Finite Group Databasebeta

← #386115

Subgroups — #386115

Order 512.

940 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic
2#21334C212yesmaximal
3#21037C212yesmaximal
4#11388C212yesmaximal
5#11364C212yesmaximal
6#11213C212yesmaximal
7#10903C212yesmaximal
8#10814C211yesmaximal, characteristic
9#10803C211yesmaximal, characteristic
10C2×#544C2212yes—
11#3164—24no—
12#3164—24no—
13C2×#501C414yescomplemented
14C2×#500C412yescomplemented
15C2×#499C412yescomplemented
16C2×#428C2212yes—
17C2×#427C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#405C2211yescharacteristic
20#2316C2212yes—
21#2274C2212yes—
22#1996—28no—
23#1921—24no—
24#1918—24no—
25C2×#344C2212yes—
26#1742C2211yescharacteristic
27#1740C2212yes—
28#1740C2212yes—
29#1740C2212yes—
30#1740C2211yescharacteristic
31#1736C2212yes—
32#1736C2212yes—
33#1735C2212yes—
34#1733C2212yes—
35#1733C2212yes—
36C2×H(3,ℤ/4)C2211yescharacteristic
37C2×H(3,ℤ/4)C2211yescharacteristic
38#1729C2212yes—
39C22×M16C2312yes—
40C22×M16C2311yescharacteristic
41C23×C8C2311yescharacteristic
42#544—24no—
43#544—24no—
44#544—42no—
45#544—42no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2×C414yes—
48C2×#125—22no—
49C2×#125C2×C414yes—
50C2×C22⋊Q8—24no—
51C2×#117C2312yes—
52C2×#117C2312yes—
53C2×#117C2312yes—
54C2×#117C2312yes—
55C2×#117C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57#501—216no—
58#500—24no—
59#500—24no—
60#499—24no—
61#499—24no—
62#428—24no—
63#428—24no—
64#427—22no—
65#427—22no—
66C2×#107C2×C411yescharacteristic
67C2×#107C2×C411yescharacteristic
68C2×#107C2×C414yes—
69C2×#107C2×C412yes—
70C2×#107C2×C412yes—
71C2×#107D414yes—
72C2×#107D414yes—
73C2×#106C2×C414yes—
74C2×#106C2×C412yes—
75#405—22no—
76#405—22no—
77C2×#97C2×C412yes—
78C2×C4×C8C2×C412yes—
79#389—24no—
80#389—42no—
81#380—24no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—42no—
84C4×C22⋊C4—24no—
85#344—24no—
86#343—21no—
87#343—21no—
88#342—24no—
89#342—24no—
90#342—24no—
91#342D414yes—
92#340—22no—
93#340—22no—
94#340—24no—
95#340—24no—
96#339—42no—
97#339—42no—
98H(3,ℤ/4)—22no—
99H(3,ℤ/4)Q814yescomplemented
100#336—22no—
101#336—22no—
102C22×Q8—24no—
103C23×C4—22no—
104C23×C4C2411yescharacteristic
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16C2×Q812yes—
114C2×M16#4112yes—
115C22×C8—22no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8C22×C411yescharacteristic
120C22×C8C22×C411yescharacteristic
121C22×C8C22×C412yes—
122C22×C8C2×D412yes—
123C22×C8C2×D412yes—
124C4⋊Q8—28no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128#125—28no—
129#125—44no—
130#125—44no—
131#125—44no—
132C22⋊Q8—48no—
133C22⋊Q8—48no—
134#117—22no—
135#117—22no—
136#117—22no—
137#117—22no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151#117C2×Q814yes—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C414yes—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×D414yes—
167C2×C4⋊C4#4111yescharacteristic
168C2×C4⋊C4#4111yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4C2×D414yes—
173C2×C22⋊C4C2×D412yes—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—44no—
179C2×C42C22×C412yes—
180#107—21no—
181#107—21no—
182#107—21no—
183#107—21no—
184#107—28no—
185#107C22⋊C414yes—
186#107C22⋊C414yes—
187#107C22⋊C418yes—
188#106—24no—
189#106—28no—
190#97—24no—
191C4×C8—22no—
192C4×C8—22no—
193#95—44no—
194#95—44no—
195C24C22×D411yescharacteristic
196C2×Q8—48no—
197C2×Q8—164no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—162no—
207C22×C4C4×D412yes—
208C22×C4C4×D412yes—
209C22×C4C23×C411yescharacteristic
210C22×C4C22×D412yes—
211C22×C4C22×Q811yescharacteristic
212C22×C4C2×#4111yescharacteristic
213C22×C4C2×#4111yescharacteristic
214C22×C4H(5,𝔽2)12yes—
215C22×C42−1+412yes—
216C22×C42−1+411yescharacteristic
217C22×C42−1+411yescharacteristic
218C22×C42−1+412yes—
219M16—24no—
220M16—24no—
221M16—42no—
222M16—42no—
223M16—42no—
224M16—42no—
225C2×C8—22no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8C2×C22⋊C411yescharacteristic
239C2×C8C2×C22⋊C411yescharacteristic
240C2×C8C2×C22⋊C412yes—
241C2×C8C4×Q814yes—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4—48no—
253C4⋊C4—48no—
254C4⋊C4—48no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—164no—
261C4⋊C4C2×C22⋊C418yes—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—22no—
272C42—22no—
273C42—24no—
274C42—42no—
275C42—42no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—42no—
280C42—42no—
281C42—164no—
282C23—41no—
283C23—41no—
284C23C2×C4×D411yescharacteristic
285C23#53612yes—
286C23D4212yes—
287C23#54711yescharacteristic
288C23D4×Q811yescharacteristic
289Q8—168no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—28no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4—84no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4C22×C22⋊C411yescharacteristic
312C2×C4C2×C4×Q811yescharacteristic
313C2×C4#51811yescharacteristic
314C2×C4#51911yescharacteristic
315C2×C4#51911yescharacteristic
316C2×C4#51911yescharacteristic
317C2×C4#53612yes—
318C2×C4#53612yes—
319C2×C4#54112yes—
320C2×C4#54212yes—
321C2×C4#54412yes—
322C2×C4#54612yes—
323C2×C4#54812yes—
324C2×C4D4×Q812yes—
325C2×C4#55611yescharacteristic
326C2×C4#55711yescharacteristic
327C2×C4#56312yes—
328C2×C4#56312yes—
329C8—21no—
330C8—21no—
331C8—22no—
332C8—24no—
333C8—82no—
334C22—21no—
335C22—22no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22#232711yescharacteristic
342C22D4×C22⋊C411yescharacteristic
343C22#236511yescharacteristic
344C22#267412yes—
345C22#267712yes—
346C4—42no—
347C4—48no—
348C4—42no—
349C4—84no—
350C4—84no—
351C4—82no—
352C4—82no—
353C4—162no—
354C4#231011yescharacteristic
355C4#233111yescharacteristic
356C4Q8×C22⋊C411yescharacteristic
357C4#238111yescharacteristic
358C4#271312yes—
359C4#271612yes—
360C2—22no—
361C2—22no—
362C2—21no—
363C2—21no—
364C2#1442011yescharacteristic
365C2#2245511yescharacteristic
366C2#2245711yescharacteristic
subgroupG/H
center, socleC22D4×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. C2×#501C4semidirect
14. C2×#500C4semidirect
15. C2×#499C4semidirect
99. H(3,ℤ/4)Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
#213342
#210372
#113882
#113642
#112132
#109032
#108142
#108032