Finite Group Databasebeta

← #368938

Subgroups — #368938

Order 512.

1178 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3166C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic
3#21332C211yesmaximal, characteristic
4#21325C211yesmaximal, characteristic
5#21306C211yesmaximal, characteristic
6#11213C214yesmaximal
7#10903C214yesmaximal
8#10837C211yesmaximal, characteristic
9C4×#344C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×#125C2211yescharacteristic
12#3166—22no—
13#3166—22no—
14#3166—22no—
15#3166—22no—
16C2×#501C414yescomplemented
17C2×#499C414yescomplemented
18C2×#428C2214yes—
19C2×C4×M16C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2370—24no—
22#2345—24no—
23#2330—22no—
24#2316C2212yes—
25C4×#125—22no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2274C2212yes—
29C4×#117C2211yescharacteristic
30#1996—24no—
31#1992—22no—
32#1991—22no—
33#1986—24no—
34C2×#344C2212yes—
35C2×#344C2212yes—
36#1740C2212yes—
37#1740C2212yes—
38#1736C2214yes—
39#1733C2214yes—
40C2×H(3,ℤ/4)C2214yes—
41C22×M16C2312yes—
42C22×M16C2312yes—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—22no—
48#533—22no—
49C2×C4⋊Q8C2×C412yes—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125C2×C412yes—
54C2×#125C2×C412yes—
55C2×#125C2×C412yes—
56C2×#125D412yes—
57C2×#125D412yes—
58C2×#125D412yes—
59C2×#125D412yes—
60C2×C22⋊Q8—22no—
61C2×C4×Q8—22no—
62C2×#117C2312yes—
63C2×#117C2312yes—
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C2×#117C2312yes—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2312yes—
69C22×C42C2311yescharacteristic
70#501—28no—
71#501—28no—
72#499—28no—
73#499—28no—
74#428—28no—
75#428—28no—
76C2×#107C2×C418yes—
77C2×#106C2×C418yes—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—22no—
83C4×M16—22no—
84C2×#97—21no—
85C2×#97C2×C412yes—
86C2×C4×C8—21no—
87C2×C4×C8C2×C412yes—
88#389—28no—
89#389—24no—
90#389—24no—
91#387—28no—
92#387—28no—
93#384—21no—
94#384—21no—
95#382—21no—
96#382—21no—
97#380—22no—
98#380—24no—
99#380—22no—
100C4×C4⋊C4—28no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4D414yes—
106C4×C22⋊C4—24no—
107C4×C22⋊C4D414yes—
108#376—21no—
109C43—21no—
110#344—24no—
111#344Q818yescomplemented
112#342—24no—
113#342—24no—
114#340—28no—
115#339—28no—
116H(3,ℤ/4)—28no—
117C22×Q8—22no—
118C23×C4—22no—
119C23×C4C2×D412yes—
120C23×C4C2411yescharacteristic
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16C2×Q814yes—
131C2×M16C2×Q814yes—
132C2×M16#4114yes—
133C22×C8—22no—
134C22×C8C22×C414yes—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139#125—22no—
140#125—22no—
141#125—24no—
142#125—24no—
143#125—24no—
144#125—24no—
145#125—24no—
146#125—24no—
147#125—24no—
148#125C22⋊C414yes—
149#125C22⋊C414yes—
150#125C22⋊C414yes—
151#125C22⋊C414yes—
152C22⋊Q8—42no—
153C22⋊Q8—42no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—42no—
160#117—22no—
161#117—24no—
162#117—24no—
163#117—22no—
164#117—22no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—24no—
170#117—24no—
171#117C2×Q814yes—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4C22×C414yes—
191C2×C4⋊C4C22×C414yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2×D414yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—28no—
204C2×C42—24no—
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D414yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42#4112yes—
214C2×C42#4112yes—
215#107—216no—
216#106—216no—
217#97—22no—
218#97—22no—
219#97—42no—
220C4×C8—22no—
221C4×C8—22no—
222C4×C8—42no—
223#95—28no—
224#95—28no—
225#95—24no—
226#95—24no—
227#95—42no—
228C24C22×D411yescharacteristic
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—84no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—82no—
241C22×C4—84no—
242C22×C4C4×D412yes—
243C22×C4C4×D412yes—
244C22×C4#12012yes—
245C22×C4#12012yes—
246C22×C4#12112yes—
247C22×C4#12112yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×Q812yes—
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4112yes—
255C22×C4C2×#4111yescharacteristic
256C22×C42−1+412yes—
257C22×C42−1+412yes—
258M16—24no—
259M16—24no—
260M16—24no—
261M16—24no—
262M16—24no—
263M16—24no—
264M16—42no—
265M16—42no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8C4×Q818yes—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—84no—
290C4⋊C4—88no—
291C4⋊C4—84no—
292C4⋊C4C2×C22⋊C418yes—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4C4⋊Q814yes—
303C42—24no—
304C42—28no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—44no—
309C42—84no—
310C42—88no—
311C42C2×C22⋊C411yescharacteristic
312C42C2×C22⋊C411yescharacteristic
313C42C2×C22⋊C411yescharacteristic
314C42C2×C22⋊C411yescharacteristic
315C42C4×D412yes—
316C42C4×D412yes—
317C42#12112yes—
318C42#12112yes—
319C42#12312yes—
320C42#12312yes—
321C23—42no—
322C23C2×C4×D411yescharacteristic
323C23C2×#12011yescharacteristic
324C23C2×#12111yescharacteristic
325C23C2×C4⋊Q811yescharacteristic
326C23#53211yescharacteristic
327C23#53611yescharacteristic
328C23#53611yescharacteristic
329Q8—84no—
330Q8—84no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—82no—
352C2×C4#38612yes—
353C2×C4#38612yes—
354C2×C4#38612yes—
355C2×C4#38612yes—
356C2×C4C22×C22⋊C411yescharacteristic
357C2×C4C2×C4×D411yescharacteristic
358C2×C4C2×C4×Q812yes—
359C2×C4#51912yes—
360C2×C4C2×#12111yescharacteristic
361C2×C4C2×#12311yescharacteristic
362C2×C4#53312yes—
363C2×C4#54112yes—
364C2×C4#54712yes—
365C2×C4#54812yes—
366C2×C4#54812yes—
367C2×C4D4×Q812yes—
368C2×C4D4×Q812yes—
369C2×C4D4×Q812yes—
370C2×C4#55612yes—
371C2×C4#55712yes—
372C8—28no—
373C8—44no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22C2×#38611yescharacteristic
382C22C4×C4⋊Q811yescharacteristic
383C22#232711yescharacteristic
384C22#243211yescharacteristic
385C22#243411yescharacteristic
386C22#254511yescharacteristic
387C22#254611yescharacteristic
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—44no—
393C4—48no—
394C4—84no—
395C4—82no—
396C4—84no—
397C4—162no—
398C4Q8×C22⋊C412yes—
399C4#236512yes—
400C4#242912yes—
401C4#244612yes—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2#1402411yescharacteristic
409C2#2099812yes—
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. C2×#501C4semidirect
17. C2×#499C4semidirect
111. #344Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31662
#342172
#213322
#213252
#213062
#112132
#109032
#108372
C4×#3442