Finite Group Databasebeta

← #160652

Subgroups — #160652

Order 512.

826 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
1#34328C211yesmaximal, characteristic
2#22734C211yesmaximal, characteristic
3#22018C211yesmaximal, characteristic
4#7602C212yesmaximal
5#7596C212yesmaximal
6#7588C211yesmaximal, characteristic
7#7585C212yesmaximal
8#7577C212yesmaximal
9#7577C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic
12C2×#557C411yescharacteristic
13C2×C8×Q8—21no—
14#2371—22no—
15#2370C412yes—
16#2365—21no—
17#2365C411yescharacteristic
18#2356—21no—
19#2356C412yes—
20#2355C411yescharacteristic
21#2355C411yescharacteristic
22#2336—22no—
23#2327—21no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×#125C2212yes—
26#2295C2211yescharacteristic
27#2295C2212yes—
28#2276—21no—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1939—22no—
33#1939—21no—
34#1916—22no—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38#1831C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2212yes—
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53#557—24no—
54#557C818yescomplemented
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125—22no—
58C2×#125C2×C412yes—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C8×Q8—44no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#391—21no—
79#391—21no—
80#391—21no—
81#391C2×C412yes—
82#391C2×C411yescharacteristic
83#391C2311yescharacteristic
84#389—22no—
85#389—21no—
86#389C2×C412yes—
87#389C2×C411yescharacteristic
88#389C2311yescharacteristic
89#389C2312yes—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—21no—
98#387—22no—
99#387—21no—
100#387—22no—
101#387—22no—
102#387—22no—
103#384C2×C411yescharacteristic
104#383—22no—
105#382—21no—
106#382C2×C412yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126#376—21no—
127#376—22no—
128#376—22no—
129C43C2312yes—
130C43C2311yescharacteristic
131#336—22no—
132#336—22no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—21no—
144C22×Q8C22⋊C411yescharacteristic
145C22×Q8C22×C411yescharacteristic
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C4⋊Q8—28no—
153C4⋊Q8—22no—
154C4⋊Q8C2×C814yes—
155#125—216no—
156#125—24no—
157#125C2×C818yes—
158C4×Q8—28no—
159C4×Q8—24no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—42no—
164C4×Q8—44no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8C2×C818yes—
168C4×Q8C2×C814yes—
169C4×Q8C2×C814yes—
170C4×Q8M1614yes—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42C22×C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—42no—
214C4⋊C8—44no—
215C4⋊C8—44no—
216C4⋊C8—42no—
217C4⋊C8—42no—
218C4⋊C8—82no—
219C4×C8—42no—
220C4×C8—44no—
221C4×C8—42no—
222C4×C8—81no—
223#95—21no—
224#95—22no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243C2×Q8—21no—
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8C22⋊C814yes—
252C2×Q8C22×C812yes—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C4⋊C4—21no—
289C4⋊C4—22no—
290C4⋊C4—28no—
291C4⋊C4—24no—
292C4⋊C4—28no—
293C4⋊C4—24no—
294C4⋊C4—28no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—81no—
309C4⋊C4—81no—
310C4⋊C4C22⋊C814yes—
311C4⋊C4C22⋊C818yes—
312C4⋊C4C22×C812yes—
313C4⋊C4C22×C814yes—
314C42—22no—
315C42—22no—
316C42—21no—
317C42—22no—
318C42—21no—
319C42—22no—
320C42—28no—
321C42—24no—
322C42—22no—
323C42—42no—
324C42—41no—
325C42—41no—
326C42—42no—
327C42—42no—
328C42—41no—
329C42—44no—
330C42—44no—
331C42—42no—
332C42—82no—
333C42—82no—
334C42C22×C814yes—
335C42C22×C812yes—
336C42C2×M1614yes—
337C42C2×M1612yes—
338C42AGL(1,ℤ/8)12yes—
339C42#13712yes—
340C23C22×C22⋊C411yescharacteristic
341Q8—28no—
342Q8—42no—
343Q8—42no—
344Q8—48no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4C2×C22⋊C814yes—
387C2×C4C2×C22⋊C812yes—
388C2×C4C2×C22⋊C814yes—
389C2×C4C2×C22⋊C812yes—
390C2×C4#41712yes—
391C2×C4#41712yes—
392C2×C4#42112yes—
393C2×C4C23×C811yescharacteristic
394C2×C4C22×M1611yescharacteristic
395C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
396C2×C4C2×#13711yescharacteristic
397C2×C4#57711yescharacteristic
398C2×C4#57811yescharacteristic
399C8—42no—
400C8—81no—
401C8—82no—
402C8—81no—
403C8—82no—
404C8—81no—
405C22C22×C22⋊C811yescharacteristic
406C22#288411yescharacteristic
407C22#288911yescharacteristic
408C22C2×#41711yescharacteristic
409C22#289811yescharacteristic
410C22C2×#42111yescharacteristic
411C22#290111yescharacteristic
412C4—24no—
413C4—24no—
414C4—22no—
415C4—22no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—42no—
424C4—44no—
425C4—42no—
426C4—42no—
427C4—42no—
428C4—81no—
429C4—81no—
430C4—82no—
431C4#147714yes—
432C2#1735211yescharacteristic
433C2C2×#147711yescharacteristic
434C2#1743911yescharacteristic
435C2#1760111yescharacteristic
436C2#1762811yescharacteristic
437C2#1768211yescharacteristic
438C2#1776311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
54. #557C8semidirect

Maximal subgroups

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

subgroupindex
#343282
#227342
#220182
#76022
#75962
#75882
#75852
#75772
#75702
#75622