Finite Group Databasebeta

← #164745

Subgroups — #164745

Order 512.

711 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36314C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic
3#20774C211yesmaximal, characteristic
4#7717C212yesmaximal, complemented
5#7685C211yesmaximal, characteristic
6#7668C211yesmaximal, characteristic
7#7657C212yesmaximal, complemented
8#7644C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic
10#7581C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic
12#2864C2212yes—
13#2857C2211yescharacteristic
14#2571C411yescharacteristic
15#2568C411yescharacteristic
16#2533—22no—
17#2530—22no—
18#2511—21no—
19#2499—21no—
20#2495C412yes—
21#2421C412yes—
22#2365C411yescharacteristic
23#2362—21no—
24#2356—21no—
25Q8×C4⋊C4C411yescharacteristic
26C4×C22⋊Q8C2211yescharacteristic
27#2295C2212yes—
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1923—22no—
31#1840—22no—
32#1839C2212yes—
33#1839C2212yes—
34#1831C2212yes—
35#1797C2211yescharacteristic
36C8×C4⋊C4—21no—
37C4×C22⋊C8C2211yescharacteristic
38#1744C2212yes—
39#1729—21no—
40#1729—21no—
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
64C2×C4⋊C8—21no—
65C2×C4⋊C8—22no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C4×C8—22no—
69C2×C4×C8—21no—
70#400—22no—
71#400—22no—
72#400—22no—
73#400—22no—
74#397—22no—
75#397—22no—
76#391—21no—
77#391—21no—
78#391—21no—
79#391—21no—
80#391C2×C412yes—
81#391C2311yescharacteristic
82#389—21no—
83#389—21no—
84#389—22no—
85#389C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2312yes—
89#389C2311yescharacteristic
90#388—22no—
91#388—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—21no—
99#387—22no—
100#387—22no—
101#385—21no—
102#384—21no—
103#384C2×C411yescharacteristic
104#383—22no—
105#382—21no—
106#382C2×C412yes—
107#382C2×C412yes—
108#380—21no—
109#380C2×C412yes—
110#380C2×C411yescharacteristic
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C22⋊C4—21no—
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4D412yes—
123#376—22no—
124#376D412yes—
125C43C2312yes—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—21no—
132#336—21no—
133#336—22no—
134#336C2312yes—
135#336C2311yescharacteristic
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140C22×Q8—21no—
141C22×Q8C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C2×M16—42no—
146C2×M16—42no—
147C2×M16—42no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—42no—
155C22×C8—42no—
156C22×C8—42no—
157C4⋊Q8—42no—
158C22⋊Q8—22no—
159C22⋊Q8—41no—
160C22⋊Q8—42no—
161C22⋊Q8—42no—
162C4×Q8—24no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—44no—
166#117—22no—
167#117—22no—
168#117—42no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22⋊C412yes—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—42no—
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C412yes—
201C2×C42C22×C412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2411yescharacteristic
209C4⋊C8—42no—
210C4⋊C8—82no—
211C22⋊C8—42no—
212C22⋊C8—81no—
213C4×C8—42no—
214C4×C8—44no—
215#95—22no—
216#95—22no—
217#95—21no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236C24C2×C22⋊C411yescharacteristic
237C2×Q8—21no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8#13112yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4#13512yes—
268C22×C4#13512yes—
269C22×C4C23×C411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272M16—84no—
273C2×C8—22no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—84no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—81no—
316C22⋊C4—24no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—44no—
320C22⋊C4—81no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—42no—
330C42—41no—
331C42—41no—
332C42—42no—
333C42—42no—
334C42—44no—
335C42—44no—
336C42—42no—
337C42—42no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23#41612yes—
345C23C22×C22⋊C411yescharacteristic
346Q8—42no—
347Q8—42no—
348Q8—84no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4#41012yes—
398C2×C4#41612yes—
399C2×C4#42112yes—
400C2×C4C2×#13111yescharacteristic
401C2×C4#56811yescharacteristic
402C2×C4C2×#13511yescharacteristic
403C2×C4C2×#13511yescharacteristic
404C2×C4#57711yescharacteristic
405C2×C4#57811yescharacteristic
406C8—42no—
407C8—81no—
408C8—82no—
409C8—81no—
410C8—81no—
411C8—82no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22#288011yescharacteristic
420C22C2×#41011yescharacteristic
421C22#288911yescharacteristic
422C22C2×#41611yescharacteristic
423C22#289811yescharacteristic
424C22C2×#42111yescharacteristic
425C22#290111yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—41no—
433C4—42no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—41no—
439C4—81no—
440C4—82no—
441C4—81no—
442C4—82no—
443C4—81no—
444C4#152114yes—
445C2—42no—
446C2#1741311yescharacteristic
447C2#1751011yescharacteristic
448C2#1752511yescharacteristic
449C2C2×#152111yescharacteristic
450C2#1760511yescharacteristic
451C2#1768311yescharacteristic
452C2#1772511yescharacteristic
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. #7717C2semidirect
7. #7657C2semidirect
8. #7644C2semidirect
10. #7581C2semidirect

Maximal subgroups

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

subgroupindex
#363142
#226322
#207742
#77172
#76852
#76682
#76572
#76442
#76172
#75812
#75662