Finite Group Databasebeta

← #163988

Subgroups — #163988

Order 512.

853 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35312C211yesmaximal, characteristic
2#22643C211yesmaximal, characteristic, complemented
3#22632C211yesmaximal, characteristic, complemented
4#7722C212yesmaximal, complemented
5#7696C212yesmaximal, complemented
6#7682C212yesmaximal, complemented
7#7656C212yesmaximal, complemented
8#7597C211yesmaximal, characteristic
9#7575C211yesmaximal, characteristic, complemented
10#7575C211yesmaximal, characteristic, complemented
11#7563C211yesmaximal, characteristic
12C2×#544C411yescharacteristic
13C2×#536C411yescharacteristic
14#2862C2212yes—
15#2861C2212yes—
16#2642C412yes—
17#2640C412yes—
18#2492—21no—
19#2492—21no—
20#2489—21no—
21#2489—21no—
22#2488—22no—
23#2488—22no—
24#2311C411yescharacteristic
25#2310C411yescharacteristic
26#2297C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1924—22no—
32#1923—22no—
33#1839C2212yes—
34#1831C2212yes—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1770—21no—
38C8×C4⋊C4—21no—
39#1767C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42#1744C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1272C2212yes—
48#1272C2211yescharacteristic
49#1272C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×#41C2×C411yescharacteristic
54C23×Q8C2×C411yescharacteristic
55#544—44no—
56#536—44no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8C2×C412yes—
60C2×#125—21no—
61C2×#125C2×C412yes—
62C2×#124C2×C412yes—
63C2×#123—22no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—22no—
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×#97—22no—
81C2×C4×C8—22no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#400—22no—
85#400—22no—
86#400—22no—
87#400—22no—
88#397—22no—
89#397—22no—
90#393—22no—
91#393—22no—
92#391—21no—
93#391—21no—
94#391—22no—
95#391—22no—
96#391—21no—
97#391—21no—
98#391C2×C412yes—
99#391C2×C412yes—
100#391C2×C412yes—
101#391C2×C412yes—
102#391C2×C411yescharacteristic
103#391C2312yes—
104#391C2311yescharacteristic
105#390C2×C412yes—
106#390C2×C411yescharacteristic
107#390C2311yescharacteristic
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#386—21no—
113#386—21no—
114#386—22no—
115#382—22no—
116#382—22no—
117#381—22no—
118#380—21no—
119#380—21no—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4D412yes—
127C4×C22⋊C4D412yes—
128#376C2×C412yes—
129C2×#95—21no—
130C2×#95—21no—
131C43C2312yes—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C2×#41—24no—
143C2×#41—42no—
144C2×#41—42no—
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—24no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8C22×C412yes—
151C22×Q8C22×C411yescharacteristic
152C22×D4—21no—
153C22×D4C22×C411yescharacteristic
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169#125—44no—
170#125—44no—
171#124—44no—
172#123—48no—
173C22⋊Q8—48no—
174C22⋊Q8—44no—
175C22⋊Q8—44no—
176#117—24no—
177#117—24no—
178#117—42no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C42—21no—
205C2×C42—42no—
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—42no—
217C4⋊C8—42no—
218C4⋊C8—44no—
219C4⋊C8—82no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222C22⋊C8—81no—
223C22⋊C8—81no—
224#97—82no—
225C4×C8—42no—
226C4×C8—42no—
227C4×C8—44no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246C24—21no—
247C24C2×C22⋊C411yescharacteristic
248#41—88no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—42no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—44no—
259C2×Q8—82no—
260C2×Q8—82no—
261C2×Q8—88no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4#13114yes—
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—41no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4#10014yes—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—44no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—44no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—81no—
346C22⋊C4—81no—
347C42—24no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—24no—
355C42—42no—
356C42—42no—
357C42—41no—
358C42—41no—
359C42—42no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23#40812yes—
374C23C22×C22⋊C411yescharacteristic
375Q8—84no—
376Q8—82no—
377Q8—84no—
378Q8—164no—
379D4—82no—
380D4—84no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—28no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4—162no—
427C2×C4#40812yes—
428C2×C4#41012yes—
429C2×C4C2×#10014yes—
430C2×C4C2×#10012yes—
431C2×C4C2×#13112yes—
432C2×C4#57711yescharacteristic
433C2×C4#57711yescharacteristic
434C2×C4#57811yescharacteristic
435C2×C4#57811yescharacteristic
436C8—42no—
437C8—42no—
438C8—81no—
439C8—82no—
440C8—82no—
441C8—81no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22C2×#40811yescharacteristic
457C22C2×#41011yescharacteristic
458C22C22×#10011yescharacteristic
459C22#289811yescharacteristic
460C22#289811yescharacteristic
461C22#290111yescharacteristic
462C22#290111yescharacteristic
463C4—24no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—22no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—41no—
473C4—41no—
474C4—82no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—82no—
479C4#146914yes—
480C2—42no—
481C2—161no—
482C2C2×#146911yescharacteristic
483C2#1746911yescharacteristic
484C2#1748411yescharacteristic
485C2#1760511yescharacteristic
486C2#1761211yescharacteristic
487C2#1768511yescharacteristic
488C2#1777311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×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
2. #22643C2semidirect
3. #22632C2semidirect
4. #7722C2semidirect
5. #7696C2semidirect
6. #7682C2semidirect
7. #7656C2semidirect
9. #7575C2semidirect
10. #7575C2semidirect

Maximal subgroups

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

subgroupindex
#353122
#226432
#226322
#77222
#76962
#76822
#76562
#75972
#75752
#75632