26#2354C2214yes—
27#2314—21no—
28C4×C4⋊Q8C2211yescharacteristic
29C4×#127C2212yes—
30#2293C2211yescharacteristic
31#2099C2212yes—
32#2094C2214yes—
33#2093C2212yes—
34#2092C2214yes—
35#2084C2214yes—
36#2073C2214yes—
37#2071C2214yes—
38#2065C2214yes—
39#2054C2214yes—
40#2041C2214yes—
41#2041C2212yes—
42#2041C2214yes—
43#2011C2214yes—
44#2011C2212yes—
45#1985C2214yes—
46#1982C2214yes—
47#1972C2214yes—
48#1961C2214yes—
49#1949C2214yes—
50#1936C2214yes—
51#1926C2214yes—
52#1926C2214yes—
53#1925C2214yes—
54#1925C2214yes—
55#1919C2214yes—
56#1917C2214yes—
57#1917C2214yes—
58#1879C2214yes—
59#1879C2214yes—
60#1864C2214yes—
61#1864C2214yes—
62#1862C2214yes—
63#1839C2212yes—
64#1776C2214yes—
65C22×D8C2314yes—
66#555—48no—
67C2×C4⋊Q8D411yescharacteristic
68C2×#127C2312yes—
69C2×#127C2314yes—
70C2×#125—21no—
71C2×#124—24no—
72C2×C22⋊Q8—22no—
73C2×#121—24no—
74C2×#121—24no—
75C2×C4×Q8C2312yes—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4C2314yes—
79C2×C4×D4C2311yescharacteristic
80C2×#117—24no—
81C2×#117—21no—
82C2×#107C2314yes—
83C2×#106C2314yes—
84C2×#106C2314yes—
85C2×C4⋊C8C2314yes—
86C2×#102C2314yes—
87C2×#102C2314yes—
88C2×#102C2314yes—
89C2×#102C2314yes—
90C2×#102C2314yes—
91C2×#102C2314yes—
92C2×#102C2314yes—
93C2×#102C2314yes—
94C2×#97C2314yes—
95#400—24no—
96#398—24no—
97#397—24no—
98#397—24no—
99#396—24no—
100#394—24no—
101#394—24no—
102#392—24no—
103#392—24no—
104#391—22no—
105#391D412yes—
106#390—24no—
107#390—22no—
108#390—22no—
109#390C2314yes—
110#390C2314yes—
111#390C2314yes—
112#390C2314yes—
113#390C2312yes—
114#389—22no—
115#389D414yes—
116#389C2314yes—
117#389C2314yes—
118#389C2312yes—
119#388—24no—
120#388—22no—
121#388—24no—
122#388—24no—
123#388—24no—
124#387—24no—
125#387—24no—
126#384D411yescharacteristic
127#384C2312yes—
128#384C2312yes—
129#382—21no—
130#382—24no—
131#382C2314yes—
132#380—22no—
133#380—24no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2314yes—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C4×C22⋊C4—21no—
140C43C2311yescharacteristic
141#340C2314yes—
142#340C2314yes—
143#340C2314yes—
144#340C2314yes—
145#336C2314yes—
146#336C2314yes—
147#336C2314yes—
148#336C2314yes—
149#336C2312yes—
150#336C2312yes—
151C22×Q8C2×D412yes—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4C2414yes—
157C22×D4C2411yescharacteristic
158C23×C4—24no—
159C23×C4—21no—
160C2×D8—44no—
161C2×D8—44no—
162C2×D8—44no—
163C22×C8C2414yes—
164C22×C8C2414yes—
165C4⋊Q8—24no—
166#127—44no—
167#127—44no—
168#127—44no—
169#125—82no—
170#124—416no—
171C22⋊Q8—84no—
172#121—416no—
173#121—88no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×D4—44no—
177C4×D4—44no—
178C4×D4—42no—
179C4×D4—48no—
180C4×D4—84no—
181#117—416no—
182#117—44no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2411yescharacteristic
196C2×C4⋊C4C2412yes—
197C2×C4⋊C4C2414yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C42—24no—
211C2×C42—21no—
212C2×C42—24no—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2412yes—
217C2×C42C2411yescharacteristic
218C2×C42C2414yes—
