Finite Group Databasebeta

← #6594407

Subgroups — #6594407

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36189C212yesmaximal
2#33311C211yesmaximal, characteristic, complemented
3#33210C212yesmaximal, complemented
4#29207C212yesmaximal, complemented
5#28834C214yesmaximal, complemented
6#27949C212yesmaximal, complemented
7#27948C212yesmaximal, complemented
8#26075C218yesmaximal, complemented
9#22666C214yesmaximal
10C4×#498C211yesmaximal, characteristic, complemented
11#20800C212yesmaximal, complemented
12C4×#436C211yesmaximal, characteristic
13C2×#499C2214yes—
14C2×#498C2211yescharacteristic
15C2×#436C2212yes—
16C2×#428C2218yes—
17C2×C4×M16C2212yes—
18#2859C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2570C2214yes—
22#2511—22no—
23#2507—22no—
24#2497—22no—
25#2487—22no—
26#2442C2214yes—
27#2441C2214yes—
28#2423—22no—
29#2417—22no—
30#2360—22no—
31#2352C2214yes—
32#2345—22no—
33C4×C4⋊Q8C2212yes—
34#2285C2212yes—
35C2×#384C2212yes—
36C42×D4C2211yescharacteristic
37#2085C2218yes—
38#2084C2218yes—
39#2059C2218yes—
40#1990C2212yes—
41#1990C2212yes—
42#1990C2212yes—
43#1949C2218yes—
44#1943C2218yes—
45#1941C2214yes—
46#1854C2218yes—
47#1853C2218yes—
48#1847C2218yes—
49#1846C2214yes—
50#1845C2212yes—
51#1845C2212yes—
52#1844C2212yes—
53#1844C2211yescharacteristic
54#1844C2211yescharacteristic
55#1844C2211yescharacteristic
56#1809C2218yes—
57C4×#106C2214yes—
58#1770C2212yes—
59C4×C4⋊C8C2211yescharacteristic
60#1753C2214yes—
61C4×C22⋊C8C2212yes—
62C42×C8C2211yescharacteristic
63C22×M16C2314yes—
64C2×C4⋊Q8C2312yes—
65C2×#125C2314yes—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×D4D412yes—
70C2×C4×D4C2311yescharacteristic
71C2×#117C2314yes—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2314yes—
75C22×C42C2312yes—
76#499—48no—
77#498—24no—
78#436Q8116yescomplemented
79#428—416no—
80C2×#107C2318yes—
81C2×#107C2318yes—
82C2×#106C2314yes—
83C2×#106C2314yes—
84C2×#106C2318yes—
85C2×#106C2314yes—
86C2×#106C2314yes—
87C2×C4⋊C8C2312yes—
88C2×C22⋊C8C2314yes—
89C4×M16—28no—
90C2×#97C2314yes—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2312yes—
93C2×C4×C8C2312yes—
94C2×C4×C8C2312yes—
95C2×C4×C8C2311yescharacteristic
96#400—24no—
97#400—24no—
98#399—24no—
99#398—24no—
100#398—24no—
101#396—24no—
102#391—22no—
103#391—22no—
104#391—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—24no—
113#389C2314yes—
114#389C2314yes—
115#389C2314yes—
116#388—24no—
117#387—24no—
118#387—24no—
119#387—24no—
120#387—24no—
121#386—24no—
122#385—24no—
123#384—22no—
124#384—22no—
125#384C2312yes—
126#384C2314yes—
127#384C2312yes—
128#384C2312yes—
129#382C2314yes—
130#382C2314yes—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380—24no—
136#380C2314yes—
137#380C2314yes—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4D412yes—
142C4×C4⋊C4C2314yes—
143C4×C4⋊C4C2311yescharacteristic
144C4×C22⋊C4—24no—
145C4×C22⋊C4D412yes—
146C4×C22⋊C4D412yes—
147C4×C22⋊C4C2312yes—
148C43C2311yescharacteristic
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#336C2312yes—
153#336C2314yes—
154C22×Q8—22no—
155C22×D4C2×D411yescharacteristic
156C23×C4—24no—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2412yes—
160C2×M16—28no—
161C2×M16—28no—
162C2×M16C2×Q8116yes—
163C22×C8C2414yes—
164C22×C8C2412yes—
165C22×C8C2412yes—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168#125—48no—
169C22⋊Q8—48no—
170C22⋊Q8—84no—
171C4×Q8—84no—
172C4×D4—44no—
173C4×D4C2×Q818yes—
