Finite Group Databasebeta

← #6434349

Subgroups — #6434349

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic, complemented
2#34070C211yesmaximal, characteristic
3#30090C214yesmaximal, complemented
4#30089C214yesmaximal, complemented
5#30086C214yesmaximal, complemented
6#30084C214yesmaximal, complemented
7#23734C212yesmaximal, complemented
8#23657C212yesmaximal, complemented
9#23631C212yesmaximal, complemented
10#23508C212yesmaximal
11#20875C214yesmaximal, complemented
12#20773C211yesmaximal, characteristic, complemented
13C2×#532C2211yescharacteristic
14C2×C4×#41C2212yes—
15C2×#428C2214yes—
16C2×#417C2214yes—
17C2×#417C2214yes—
18#2647—24no—
19#2647C2214yes—
20#2647C2214yes—
21C2×#400—22no—
22C2×#398—22no—
23#2328C2212yes—
24#2327C2212yes—
25#2326C2212yes—
26#2326C2212yes—
27#2325—22no—
28#2325—22no—
29#2297C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32#2290—24no—
33#2285C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#376C2211yescharacteristic
37#2096C2214yes—
38#2095C2214yes—
39#2094C2214yes—
40#2093C2214yes—
41#2085C2218yes—
42#2084C2214yes—
43#2083C2214yes—
44#2074C2214yes—
45#2074C2214yes—
46#2073C2214yes—
47#2073C2214yes—
48#2043C2214yes—
49#2042C2218yes—
50#2041C2214yes—
51#1865C2218yes—
52#1864C2218yes—
53#1863C2218yes—
54#1862C2218yes—
55#1840C2214yes—
56#1840C2214yes—
57#1736C2214yes—
58#1728C2212yes—
59#1728C2212yes—
60#1728C2212yes—
61C22×#41C2312yes—
62C22×M16C2314yes—
63#532—28no—
64C2×C4⋊Q8C2311yescharacteristic
65C2×#127C2311yescharacteristic
66C2×#124C2312yes—
67C2×#121—22no—
68C4×#41—48no—
69C2×C4×Q8C2312yes—
70C2×C4×Q8C2312yes—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4C2312yes—
75C2×C4×D4C2312yes—
76C2×#117—22no—
77C2×#117C2312yes—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C42—22no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#428—48no—
88C2×#107C2318yes—
89C2×#106C2318yes—
90#417—216no—
91#417—48no—
92C2×#103C2318yes—
93C2×#103C2318yes—
94C2×#102C2318yes—
95C2×#102C2318yes—
96#400—28no—
97#400—28no—
98#398—28no—
99#398—28no—
100#391C2312yes—
101#391C2312yes—
102#390C2312yes—
103#390C2312yes—
104#389—22no—
105#389—22no—
106#389—24no—
107#389C2314yes—
108#389C2314yes—
109#389C2318yes—
110#388—24no—
111#388—24no—
112#387—24no—
113#387—24no—
114#386—24no—
115#386—24no—
116#386—24no—
117#385—24no—
118#385—24no—
119#384—22no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2314yes—
123#382—22no—
124#382C2312yes—
125#382C2314yes—
126#380—22no—
127C4×C4⋊C4—22no—
128C4×C22⋊C4—24no—
129#376D414yes—
130#376C2312yes—
131#376C2312yes—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95D412yes—
136C2×#95D412yes—
137#340C2318yes—
138#340C2318yes—
139#336C2314yes—
140#336C2314yes—
141#336C2314yes—
142#336C2314yes—
143#336C2318yes—
144C2×#41—24no—
145C2×#41—28no—
146C2×#41—28no—
147C2×#41#4118yes—
148C22×Q8C2412yes—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4C2412yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C2×M16—28no—
163C2×M16—28no—
164C2×M16—28no—
165C22×C8C2418yes—
166C4⋊Q8—24no—
167#127—24no—
168#124—24no—
169#124—24no—
170#121—216no—
171C4×Q8—44no—
172C4×Q8—44no—
173C4×D4—216no—
174C4×D4—44no—
175C4×D4—44no—
176C4×D4—48no—
177C4×D4—48no—
178#117—28no—
179#117—48no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—84no—
