Finite Group Databasebeta

← #213721

Subgroups — #213721

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34199C211yesmaximal, characteristic
2#17350C212yesmaximal
3#9108C214yesmaximal, complemented
4#8966C214yesmaximal, complemented
5#8734C212yesmaximal, complemented
6#8300C212yesmaximal, complemented
7C22×#123C411yescharacteristic
8C2×#518C2211yescharacteristic
9C22×#117C2211yescharacteristic
10#2972C2214yes—
11#2359—22no—
12#2359—22no—
13#2344—22no—
14#2342—22no—
15#2322C412yes—
16#2322C2212yes—
17#2315C411yescharacteristic
18#2308C411yescharacteristic
19C4×#123C412yes—
20C4×#123C2212yes—
21C2×#381C411yescharacteristic
22#2273C2211yescharacteristic
23#1817C2214yescomplemented
24#1460—22no—
25#1459—22no—
26#1459C2214yes—
27#1458—22no—
28C2×#324C2214yescomplemented
29C2×#323—22no—
30C2×#323C2214yes—
31#1282C2218yescomplemented
32C23×D4C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C2×#123—22no—
35C2×#123—24no—
36C2×#123C818yescomplemented
37C2×#123C2×C412yes—
38C2×#123C2×C412yes—
39#518—44no—
40C2×C4×D4C2×C412yes—
41C2×C4×D4D412yes—
42C2×C4×D4D412yes—
43C2×C4×D4C2312yes—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—44no—
47C2×#117D412yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C22⋊C4—21no—
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4D412yes—
57C22×C22⋊C4C2311yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2311yescharacteristic
60C8×D4—48no—
61#431—24no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8C2318yes—
64#389—22no—
65#389—22no—
66#388—22no—
67#388—22no—
68#388—22no—
69#388—22no—
70#387—22no—
71#387—22no—
72#387—22no—
73#387—22no—
74#386—22no—
75#386C2×C412yes—
76#386C2×C412yes—
77#385—21no—
78#385—21no—
79#385—22no—
80#385—22no—
81#385C2×C412yes—
82#385C2×C412yes—
83#381—21no—
84#381—22no—
85#381C2×C412yes—
86#381C2×C412yes—
87#380—22no—
88#379—21no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4C2×C412yes—
96C4×C22⋊C4C2312yes—
97C2×#95—21no—
98C2×#95—21no—
99#336—44no—
100#324—24no—
101#324—44no—
102#323—24no—
103#323—24no—
104#323—24no—
105C25C22⋊C411yescharacteristic
106C25C22×C411yescharacteristic
107C22×D4—42no—
108C22×D4—42no—
109C22×D4C22⋊C412yes—
110C22×D4C2×C814yes—
111C22×D4M1614yes—
112C22×D4C22×C412yes—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22⋊C412yes—
124C23×C4C2×C814yescomplemented
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2411yescharacteristic
132C22×C8—28no—
133C22×C8—44no—
134#123—216no—
135#123—24no—
136#123—24no—
137#123C2×C818yescomplemented
138#123C2×C818yescomplemented
139C4×D4—28no—
140C4×D4—44no—
141C4×D4—44no—
142C4×D4—44no—
143#117—44no—
144#117—48no—
145#117—48no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4C22⋊C412yes—
173C2×C22⋊C4C22⋊C412yes—
174C2×C22⋊C4C2×C814yescomplemented
175C2×C22⋊C4C22×C412yes—
176C2×C22⋊C4C22×C412yes—
177C2×C22⋊C4C22×C412yes—
178C2×C22⋊C4C2×D412yes—
179C2×C22⋊C4C2×D412yes—
180C2×C22⋊C4C2×D412yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—24no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—42no—
187C2×C42C22⋊C412yes—
188C2×C42C2×D412yes—
189C4⋊C8—44no—
190C22⋊C8—28no—
191C22⋊C8—28no—
192C22⋊C8—44no—
193C4×C8—44no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204C24—22no—
205C24—22no—
206C24—41no—
207C24—41no—
208C24—81no—
209C24—81no—
210C24—81no—
211C24—81no—
212C24C22⋊C814yes—
213C24C2×C22⋊C412yes—
214C24C2×C22⋊C411yescharacteristic
215C24C2×C22⋊C411yescharacteristic
216C24C22×C812yes—
217C24C2×M1612yes—
218C24C23×C411yescharacteristic
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—48no—
224C2×D4—48no—
225C2×D4C22⋊C818yes—
226C2×D4C22×C814yes—
227C2×D4C2×M1614yes—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—48no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—48no—
258C22×C4—42no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—82no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—82no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4C22⋊C814yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C22×C812yes—
277C22×C4C22×C812yes—
278C22×C4C22×D412yes—
279C22×C4H(5,𝔽2)14yes—
280C2×C8—48no—
281C2×C8—88no—
282C2×C8—164no—
283C4⋊C4—24no—
284C4⋊C4—28no—
285C4⋊C4—28no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C22⋊C4—24no—
292C22⋊C4—22no—
293C22⋊C4—28no—
294C22⋊C4—28no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4C22×C814yes—
308C42—24no—
309C42—42no—
310C42—44no—
311C42—44no—
312C42—82no—
313C42—82no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—21no—
318C23—21no—
319C23—22no—
320C23—24no—
321C23—24no—
322C23—41no—
323C23—41no—
324C23—44no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—42no—
330C23—42no—
331C23—82no—
332C23—82no—
333C23—82no—
334C23—82no—
335C23—84no—
336C23—84no—
337C23—161no—
338C23—161no—
339C23—162no—
340C23C2×C22⋊C814yes—
341C23C2×C22⋊C812yes—
342C23C2×C22⋊C812yes—
343C23C22×C22⋊C411yescharacteristic
344C23#51812yes—
345C23C23×C811yescharacteristic
346C23C22×M1611yescharacteristic
347C23#56812yes—
348D4—28no—
349D4—28no—
350D4—48no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—84no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4—82no—
389C2×C4—161no—
390C2×C4—162no—
391C2×C4—161no—
392C2×C4—162no—
393C2×C4C2×C22⋊C814yes—
394C2×C4#53414yes—
395C8—88no—
396C22—21no—
397C22—22no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—24no—
403C22—41no—
404C22—41no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—44no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—42no—
416C22—41no—
417C22—82no—
418C22—84no—
419C22—84no—
420C22—164no—
421C22#230811yescharacteristic
422C22C22×C22⋊C811yescharacteristic
423C22#287911yescharacteristic
424C22#297212yes—
425C22#297312yes—
426C4—24no—
427C4—24no—
428C4—44no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—82no—
435C4—81no—
436C4—81no—
437C2—21no—
438C2—21no—
439C2—22no—
440C2—22no—
441C2—42no—
442C2—42no—
443C2—44no—
444C2#2173811yescharacteristic
445C2#2177711yescharacteristic
446C2#2177911yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9108C2semidirect
4. #8966C2semidirect
5. #8734C2semidirect
6. #8300C2semidirect
23. #1817C22semidirect
28. C2×#324C22semidirect
31. #1282C22semidirect
36. C2×#123C8semidirect
124. C23×C4C2×C8semidirect
137. #123C2×C8semidirect
138. #123C2×C8semidirect
174. C2×C22⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
#341992
#173502
#91082
#89662
#87342
#83002