Finite Group Databasebeta

← #160900

Subgroups — #160900

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34334C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic
3#20568C211yesmaximal, characteristic
4#7634C212yesmaximal, complemented
5#7629C212yesmaximal, complemented
6#7623C212yesmaximal, complemented
7#7609C212yesmaximal, complemented
8#7590C211yesmaximal, characteristic, complemented
9#7577C211yesmaximal, characteristic, complemented
10#7571C211yesmaximal, characteristic, complemented
11#7562C211yesmaximal, characteristic, complemented
12C2×D4×Q8C411yescharacteristic
13C2×#436—21no—
14#2369—21no—
15#2368—22no—
16#2365C411yescharacteristic
17#2360C412yes—
18#2355C411yescharacteristic
19#2349—21no—
20D4×C4⋊C4C411yescharacteristic
21Q8×C22⋊C4C412yes—
22#2331—21no—
23#2329—22no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C22⋊Q8C2212yes—
26#2295C2211yescharacteristic
27C2×#391C2212yes—
28#2275—21no—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1924—22no—
32#1831C2211yescharacteristic
33#1817C2212yes—
34#1770—21no—
35#1770—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1753—22no—
39#1749C2212yes—
40#1749C2212yes—
41#1744C2211yescharacteristic
42#1728C2212yes—
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1272C2212yes—
47#1272C2212yes—
48#1272C2212yes—
49#1272C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C23×Q8C2×C412yes—
54C22×M16—22no—
55D4×Q8—24no—
56D4×Q8—24no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8C2×C412yes—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C412yes—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C22×C4⋊C4C2×C412yes—
70C22×C42C2312yes—
71#436—28no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×#97—22no—
79C2×#97—22no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83#391—22no—
84#391—22no—
85#391—22no—
86#391—22no—
87#391C2×C411yescharacteristic
88#391C2×C412yes—
89#391C2×C412yes—
90#391C2312yes—
91#391C2311yescharacteristic
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—22no—
97#389C2×C411yescharacteristic
98#389C2311yescharacteristic
99#387—21no—
100#387—22no—
101#387—21no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#384C2×C411yescharacteristic
109#382—21no—
110#380—22no—
111#380C2×C412yes—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129#376—21no—
130C2×#95—22no—
131C2×#95—22no—
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—24no—
144C22×Q8—24no—
145C22×Q8—22no—
146C22×Q8—82no—
147C22×Q8—82no—
148C22×Q8C22×C412yes—
149C22×Q8C22×C411yescharacteristic
150C22×D4C22⋊C411yescharacteristic
151C23×C4—22no—
152C23×C4—22no—
153C23×C4C22×C412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C2×M16—28no—
157C2×M16—82no—
158C2×M16—82no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C4⋊Q8—28no—
166C4⋊Q8—22no—
167C4⋊Q8M1614yescomplemented
168C22⋊Q8—216no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C4×Q8—22no—
172C4×Q8—82no—
173C4×Q8—82no—
174C4×Q8M1614yescomplemented
175C4×D4—28no—
176C4×D4—24no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—44no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4C22⋊C412yes—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—82no—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C412yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2411yescharacteristic
222C4⋊C8—24no—
223C4⋊C8—42no—
224C4⋊C8—42no—
225C22⋊C8—28no—
226C22⋊C8—44no—
227C22⋊C8—82no—
228#97—82no—
229#97—82no—
230C4×C8—24no—
231C4×C8—42no—
232C4×C8—81no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24C2×C22⋊C412yes—
252C2×Q8—28no—
253C2×Q8—21no—
254C2×Q8—22no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—82no—
260C2×Q8—82no—
261C2×Q8—88no—
262C2×Q8—88no—
263C2×Q8C2×M1612yes—
264C2×Q8C2×M1614yes—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4#10414yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—28no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302M16—88no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C4⋊C4—28no—
322C4⋊C4—21no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—81no—
336C4⋊C4—81no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4C2×M1612yes—
341C22⋊C4—28no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—44no—
346C42—21no—
347C42—21no—
348C42—24no—
349C42—22no—
350C42—42no—
351C42—42no—
352C42—41no—
353C42—41no—
354C42—42no—
355C42—41no—
356C42—42no—
357C42—82no—
358C42—82no—
359C42—82no—
360C42C2×M1612yes—
361C42C2×M1612yes—
362C42#13712yes—
363C42#13712yes—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23C2×#10414yes—
371C23C22×C22⋊C411yescharacteristic
372Q8—42no—
373Q8—44no—
374Q8—44no—
375Q8—42no—
376Q8—88no—
377D4—28no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
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—21no—
402C2×C4—22no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—88no—
425C2×C4—164no—
426C2×C4#40712yes—
427C2×C4#40712yes—
428C2×C4#41912yes—
429C2×C4#41912yes—
430C2×C4C2×#10412yes—
431C2×C4C22×M1611yescharacteristic
432C2×C4C22×M1611yescharacteristic
433C2×C4C2×#13711yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4#57711yescharacteristic
436C2×C4#57811yescharacteristic
437C8—42no—
438C8—42no—
439C8—44no—
440C8—42no—
441C8—81no—
442C8—82no—
443C22—24no—
444C22—24no—
445C22—24no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22C2×#40711yescharacteristic
451C22#288311yescharacteristic
452C22#288811yescharacteristic
453C22C2×#41911yescharacteristic
454C22#289811yescharacteristic
455C22C22×#10411yescharacteristic
456C22#290111yescharacteristic
457C4—22no—
458C4—22no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—44no—
470C4—42no—
471C4—41no—
472C4—81no—
473C4—82no—
474C4—81no—
475C4#149114yes—
476C2—44no—
477C2#1736911yescharacteristic
478C2C2×#149111yescharacteristic
479C2#1745411yescharacteristic
480C2#1754811yescharacteristic
481C2#1762711yescharacteristic
482C2#1765211yescharacteristic
483C2#1776011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7634C2semidirect
5. #7629C2semidirect
6. #7623C2semidirect
7. #7609C2semidirect
8. #7590C2semidirect
9. #7577C2semidirect
10. #7571C2semidirect
11. #7562C2semidirect
167. C4⋊Q8M16semidirect
174. C4×Q8M16semidirect

Maximal subgroups

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

subgroupindex
#343342
#228392
#205682
#76342
#76292
#76232
#76092
#75902
#75772
#75712
#75622