219#107—44no—
220#106—44no—
221#106—44no—
222C4⋊C8—44no—
223#102—44no—
224#102—44no—
225#102—44no—
226#102—44no—
227#102—44no—
228#102—44no—
229#102—44no—
230#102—44no—
231#97—44no—
232#95—22no—
233#95—24no—
234#95—24no—
235#95—21no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95C2×D414yes—
241#95C2×D414yes—
242C24—21no—
243C24—24no—
244C2×Q8—24no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—24no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—48no—
259C2×D4—44no—
260C2×D4—82no—
261C2×D4—82no—
262C2×D4—84no—
263C2×D4—84no—
264C2×D4—84no—
265C2×D4—84no—
266C2×D4—84no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—42no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—42no—
275C22×C4—48no—
276C22×C4—42no—
277C22×C4C22×D414yes—
278C22×C4C22×D414yes—
279C22×C4C22×D412yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D412yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C2511yescharacteristic
284D8—88no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C4⋊C4—28no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C22⋊C4—48no—
309C22⋊C4—48no—
310C22⋊C4—42no—
311C22⋊C4—48no—
312C22⋊C4—44no—
313C22⋊C4—48no—
314C22⋊C4—82no—
315C22⋊C4—84no—
316C22⋊C4—84no—
317C22⋊C4—84no—
318C22⋊C4—84no—
319C22⋊C4—84no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—22no—
324C42—42no—
325C42—41no—
326C42—48no—
327C42—42no—
328C42—84no—
329C42AGL(1,ℤ/8)14yes—
330C23—41no—
331C23—42no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—41no—
336C23—44no—
337C23—44no—
338C23—44no—
339C23—42no—
340C23—44no—
341C23—41no—
342C23C23×D411yescharacteristic
343Q8—84no—
344D4—44no—
345D4—44no—
346D4—44no—
347D4—44no—
348D4—44no—
349D4—88no—
350D4—88no—
351D4—168no—
352D4—168no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—21no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—84no—
381C2×C4—816no—
382C2×C4C2×AGL(1,ℤ/8)12yes—
383C2×C4C2×AGL(1,ℤ/8)14yes—
384C2×C4#58512yes—
385C2×C4#58514yes—
386C8—44no—
387C8—44no—
388C22—81no—
389C22—84no—
390C22—84no—
391C22—81no—
392C22—81no—
393C22—84no—
394C22—84no—
395C22—84no—
396C22—82no—
397C22—84no—
398C22—82no—
399C22—84no—
400C22#348511yescharacteristic
401C22#348712yes—
402C22C22×AGL(1,ℤ/8)11yescharacteristic
403C22#358511yescharacteristic
404C22#358512yes—
405C4—22no—
406C4—24no—
407C4—44no—
408C4—41no—
409C4—44no—
410C4—42no—
411C4—81no—
412C4—82no—
413C4—84no—
414C4—84no—
415C4#356614yes—
416C2—161no—
417C2—164no—
418C2#5208411yescharacteristic
419C2#6172011yescharacteristic
420C2#6179711yescharacteristic
421C2#6189012yes—
422C2#6193812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59822C2semidirect
2. #39506C2semidirect
3. #32623C2semidirect
4. #32178C2semidirect
5. #32096C2semidirect
6. #30076C2semidirect
7. #29364C2semidirect
8. #27504C2semidirect
9. #27374C2semidirect
10. #26334C2semidirect
11. #23366C2semidirect

Maximal subgroups

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

subgroupindex
#598222
#395062
#326232
#321782
#320962
#300762
#293642
#275042
#273742
#263342
#233662