195C2×C4⋊C4—84no—
196C2×C4⋊C4—84no—
197C2×C4⋊C4C2414yes—
198C2×C4⋊C4C2414yes—
199C2×C4⋊C4C2412yes—
200C2×C4⋊C4C2412yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—48no—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2412yes—
218C2×C42C2411yescharacteristic
219C2×C42C2411yescharacteristic
220C2×C42C2412yes—
221#107—48no—
222#106—48no—
223#103—216no—
224#103—48no—
225#102—216no—
226#102—48no—
227#95—28no—
228#95—24no—
229#95—24no—
230#95—28no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95C2×D414yes—
235#95C2×D414yes—
236#95C2×D414yes—
237#95C2×D414yes—
238C24—22no—
239C24C22×D411yescharacteristic
240#41—232no—
241#41—416no—
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8C2×#4114yes—
246C2×D4—28no—
247C2×D4—28no—
248C2×D4—22no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—82no—
254C2×D4—82no—
255C2×D4C2×#4114yes—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—28no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4C22×D412yes—
285C22×C4C22×D414yes—
286C22×C4C22×D412yes—
287C22×C4C22×D412yes—
288C22×C4C22×D412yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C2×#4114yes—
291C22×C4H(5,𝔽2)14yes—
292C22×C42−1+414yes—
293C22×C4C2511yescharacteristic
294M16—416no—
295C2×C8—28no—
296C2×C8—28no—
297C2×C8—28no—
298C4⋊C4—28no—
299C4⋊C4—28no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—88no—
311C4⋊C4—84no—
312C4⋊C4—88no—
313C4⋊C4—84no—
314C22⋊C4—216no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—84no—
319C22⋊C4—84no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—22no—
324C42—22no—
325C42—24no—
326C42—24no—
327C42—44no—
328C42—44no—
329C42—44no—
330C23—22no—
331C23—22no—
332C23—24no—
333C23—44no—
334C23—44no—
335C23—42no—
336C23—42no—
337C23#52512yes—
338C23#53512yes—
339C23#53612yes—
340C23C23×D411yescharacteristic
341Q8—28no—
342Q8—44no—
343D4—28no—
344D4—416no—
345D4—44no—
346D4—88no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—48no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—416no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4—88no—
379C2×C4—88no—
380C2×C4—88no—
381C2×C4—88no—
382C2×C4#52512yes—
383C2×C4#53512yes—
384C2×C4#53612yes—
385C2×C4#539116yes—
386C2×C4C22×#4112yes—
387C2×C4C2×H(5,𝔽2)12yes—
388C2×C4C2×2−1+412yes—
389C8—48no—
390C22—22no—
391C22—22no—
392C22—24no—
393C22—24no—
394C22—44no—
395C22—48no—
396C22—82no—
397C22—82no—
398C22—84no—
399C22#263314yes—
400C22C2×#52511yescharacteristic
401C22C2×#53511yescharacteristic
402C22C2×#53611yescharacteristic
403C22#358512yes—
404C22#358612yes—
405C4—44no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—44no—
411C4—82no—
412C4—84no—
413C4—82no—
414C4—84no—
415C4#263314yes—
416C4C2×#53914yes—
417C2—24no—
418C2—84no—
419C2C2×#263311yescharacteristic
420C2#6056712yes—
421C2#6080811yescharacteristic
422C2#6080911yescharacteristic
423C2#6086712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2647C2semidirect
3. #30090C2semidirect
4. #30089C2semidirect
5. #30086C2semidirect
6. #30084C2semidirect
7. #23734C2semidirect
8. #23657C2semidirect
9. #23631C2semidirect
11. #20875C2semidirect
12. #20773C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
#340702
#300902
#300892
#300862
#300842
#237342
#236572
#236312
#235082
#208752
#207732