Finite Group Databasebeta

← #7500215

Subgroups — #7500215

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59822C211yesmaximal, characteristic, complemented
2#39379C211yesmaximal, characteristic, complemented
3#32596C214yesmaximal, complemented
4#32162C214yesmaximal, complemented
5#32078C214yesmaximal, complemented
6#30075C212yesmaximal, complemented
7#29364C212yesmaximal, complemented
8#27486C214yesmaximal, complemented
9#27358C214yesmaximal, complemented
10#26316C214yesmaximal, complemented
11#23363C211yesmaximal, characteristic, complemented
12C2×#555—21no—
13C2×#541—21no—
14#2859C2211yescharacteristic
15#2733—22no—
16#2725—24no—
17#2629—21no—
18#2548C2214yes—
19#2541C2214yes—
20#2538C2212yes—
21#2537—21no—
22#2512C2214yes—
23#2506C2214yes—
24#2421—22no—
25#2414—24no—
26#2354C2214yes—
27Q8×C4⋊C4C2212yes—
28C4×C4⋊Q8C2211yescharacteristic
29C4×#127C2212yes—
30#2293C2211yescharacteristic
31#2099C2212yes—
32#2093C2214yes—
33#2093C2214yes—
34#2093C2212yes—
35#2083C2214yes—
36#2071C2214yes—
37#2071C2214yes—
38#2067C2214yes—
39#2054C2214yes—
40#2041C2214yes—
41#2041C2212yes—
42#2041C2214yes—
43#2011C2214yes—
44#2011C2212yes—
45#1984C2214yes—
46#1982C2214yes—
47#1970C2214yes—
48#1957C2214yes—
49#1947C2214yes—
50#1934C2214yes—
51#1925C2214yes—
52#1925C2214yes—
53#1925C2214yes—
54#1925C2214yes—
55#1918C2214yes—
56#1916C2214yes—
57#1916C2214yes—
58#1879C2214yes—
59#1879C2214yes—
60#1862C2214yes—
61#1862C2214yes—
62#1862C2214yes—
63#1839C2212yes—
64C4×#107C2214yes—
65C22×D8C2314yes—
66#555—48no—
67#541—48no—
68C2×C4⋊Q8D411yescharacteristic
69C2×#127C2312yes—
70C2×#127C2314yes—
71C2×#126—24no—
72C2×#125—24no—
73C2×#125—21no—
74C2×#124—24no—
75C2×C22⋊Q8—22no—
76C2×#121—24no—
77C2×#121—24no—
78C2×C4×Q8—22no—
79C2×C4×Q8C2312yes—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4C2314yes—
83C2×C4×D4C2311yescharacteristic
84C2×#117—24no—
85C2×#117—21no—
86C2×#107C2314yes—
87C2×#107C2314yes—
88C2×#107C2314yes—
89C2×C4⋊C8C2314yes—
90C2×#102C2314yes—
91C2×#102C2314yes—
92C2×#102C2314yes—
93C2×#102C2314yes—
94C2×#102C2314yes—
95C2×#102C2314yes—
96C2×#102C2314yes—
97C2×#102C2314yes—
98C2×C4×C8C2314yes—
99#400—24no—
100#398—24no—
101#397—24no—
102#396—24no—
103#396—24no—
104#394—24no—
105#394—24no—
106#392—24no—
107#392—24no—
108#391D412yes—
109#390—24no—
110#390—22no—
111#390—22no—
112#390C2314yes—
113#390C2314yes—
114#390C2314yes—
115#390C2314yes—
116#390C2312yes—
117#389—22no—
118#389—24no—
119#389—22no—
120#389D414yes—
121#389C2314yes—
122#389C2314yes—
123#389C2312yes—
124#388—22no—
125#388—24no—
126#388—24no—
127#388—24no—
128#387—24no—
129#384D411yescharacteristic
130#384C2312yes—
131#384C2312yes—
132#384C2314yes—
133#382—21no—
134#380—24no—
135#380—22no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C4×C4⋊C4C2314yes—
140C4×C22⋊C4—21no—
141C43C2311yescharacteristic
142#340C2314yes—
143#340C2314yes—
144#340C2314yes—
145#340C2314yes—
146#336C2314yes—
147#336C2314yes—
148#336C2314yes—
149#336C2312yes—
150#336C2314yes—
151#336C2312yes—
152C22×Q8C2×D412yes—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4C2414yes—
158C22×D4C2411yescharacteristic
159C23×C4—24no—
160C23×C4—21no—
161C2×D8—44no—
162C2×D8—44no—
163C2×D8—44no—
164C22×C8C2414yes—
165C22×C8C2414yes—
166C4⋊Q8D818yescomplemented
167#127—44no—
168#127—44no—
169#127—44no—
170#126—416no—
171#125—416no—
172#125—44no—
173#124—416no—
174C22⋊Q8—48no—
175#121—416no—
176#121—88no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—48no—
180C4×D4—44no—
181C4×D4—44no—
182C4×D4—42no—
183C4×D4—48no—
