Finite Group Databasebeta

← #201792

Subgroups — #201792

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60284C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#21799C212yesmaximal, complemented
4#17366C211yesmaximal, characteristic
5#9506C218yesmaximal, complemented
6#8105C212yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
8C2×C4×#41C2211yescharacteristic
9#2991—24no—
10#2982—21no—
11#2975—21no—
12#2975C2214yes—
13#2971—24no—
14#2920—22no—
15#2920C2212yes—
16#2920C2212yes—
17#2857—22no—
18#2628—22no—
19#2568—21no—
20#2567—21no—
21#2563—21no—
22#2542—21no—
23#2487C414yes—
24#2409C2214yes—
25#2317C412yes—
26#2311C412yes—
27#2285C2211yescharacteristic
28#1966—24no—
29#1966—24no—
30#1784C2218yes—
31#1465C2214yes—
32#1464—22no—
33#1461—21no—
34#1460—21no—
35#1305C2218yes—
36C22×#41C2×C411yescharacteristic
37C23×Q8C2312yes—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C22⋊Q8—24no—
47C4×#41—22no—
48C4×#41—22no—
49C4×#41—22no—
50C4×#41—22no—
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×D4C2×C412yes—
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4C2312yes—
57C2×#117C2×C412yes—
58C2×#117C2×C411yescharacteristic
59C2×#117D412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C42D412yes—
63C22×C42C2311yescharacteristic
64C8×Q8—28no—
65#436—28no—
66#436—216no—
67#432—24no—
68#432—22no—
69#432—22no—
70#432—22no—
71#432—22no—
72#431—24no—
73#431—24no—
74#431—22no—
75#431—22no—
76#431—22no—
77#431—22no—
78#423—24no—
79#423—22no—
80#423D414yes—
81C2×#101—216no—
82#407—24no—
83#407C2314yes—
84#407C2314yes—
85C4×M16—24no—
86C4×M16—22no—
87C4×M16D414yes—
88#400—24no—
89#399—24no—
90#395—24no—
91#393—24no—
92#391—22no—
93#391—24no—
94#390—22no—
95#389—24no—
96#389—24no—
97#386C2×C418yes—
98#385C2×C418yes—
99#384—22no—
100#384—21no—
101#382—22no—
102#382—21no—
103#380—24no—
104#380—24no—
105#380C2×C412yes—
106#380C2312yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109#324—24no—
110#323—24no—
111C2×#41—24no—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—42no—
115C22×Q8—22no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8—84no—
120C22×Q8C22⋊C411yescharacteristic
121C22×D4C22⋊C411yescharacteristic
122C22×D4C22×C412yes—
123C23×C4C22⋊C412yes—
124C23×C4C22⋊C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C414yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2411yescharacteristic
130C2×M16—28no—
131C2×M16—24no—
132C2×M16C2×D418yes—
133C4⋊Q8—42no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—42no—
143C22⋊Q8—88no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156#117—22no—
157#117—22no—
158#117—24no—
159#117—24no—
160#117—44no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C22⋊C4—28no—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C22×C412yes—
180C2×C22⋊C4C22×C414yes—
181C2×C22⋊C4C2×D412yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42#4112yes—
197C2×C42#4112yes—
198C4⋊C8—28no—
199C4⋊C8—24no—
200C4⋊C8—24no—
201C4⋊C8—42no—
202C4⋊C8—42no—
203#101—816no—
204C22⋊C8—28no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208C22⋊C8—24no—
209#97—24no—
210#97—42no—
211C4×C8—28no—
212C4×C8—24no—
213C4×C8—42no—
214#95—22no—
215#95—28no—
216#95—28no—
217#95—22no—
218C24C2×C22⋊C412yes—
219C24C23×C411yescharacteristic
220#41—48no—
221#41—48no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—416no—
225C2×Q8—48no—
226C2×Q8—42no—
227C2×Q8—44no—
228C2×Q8—42no—
229C2×Q8—84no—
230C2×Q8—88no—
231C2×Q8#10114yescomplemented
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—22no—
237C2×D4—42no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—84no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C4×D412yes—
264C22×C4C4×D412yes—
265C22×C4C4×D412yes—
266C22×C4C4×D412yes—
267C22×C4C22×D412yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C2×#4112yes—
271M16—48no—
272M16—48no—
273M16—88no—
274C2×C8—28no—
275C2×C8—44no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—88no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—44no—
294C22⋊C4—88no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—44no—
302C42#12012yes—
303C42#12011yescharacteristic
304C42#12011yescharacteristic
305C42#12112yes—
306C42#12112yes—
307C42#12112yes—
308C42#12112yes—
309C42#12311yescharacteristic
310C42#12311yescharacteristic
311C42#12312yes—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—42no—
320C23—42no—
321C23C22×C22⋊C411yescharacteristic
322C23C2×C4×D412yes—
323Q8—28no—
324Q8—88no—
325Q8—84no—
326Q8—84no—
327D4—48no—
328D4—44no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—48no—
353C2×C4—44no—
354C2×C4—416no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4#38611yescharacteristic
360C2×C4#38611yescharacteristic
361C2×C4#38611yescharacteristic
362C2×C4#38612yes—
363C2×C4#38612yes—
364C2×C4#38611yescharacteristic
365C2×C4C2×#10114yes—
366C2×C4C2×#10112yes—
367C2×C4#41312yes—
368C2×C4C2×#12011yescharacteristic
369C2×C4C2×#12112yes—
370C2×C4C2×#12311yescharacteristic
371C8—48no—
372C8—84no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—42no—
379C22—42no—
380C22—44no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—82no—
385C22C2×#38611yescharacteristic
386C22C22×#10111yescharacteristic
387C22C2×#41311yescharacteristic
388C4—21no—
389C4—22no—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—24no—
395C4—42no—
396C4—42no—
397C4—84no—
398C4—84no—
399C4#190114yes—
400C4#191512yes—
401C4#302712yes—
402C2—21no—
403C2—21no—
404C2—42no—
405C2—82no—
406C2#2096111yescharacteristic
407C2C2×#190111yescharacteristic
408C2C2×#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. #21799C2semidirect
5. #9506C2semidirect
6. #8105C2semidirect
231. C2×Q8#101semidirect

Maximal subgroups

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

subgroupindex
#602842
#352792
#217992
#173662
#95062
#81052