Finite Group Databasebeta

← #164204

Subgroups — #164204

Order 512.

829 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#36301C211yesmaximal, characteristic
2#22636C211yesmaximal, characteristic, complemented
3#20430C211yesmaximal, characteristic, complemented
4#7697C212yesmaximal, complemented
5#7688C211yesmaximal, characteristic, complemented
6#7677C212yesmaximal, complemented
7#7665C211yesmaximal, characteristic, complemented
8#7628C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic
10#7574C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic
12C2×#424—21no—
13C2×#405—21no—
14#2864C2212yes—
15#2857C2211yescharacteristic
16#2571C411yescharacteristic
17#2563C411yescharacteristic
18#2527—22no—
19#2494—21no—
20#2494C412yes—
21#2485—21no—
22#2477—22no—
23#2420C412yes—
24#2369C411yescharacteristic
25#2352—21no—
26D4×C4⋊C4C411yescharacteristic
27#2344—21no—
28C4×#121C2211yescharacteristic
29#2293C2212yes—
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#1271C2212yes—
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×M16—21no—
54C23×C8—21no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C22⋊Q8—21no—
57C2×#121C2×C411yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C411yescharacteristic
61C2×#117—22no—
62C2×#117D411yescharacteristic
63C2×#117D411yescharacteristic
64C22×C4⋊C4—21no—
65C22×C42C2311yescharacteristic
66#424—28no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73#405—44no—
74C2×#97—22no—
75C2×#97—22no—
76C2×C4×C8—22no—
77C2×C4×C8—21no—
78#401—22no—
79#400—22no—
80#397—22no—
81#397—22no—
82#394—22no—
83#393—22no—
84#391—21no—
85#391—22no—
86#390C2×C412yes—
87#390C2311yescharacteristic
88#389—21no—
89#389—21no—
90#389—21no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2312yes—
95#389C2311yescharacteristic
96#388—22no—
97#388—22no—
98#388—22no—
99#388—21no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#386—21no—
106#386—22no—
107#385—22no—
108#384C2×C411yescharacteristic
109#382—21no—
110#382C2×C412yes—
111#382C2×C412yes—
112#381—22no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380C2×C412yes—
118#380C2×C411yescharacteristic
119C4×C4⋊C4—21no—
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⋊C4—22no—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
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×D4—21no—
144C22×D4C22×C411yescharacteristic
145C23×C4—21no—
146C23×C4—22no—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—42no—
153C22×C8—24no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—42no—
161C22×C8—42no—
162C4⋊Q8—42no—
163C22⋊Q8—81no—
164C22⋊Q8—81no—
165#121—22no—
166#121—41no—
167C4×D4—24no—
168C4×D4—42no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—42no—
172#117—22no—
173#117—22no—
174#117—24no—
175#117—82no—
176#117—82no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—44no—
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×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4C22⋊C412yes—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—42no—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C412yes—
211C2×C42C22×C412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2411yescharacteristic
219C4⋊C8—28no—
220C4⋊C8—28no—
221C4⋊C8—42no—
222C4⋊C8—82no—
223C22⋊C8—42no—
224C22⋊C8—81no—
225#97—44no—
226#97—44no—
227C4×C8—42no—
228C4×C8—44no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246C24—21no—
247C24C2×C22⋊C411yescharacteristic
248C2×Q8—42no—
249C2×Q8—81no—
250C2×Q8—81no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—81no—
257C2×D4#13112yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—82no—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4AGL(1,ℤ/8)12yes—
287C22×C4#13712yes—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291M16—48no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—48no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—81no—
331C4⋊C4#10414yes—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—44no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—24no—
354C42—42no—
355C42—41no—
356C42—41no—
357C42—42no—
358C42—42no—
359C42—82no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—22no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23#41712yes—
374C23C22×C22⋊C411yescharacteristic
375Q8—162no—
376D4—42no—
377D4—42no—
378D4—84no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—88no—
424C2×C4—84no—
425C2×C4#41012yes—
426C2×C4#41712yes—
427C2×C4C2×#10414yes—
428C2×C4C2×#10412yes—
429C2×C4C2×#13111yescharacteristic
430C2×C4#56811yescharacteristic
431C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
432C2×C4C2×#13711yescharacteristic
433C2×C4#57611yescharacteristic
434C2×C4#57711yescharacteristic
435C8—42no—
436C8—42no—
437C8—42no—
438C8—44no—
439C8—81no—
440C8—82no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22#288011yescharacteristic
456C22C2×#41011yescharacteristic
457C22#288911yescharacteristic
458C22C2×#41711yescharacteristic
459C22#289711yescharacteristic
460C22C22×#10411yescharacteristic
461C22#290111yescharacteristic
462C4—21no—
463C4—21no—
464C4—24no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—41no—
470C4—42no—
471C4—42no—
472C4—41no—
473C4—42no—
474C4—81no—
475C4—81no—
476C4—82no—
477C4—81no—
478C4—82no—
479C4#150914yes—
480C2—42no—
481C2—82no—
482C2#1741311yescharacteristic
483C2#1751211yescharacteristic
484C2#1751711yescharacteristic
485C2C2×#150911yescharacteristic
486C2#1761011yescharacteristic
487C2#1765411yescharacteristic
488C2#1776511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×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. #22636C2semidirect
3. #20430C2semidirect
4. #7697C2semidirect
5. #7688C2semidirect
6. #7677C2semidirect
7. #7665C2semidirect
8. #7628C2semidirect
10. #7574C2semidirect

Maximal subgroups

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

subgroupindex
#363012
#226362
#204302
#76972
#76882
#76772
#76652
#76282
#76162
#75742
#75652