Finite Group Databasebeta

← #201795

Subgroups — #201795

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60284C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#21829C212yesmaximal, complemented
4#17366C211yesmaximal, characteristic
5#9510C218yesmaximal, 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#2487C2214yes—
24#2409C414yes—
25#2317C412yes—
26#2311C412yes—
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—
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⋊C4C2312yes—
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—
81#407—24no—
82#407C2314yes—
83#407C2314yes—
84C4×M16—24no—
85C4×M16—22no—
86C4×M16D414yes—
87#400—24no—
88#399—24no—
89#395—24no—
90#393—24no—
91#391—22no—
92#391—24no—
93#390—22no—
94#389—24no—
95#389—24no—
96#386C2×C418yes—
97#385C2×C418yes—
98#384—22no—
99#384—21no—
100#382—22no—
101#382—21no—
102#380—24no—
103#380—24no—
104#380C2×C412yes—
105#380C2312yes—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108#343—216no—
109#324—24no—
110#323—24no—
111C2×#41—24no—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—42no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—24no—
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—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
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—
203C22⋊C8—28no—
204C22⋊C8—24no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208#97—24no—
209#97—42no—
210C4×C8—28no—
211C4×C8—24no—
212C4×C8—42no—
213#95—22no—
214#95—28no—
215#95—28no—
216#95—22no—
217C24C2×C22⋊C412yes—
218C24C23×C411yescharacteristic
219#41—48no—
220#41—48no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—48no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—42no—
227C2×Q8—88no—
228C2×Q8—84no—
229C2×Q8—88no—
230C2×Q8#10114yes—
231C2×D4—24no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—42no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—84no—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C4×D412yes—
263C22×C4C4×D412yes—
264C22×C4C4×D412yes—
265C22×C4C4×D412yes—
266C22×C4C22×D412yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4112yes—
270M16—48no—
271M16—48no—
272M16—88no—
273C2×C8—28no—
274C2×C8—44no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—88no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—44no—
293C22⋊C4—88no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—44no—
301C42#12012yes—
302C42#12011yescharacteristic
303C42#12011yescharacteristic
304C42#12112yes—
305C42#12112yes—
306C42#12112yes—
307C42#12112yes—
308C42#12311yescharacteristic
309C42#12311yescharacteristic
310C42#12312yes—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—41no—
317C23—41no—
318C23—42no—
319C23—42no—
320C23C22×C22⋊C411yescharacteristic
321C23C2×C4×D412yes—
322Q8—28no—
323Q8—84no—
324Q8—88no—
325Q8—84no—
326D4—48no—
327D4—44no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—88no—
358C2×C4#38611yescharacteristic
359C2×C4#38611yescharacteristic
360C2×C4#38611yescharacteristic
361C2×C4#38612yes—
362C2×C4#38612yes—
363C2×C4#38611yescharacteristic
364C2×C4C2×#10114yes—
365C2×C4C2×#10112yes—
366C2×C4#41312yes—
367C2×C4C2×#12011yescharacteristic
368C2×C4C2×#12112yes—
369C2×C4C2×#12311yescharacteristic
370C8—48no—
371C8—84no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—42no—
378C22—42no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—82no—
384C22C2×#38611yescharacteristic
385C22C22×#10111yescharacteristic
386C22C2×#41311yescharacteristic
387C4—21no—
388C4—22no—
389C4—22no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—24no—
394C4—42no—
395C4—42no—
396C4—84no—
397C4—84no—
398C4#190114yes—
399C4#191412yes—
400C4#302812yes—
401C2—21no—
402C2—21no—
403C2—42no—
404C2—82no—
405C2#2096211yescharacteristic
406C2C2×#190111yescharacteristic
407C2C2×#191411yescharacteristic
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. #9510C2semidirect
6. #8105C2semidirect

Maximal subgroups

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

subgroupindex
#602842
#352792
#218292
#173662
#95102
#81052