174#117—48no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—48no—
192C2×C4⋊C4—84no—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2414yes—
196C2×C4⋊C4C2414yes—
197C2×C4⋊C4C2414yes—
198C2×C4⋊C4C2414yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4C2×D412yes—
203C2×C22⋊C4C2×D412yes—
204C2×C22⋊C4C2×D412yes—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2411yescharacteristic
214C2×C42C2412yes—
215C2×C42C2411yescharacteristic
216C2×C42C2411yescharacteristic
217#107—216no—
218#107—48no—
219#106—28no—
220#106—28no—
221#106—48no—
222#106—44no—
223#106—44no—
224C4⋊C8C2×Q818yes—
225C22⋊C8C2×Q8116yes—
226#97—28no—
227C4×C8—24no—
228C4×C8—24no—
229C4×C8—22no—
230C4×C8C2×Q818yes—
231C4×C8#4114yes—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95C2×D411yescharacteristic
243#95C2×D411yescharacteristic
244#95C2×D412yes—
245#95C2×D412yes—
246C24C22×D412yes—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×D4—42no—
251C2×D4C4⋊Q814yes—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—48no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4C22×D412yes—
269C22×C4C22×D412yes—
270C22×C4C22×D412yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D412yes—
274C22×C4C22×D412yes—
275C22×C4C22×Q818yes—
276C22×C4C2511yescharacteristic
277M16—232no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—22no—
282C2×C8C22×Q814yes—
283C2×C8C22×Q818yes—
284C2×C8C22×Q814yes—
285C2×C8C2×#4114yes—
286C2×C8H(5,𝔽2)14yes—
287C4⋊C4—24no—
288C4⋊C4—28no—
289C4⋊C4—28no—
290C4⋊C4—28no—
291C4⋊C4—24no—
292C4⋊C4—42no—
293C4⋊C4—48no—
294C4⋊C4—48no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C4⋊C4C4⋊Q814yes—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—48no—
312C22⋊C4—84no—
313C22⋊C4—84no—
314C22⋊C4C4⋊Q818yes—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—84no—
323C42—84no—
324C42C22×Q814yes—
325C42C2×#4112yes—
326C422−1+412yes—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23C2×C4⋊Q814yes—
334C23C23×D411yescharacteristic
335Q8—88no—
336D4—44no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—48no—
353C2×C4—48no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—168no—
367C2×C4C2×C4⋊Q812yes—
368C2×C4C2×C4⋊Q812yes—
369C2×C4C2×C4⋊Q814yes—
370C2×C4#54714yes—
371C2×C4#57514yes—
372C2×C4#57714yes—
373C2×C4C23×Q812yes—
374C2×C4C22×#4111yescharacteristic
375C2×C4C2×H(5,𝔽2)12yes—
376C2×C4C2×2−1+411yescharacteristic
377C8—28no—
378C8—24no—
379C8#55418yes—
380C22—24no—
381C22—24no—
382C22—24no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22C22×C4⋊Q811yescharacteristic
388C22C2×#54711yescharacteristic
389C22#349811yescharacteristic
390C22C2×#57512yes—
391C22C2×#57712yes—
392C4—24no—
393C4—22no—
394C4—22no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—84no—
399C4—84no—
400C4—84no—
401C4—84no—
402C4#335118yes—
403C4C2×#55412yes—
404C4#350312yes—
405C2—44no—
406C2#3618011yescharacteristic
407C2#6066311yescharacteristic
408C2#6066511yescharacteristic
409C2C2×#335112yes—
410C2#6242712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33311C2semidirect
3. #33210C2semidirect
4. #29207C2semidirect
5. #28834C2semidirect
6. #27949C2semidirect
7. #27948C2semidirect
8. #26075C2semidirect
10. C4×#498C2semidirect
11. #20800C2semidirect
78. #436Q8semidirect

Maximal subgroups

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

subgroupindex
#361892
#333112
#332102
#292072
#288342
#279492
#279482
#260752
#226662
C4×#4982
#208002
C4×#4362