Finite Group Databasebeta

← #215645

Subgroups — #215645

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2608C211yesmaximal, characteristic
2#21904C211yesmaximal, characteristic
3#21904C211yesmaximal, characteristic
4#9883C212yesmaximal, complemented
5#9881C212yesmaximal, complemented
6#9880C212yesmaximal, complemented
7#9879C212yesmaximal, complemented
8#8702C212yesmaximal
9#8700C212yesmaximal
10#2608—22no—
11#2608—22no—
12#2608—22no—
13#2608—22no—
14C2×#396C412yes—
15C2×#396C412yes—
16C2×#389C412yes—
17C2×#389C2212yes—
18C2×#387C412yes—
19C2×#387C2212yes—
20C2×#381C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23#1924—22no—
24#1924—22no—
25#1923—22no—
26#1923—22no—
27#1797C2212yes—
28#1797C2212yes—
29#1779C2212yes—
30#1779C2212yes—
31#1749C2211yescharacteristic
32#1749C2211yescharacteristic
33C4×C22⋊C8C2211yescharacteristic
34C4×C22⋊C8C2211yescharacteristic
35#1280C2212yes—
36#1280C2212yes—
37#1280C2212yes—
38#1280C2212yes—
39#1280C2212yes—
40#1280C2212yes—
41#1280C2212yes—
42#1280C2212yes—
43C24×C4C2311yescharacteristic
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C412yes—
50C22×C22⋊C4C2×C412yes—
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C412yes—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8C2312yes—
65C2×C22⋊C8C2312yes—
66C2×C22⋊C8C2312yes—
67C2×C22⋊C8C2312yes—
68C2×#97—21no—
69C2×#97—21no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72#396—22no—
73#396—22no—
74#396—22no—
75#396—22no—
76#396—22no—
77#396—22no—
78#396—22no—
79#396—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#381—22no—
97#381—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4D412yes—
101C4×C22⋊C4D412yes—
102C4×C22⋊C4D412yes—
103C4×C22⋊C4D412yes—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C2×#95C2×C412yes—
108C2×#95C2312yes—
109C2×#95C2312yes—
110#336—22no—
111#336—22no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116C25C22×C411yescharacteristic
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—81no—
120C23×C4—81no—
121C23×C4C22×C412yes—
122C23×C4C22×C412yes—
123C23×C4C22×C412yes—
124C23×C4C22×C412yes—
125C23×C4C22×C412yes—
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—81no—
163C2×C22⋊C4—81no—
164C2×C22⋊C4C22⋊C412yes—
165C2×C22⋊C4C22⋊C412yes—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C22⋊C412yes—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—81no—
171C2×C42—81no—
172C2×C42—81no—
173C2×C42—81no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D412yes—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D412yes—
180C4⋊C8—82no—
181C4⋊C8—82no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—82no—
191C22⋊C8—82no—
192C22⋊C8—82no—
193C22⋊C8—82no—
194#97—81no—
195#97—81no—
196C4×C8—81no—
197C4×C8—81no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95C22⋊C412yes—
209#95C22⋊C412yes—
210#95C22⋊C412yes—
211#95C22⋊C412yes—
212C24—21no—
213C24—21no—
214C24—22no—
215C24—22no—
216C24—42no—
217C24—41no—
218C24—41no—
219C24C2×C22⋊C412yes—
220C24C23×C411yescharacteristic
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—84no—
261C22×C4—84no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4#13114yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C42−1+412yes—
271C22×C42−1+412yes—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C42—41no—
305C42—41no—
306C42—41no—
307C42—41no—
308C42—42no—
309C42—42no—
310C42—82no—
311C42—82no—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—41no—
328C23—41no—
329C23—82no—
330C23—82no—
331C23—82no—
332C23—82no—
333C23C22×C22⋊C411yescharacteristic
334C23#51911yescharacteristic
335C23#51911yescharacteristic
336C23C2×#13112yes—
337C23#56812yes—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4#41314yes—
387C2×C4#53612yes—
388C2×C4#53612yes—
389C8—82no—
390C8—82no—
391C8—82no—
392C8—82no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—84no—
409C22—82no—
410C22—82no—
411C22#231011yescharacteristic
412C22#288012yes—
413C22#288311yescharacteristic
414C22#288311yescharacteristic
415C22C2×#41312yes—
416C22#298512yes—
417C22#298611yescharacteristic
418C22#298711yescharacteristic
419C4—24no—
420C4—22no—
421C4—22no—
422C4—42no—
423C4—42no—
424C4—44no—
425C4—42no—
426C4—42no—
427C4—82no—
428C4—82no—
429C4—82no—
430C2—22no—
431C2—21no—
432C2—21no—
433C2—82no—
434C2#1742412yes—
435C2#1742712yes—
436C2#2182411yescharacteristic
437C2#2182711yescharacteristic
438C2#2183111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9883C2semidirect
5. #9881C2semidirect
6. #9880C2semidirect
7. #9879C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26082
#219042
#98832
#98812
#98802
#98792
#87022
#87002