Finite Group Databasebeta

← #164276

Subgroups — #164276

Order 512.

779 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36314C211yesmaximal, characteristic
2#22636C211yesmaximal, characteristic
3#20430C211yesmaximal, characteristic
4#7697C212yesmaximal, complemented
5#7685C211yesmaximal, characteristic
6#7677C212yesmaximal, complemented
7#7668C211yesmaximal, characteristic
8#7630C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic
10#7577C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic
12C2×#424—21no—
13C2×#405—21no—
14#2864C2212yes—
15#2857C2211yescharacteristic
16#2571C411yescharacteristic
17#2568C411yescharacteristic
18#2533—22no—
19#2530—22no—
20#2511—21no—
21#2499—21no—
22#2495C412yes—
23#2421C412yes—
24#2365C411yescharacteristic
25#2362—21no—
26#2356—21no—
27Q8×C4⋊C4C411yescharacteristic
28C4×C22⋊Q8C2211yescharacteristic
29#2295C2212yes—
30C4×#117C2211yescharacteristic
31#1939—21no—
32#1924—22no—
33#1831C2212yes—
34#1797C2211yescharacteristic
35#1770—21no—
36#1768—22no—
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C22⋊C8C2211yescharacteristic
40#1745C2212yes—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2212yes—
53C22×M16—21no—
54C23×C8—21no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C411yescharacteristic
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C411yescharacteristic
61C2×#117D411yescharacteristic
62C2×#117D411yescharacteristic
63C22×C42C2311yescharacteristic
64#424—28no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71#405—44no—
72C2×#97—22no—
73C2×#97—22no—
74C2×C4×C8—22no—
75C2×C4×C8—21no—
76#400—22no—
77#400—22no—
78#400—22no—
79#400—22no—
80#397—22no—
81#397—22no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391—21no—
86#391C2×C412yes—
87#391C2311yescharacteristic
88#389—21no—
89#389—21no—
90#389—22no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2312yes—
95#389C2311yescharacteristic
96#388—22no—
97#388—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—21no—
105#387—22no—
106#387—22no—
107#385—21no—
108#384—21no—
109#384C2×C411yescharacteristic
110#383—22no—
111#382—21no—
112#382C2×C412yes—
113#382C2×C412yes—
114#380—21no—
115#380C2×C412yes—
116#380C2×C411yescharacteristic
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129#376—22no—
130#376D412yes—
131C43C2312yes—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×Q8C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—42no—
150C22×C8—24no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—42no—
158C22×C8—42no—
159C4⋊Q8—42no—
160C22⋊Q8—22no—
161C22⋊Q8—41no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C4×Q8—24no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8—44no—
168#117—22no—
169#117—22no—
170#117—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C22⋊C412yes—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—42no—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C412yes—
203C2×C42C22×C412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2411yescharacteristic
211C4⋊C8—28no—
212C4⋊C8—28no—
213C4⋊C8—42no—
214C4⋊C8—82no—
215C22⋊C8—42no—
216C22⋊C8—81no—
217#97—44no—
218#97—44no—
219C4×C8—42no—
220C4×C8—44no—
221#95—22no—
222#95—22no—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242C24C2×C22⋊C411yescharacteristic
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8#13112yes—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4AGL(1,ℤ/8)12yes—
274C22×C4#13712yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278M16—48no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—48no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—41no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—81no—
321C4⋊C4#10414yes—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—42no—
325C22⋊C4—44no—
326C22⋊C4—81no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—24no—
336C42—42no—
337C42—41no—
338C42—41no—
339C42—42no—
340C42—44no—
341C42—42no—
342C42—42no—
343C42—44no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23#41712yes—
351C23C22×C22⋊C411yescharacteristic
352Q8—42no—
353Q8—42no—
354Q8—84no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4#41012yes—
403C2×C4#41712yes—
404C2×C4C2×#10414yes—
405C2×C4C2×#10412yes—
406C2×C4C2×#13111yescharacteristic
407C2×C4#56811yescharacteristic
408C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
409C2×C4C2×#13711yescharacteristic
410C2×C4#57711yescharacteristic
411C2×C4#57811yescharacteristic
412C8—42no—
413C8—42no—
414C8—42no—
415C8—44no—
416C8—81no—
417C8—82no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22#288011yescharacteristic
426C22C2×#41011yescharacteristic
427C22#288911yescharacteristic
428C22C2×#41711yescharacteristic
429C22#289811yescharacteristic
430C22C22×#10411yescharacteristic
431C22#290111yescharacteristic
432C4—21no—
433C4—21no—
434C4—24no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—41no—
444C4—42no—
445C4—82no—
446C4—81no—
447C4—82no—
448C4—81no—
449C4—81no—
450C4#150914yes—
451C2—42no—
452C2#1741311yescharacteristic
453C2#1751511yescharacteristic
454C2#1751911yescharacteristic
455C2C2×#150911yescharacteristic
456C2#1760511yescharacteristic
457C2#1765411yescharacteristic
458C2#1776311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7697C2semidirect
6. #7677C2semidirect
8. #7630C2semidirect
10. #7577C2semidirect

Maximal subgroups

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

subgroupindex
#363142
#226362
#204302
#76972
#76852
#76772
#76682
#76302
#76172
#75772
#75662