Finite Group Databasebeta

← #214047

Subgroups — #214047

Order 512.

724 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37392C211yesmaximal, characteristic
2#17410C211yesmaximal, characteristic
3#17410C211yesmaximal, characteristic
4#9456C212yesmaximal, complemented
5#9454C212yesmaximal, complemented
6#9450C212yesmaximal, complemented
7#9450C212yesmaximal, complemented
8#8745C211yesmaximal, characteristic
9#8744C211yesmaximal, characteristic
10#8525C211yesmaximal, characteristic
11#8525C211yesmaximal, characteristic
12C2×#544C2211yescharacteristic
13C2×#541C2211yescharacteristic
14#2983C2212yes—
15#2983C2212yes—
16#2862C411yescharacteristic
17#2859C411yescharacteristic
18#2719—22no—
19#2713—22no—
20#2698—22no—
21#2689—22no—
22#2660C411yescharacteristic
23#2642C411yescharacteristic
24#2568C2211yescharacteristic
25#2567C2211yescharacteristic
26#2565C2211yescharacteristic
27#2565C2211yescharacteristic
28#2503C411yescharacteristic
29#2502C411yescharacteristic
30#2499C411yescharacteristic
31#2499C411yescharacteristic
32#2298C2211yescharacteristic
33#1556C2212yes—
34#1556C2212yes—
35#1534C2212yes—
36#1534C2212yes—
37#1473—22no—
38#1473—22no—
39#1470—22no—
40#1470—22no—
41#1467C2212yes—
42#1467C2212yes—
43#1464C2211yescharacteristic
44#1464C2211yescharacteristic
45#1464C2211yescharacteristic
46#1464C2211yescharacteristic
47#1285C2212yes—
48#1285C2212yes—
49#1285C2212yes—
50#1285C2212yes—
51#544—42no—
52#544—42no—
53#541—42no—
54#541—42no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2311yescharacteristic
59C2×#126D412yes—
60C2×#126D412yes—
61C2×#125—21no—
62C2×#125C2×C411yescharacteristic
63C2×#125C2×C411yescharacteristic
64C2×#125D412yes—
65C2×#125D412yes—
66C2×#125C2311yescharacteristic
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8C2×C411yescharacteristic
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76#446—42no—
77#446—42no—
78#446—42no—
79#446—42no—
80#432—22no—
81#432—22no—
82#432C2312yes—
83#432C2312yes—
84#432C2312yes—
85#432C2312yes—
86#431—22no—
87#431—22no—
88#400—22no—
89#400—22no—
90#400—22no—
91#400—22no—
92#398—22no—
93#398—22no—
94#398—22no—
95#398—22no—
96#397—22no—
97#397—22no—
98#395—22no—
99#395—22no—
100#391—22no—
101#391—22no—
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#391C2×C411yescharacteristic
105#391C2×C411yescharacteristic
106#391C2×C411yescharacteristic
107#391C2×C411yescharacteristic
108#391C2×C411yescharacteristic
109#391C2×C411yescharacteristic
110#389—22no—
111#389—22no—
112#389—22no—
113#388—21no—
114#388—21no—
115#387—22no—
116#387—22no—
117#387—21no—
118#387—21no—
119#387—22no—
120#387—22no—
121#385—22no—
122#384C2×C411yescharacteristic
123#384C2×C411yescharacteristic
124#384C2×C411yescharacteristic
125#384C2×C411yescharacteristic
126#382—22no—
127#382C2×C411yescharacteristic
128#382C2×C411yescharacteristic
129#382C2×C411yescharacteristic
130#382C2×C411yescharacteristic
131#381—22no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4C2×C411yescharacteristic
135C4×C22⋊C4C2×C411yescharacteristic
136#376C2×C411yescharacteristic
137C43C2×C411yescharacteristic
138#332—42no—
139#332—42no—
140#330—42no—
141#330—42no—
142#324—21no—
143#324—21no—
144#324—21no—
145#324—21no—
146#324—24no—
147#324—24no—
148#323—21no—
149#323—21no—
150#323—21no—
151#323—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8C22×C411yescharacteristic
155C22×Q8C22×C411yescharacteristic
156C23×C4—21no—
157C23×C4C22×C411yescharacteristic
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8—44no—
165#126—42no—
166#126—42no—
167#126—42no—
168#126—42no—
169#125—22no—
170#125—22no—
171#125—41no—
172#125—41no—
173#125—41no—
174#125—41no—
175#125—42no—
176#125—42no—
177#125—42no—
178#125—42no—
179#125—44no—
180C22⋊Q8—44no—
181C22⋊Q8—44no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190#117—22no—
191#117—22no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C2×D411yescharacteristic
231C2×C22⋊C4C2×D411yescharacteristic
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2411yescharacteristic
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255#97—22no—
256#97—22no—
257#97—22no—
258#97—22no—
259#97—42no—
260#97—42no—
261C4×C8—42no—
262C4×C8—42no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—21no—
280#95—21no—
281C24C2×C22⋊C411yescharacteristic
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—82no—
293C2×Q8—82no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—82no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C412yes—
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—81no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—81no—
366C42—21no—
367C42—21no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—42no—
377C42—42no—
378C42—81no—
379C42H(5,𝔽2)12yes—
380C42H(5,𝔽2)12yes—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—41no—
385C23—41no—
386C23C22×C22⋊C411yescharacteristic
387Q8—81no—
388Q8—81no—
389Q8—81no—
390Q8—81no—
391Q8—162no—
392Q8—162no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—162no—
437C2×C4#51811yescharacteristic
438C2×C4#51811yescharacteristic
439C2×C4#53412yes—
440C2×C4#53412yes—
441C8—82no—
442C8—82no—
443C8—82no—
444C8—82no—
445C22—21no—
446C22—21no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—81no—
451C22—81no—
452C22#230811yescharacteristic
453C22#288911yescharacteristic
454C22#288911yescharacteristic
455C4—22no—
456C4—22no—
457C4—21no—
458C4—21no—
459C4—42no—
460C4—41no—
461C4—42no—
462C4—42no—
463C4—41no—
464C4—41no—
465C4—42no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—81no—
473C4—81no—
474C2—21no—
475C2—21no—
476C2—81no—
477C2#2175611yescharacteristic
478C2#2178311yescharacteristic
479C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9456C2semidirect
5. #9454C2semidirect
6. #9450C2semidirect
7. #9450C2semidirect

Maximal subgroups

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

subgroupindex
#373922
#174102
#94562
#94542
#94502
#87452
#87442
#85252