184C4×D4—84no—
185#117—416no—
186#117—44no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D414yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2411yescharacteristic
202C2×C4⋊C4C2412yes—
203C2×C4⋊C4C2414yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C42—24no—
217C2×C42—21no—
218C2×C42—24no—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2412yes—
223C2×C42C2411yescharacteristic
224C2×C42C2414yes—
225#107—44no—
226#107—44no—
227#107—44no—
228C4⋊C8—44no—
229#102—44no—
230#102—44no—
231#102—44no—
232#102—44no—
233#102—44no—
234#102—44no—
235#102—44no—
236#102—44no—
237C4×C8—44no—
238#95—22no—
239#95—24no—
240#95—21no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95C2×D414yes—
245#95C2×D414yes—
246C24—21no—
247C24—24no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8C2×D818yes—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—24no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—48no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—84no—
267C2×D4—84no—
268C2×D4—84no—
269C2×D4—84no—
270C2×D4—84no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—42no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—42no—
279C22×C4—48no—
280C22×C4—42no—
281C22×C4C22×D412yes—
282C22×C4C22×D414yes—
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D414yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D412yes—
287C22×C4C2511yescharacteristic
288D8—88no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C4⋊C4—48no—
310C4⋊C4—82no—
311C4⋊C4—84no—
312C4⋊C4—82no—
313C4⋊C4C2×D8116yes—
314C22⋊C4—42no—
315C22⋊C4—48no—
316C22⋊C4—48no—
317C22⋊C4—48no—
318C22⋊C4—44no—
319C22⋊C4—48no—
320C22⋊C4—48no—
321C22⋊C4—82no—
322C22⋊C4—84no—
323C22⋊C4—84no—
324C22⋊C4—84no—
325C22⋊C4—84no—
326C42—24no—
327C42—24no—
328C42—24no—
329C42—22no—
330C42—41no—
331C42—42no—
332C42—48no—
333C42—42no—
334C42—48no—
335C42C2×D814yes—
336C23—42no—
337C23—44no—
338C23—44no—
339C23—41no—
340C23—44no—
341C23—41no—
342C23—44no—
343C23—44no—
344C23—44no—
345C23—42no—
346C23—41no—
347C23—44no—
348C23C23×D411yescharacteristic
349Q8—48no—
350D4—44no—
351D4—44no—
352D4—44no—
353D4—44no—
354D4—44no—
355D4—88no—
356D4—88no—
357D4—168no—
358D4—168no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—21no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—41no—
385C2×C4—816no—
386C2×C4—84no—
387C2×C4C22×D818yes—
388C2×C4C22×D812yes—
389C2×C4C22×D814yes—
390C2×C4#58512yes—
391C2×C4#58514yes—
392C8—44no—
393C8—44no—
394C22—84no—
395C22—81no—
396C22—81no—
397C22—84no—
398C22—84no—
399C22—84no—
400C22—82no—
401C22—84no—
402C22—82no—
403C22—84no—
404C22—81no—
405C22—84no—
406C22#348511yescharacteristic
407C22#348712yes—
408C22C23×D811yescharacteristic
409C22#358512yes—
410C22#358611yescharacteristic
411C4—24no—
412C4—22no—
413C4—28no—
414C4—41no—
415C4—44no—
416C4—42no—
417C4—82no—
418C4—84no—
419C4—84no—
420C4—81no—
421C4#356614yes—
422C2—164no—
423C2—161no—
424C2#5208411yescharacteristic
425C2#6169411yescharacteristic
426C2#6179711yescharacteristic
427C2#6188012yes—
428C2#6193912yes—
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. #39379C2semidirect
3. #32596C2semidirect
4. #32162C2semidirect
5. #32078C2semidirect
6. #30075C2semidirect
7. #29364C2semidirect
8. #27486C2semidirect
9. #27358C2semidirect
10. #26316C2semidirect
11. #23363C2semidirect
166. C4⋊Q8D8semidirect

Maximal subgroups

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

subgroupindex
#598222
#393792
#325962
#321622
#320782
#300752
#293642
#274862
#273582
#263162
#233632