Finite Group Databasebeta

← #201794

Subgroups — #201794

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60284C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic
3#21829C212yesmaximal, complemented
4#17366C211yesmaximal, characteristic
5#9508C218yesmaximal, complemented
6#8105C212yesmaximal, complemented
7C2×#536C412yes—
8C22×C4⋊Q8C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2991—24no—
11#2982—21no—
12#2975—21no—
13#2975C2214yes—
14#2971—24no—
15#2920—22no—
16#2920C2212yes—
17#2920C2212yes—
18#2857—22no—
19#2621—22no—
20#2547—21no—
21#2547—21no—
22#2546—21no—
23#2546—21no—
24#2486C414yes—
25#2486C2214yes—
26#2310C412yes—
27#2285C2211yescharacteristic
28#1997—24no—
29#1997—24no—
30#1821C2218yes—
31#1465C2214yes—
32#1464—22no—
33#1461—21no—
34#1460—21no—
35#1305C2218yes—
36C22×#41C2×C411yescharacteristic
37C23×Q8C2×C412yes—
38#536—48no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×#124—22no—
49C2×#123C2×C418yes—
50C2×C22⋊Q8—24no—
51C4×#41—22no—
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×D4C2×C412yes—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4C2312yes—
61C2×#117C2×C412yes—
62C2×#117C2×C411yescharacteristic
63C2×#117D412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2312yes—
66C22×C42D412yes—
67C22×C42C2311yescharacteristic
68C8×Q8—28no—
69#436—28no—
70#436—216no—
71#432—24no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432—22no—
76#431—24no—
77#431—24no—
78#431—22no—
79#431—22no—
80#431—22no—
81#431—22no—
82#423—24no—
83#423—22no—
84#423D414yes—
85#407—24no—
86#407C2314yes—
87#407C2314yes—
88C4×M16—24no—
89C4×M16—22no—
90C4×M16D414yes—
91#398—24no—
92#398—24no—
93#397—24no—
94#397—24no—
95#391—22no—
96#391—22no—
97#391—24no—
98#389—24no—
99#389—24no—
100#384—21no—
101#382—21no—
102#381C2×C418yes—
103#380—24no—
104#380—24no—
105#380C2×C412yes—
106#380C2312yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109#343—216no—
110#324—24no—
111#323—24no—
112C2×#41—24no—
113C2×#41—22no—
114C2×#41—22no—
115C2×#41—42no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8—44no—
121C22×Q8—44no—
122C22×Q8C22⋊C411yescharacteristic
123C22×D4C22⋊C411yescharacteristic
124C22×D4C22×C412yes—
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C414yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2411yescharacteristic
132C2×M16—28no—
133C2×M16—24no—
134C2×M16C2×D418yes—
135C4⋊Q8—48no—
136C4⋊Q8—42no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4⋊Q8—42no—
140C4⋊Q8—42no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C4⋊Q8—42no—
146#124—48no—
147#123—416no—
148C22⋊Q8—416no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161#117—22no—
162#117—22no—
163#117—24no—
164#117—24no—
165#117—44no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C22⋊C4—28no—
183C2×C22⋊C4C22⋊C412yes—
184C2×C22⋊C4C22×C412yes—
185C2×C22⋊C4C22×C414yes—
186C2×C22⋊C4C2×D412yes—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C22⋊C412yes—
193C2×C42C22⋊C412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42#4112yes—
202C2×C42#4112yes—
203C4⋊C8—28no—
204C4⋊C8—24no—
205C4⋊C8—24no—
206C4⋊C8—42no—
207C4⋊C8—42no—
208C22⋊C8—28no—
209C22⋊C8—24no—
210C22⋊C8—24no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213#97—24no—
214#97—42no—
215C4×C8—28no—
216C4×C8—24no—
217C4×C8—42no—
218#95—22no—
219#95—28no—
220#95—28no—
221#95—22no—
222C24C2×C22⋊C412yes—
223C24C23×C411yescharacteristic
224#41—48no—
225#41—48no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—44no—
229C2×Q8—44no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—88no—
236C2×Q8—88no—
237C2×Q8#10014yes—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—42no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C4×D412yes—
271C22×C4C4×D412yes—
272C22×C4C4×D412yes—
273C22×C4C4×D412yes—
274C22×C4C22×D412yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4112yes—
278M16—48no—
279M16—48no—
280M16—88no—
281C2×C8—28no—
282C2×C8—44no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—416no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—416no—
301C22⋊C4—44no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—44no—
309C42#12012yes—
310C42#12011yescharacteristic
311C42#12011yescharacteristic
312C42#12112yes—
313C42#12112yes—
314C42#12112yes—
315C42#12112yes—
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C42#12312yes—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23C22×C22⋊C411yescharacteristic
329C23C2×C4×D412yes—
330Q8—28no—
331Q8—84no—
332Q8—84no—
333Q8—88no—
334D4—48no—
335D4—44no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—88no—
367C2×C4#38611yescharacteristic
368C2×C4#38611yescharacteristic
369C2×C4#38611yescharacteristic
370C2×C4#38612yes—
371C2×C4#38612yes—
372C2×C4#38611yescharacteristic
373C2×C4C2×#10014yes—
374C2×C4C2×#10012yes—
375C2×C4#41312yes—
376C2×C4C2×#12011yescharacteristic
377C2×C4C2×#12112yes—
378C2×C4C2×#12311yescharacteristic
379C8—48no—
380C8—84no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—42no—
387C22—42no—
388C22—44no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—82no—
393C22C2×#38611yescharacteristic
394C22C22×#10011yescharacteristic
395C22C2×#41311yescharacteristic
396C4—21no—
397C4—22no—
398C4—22no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—24no—
403C4—42no—
404C4—42no—
405C4—84no—
406C4—84no—
407C4#190014yes—
408C4#191512yes—
409C4#302812yes—
410C2—21no—
411C2—21no—
412C2—42no—
413C2—82no—
414C2#2096211yescharacteristic
415C2C2×#190011yescharacteristic
416C2C2×#191511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21829C2semidirect
5. #9508C2semidirect
6. #8105C2semidirect

Maximal subgroups

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

subgroupindex
#602842
#353162
#218292
#173662
#95082
#81052