Finite Group Databasebeta

← #182677

Subgroups — #182677

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#20533C211yesmaximal, characteristic
3C2×#1518C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#13804C211yesmaximal, characteristic
6#8221C214yesmaximal
7C4×#328C214yesmaximal
8#8026C211yesmaximal, characteristic
9#8014C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2370—22no—
14#2369C412yescomplemented
15#2365C412yescomplemented
16#2355—22no—
17#2355C412yescomplemented
18Q8×C4⋊C4C412yescomplemented
19#2329—22no—
20#2317—21no—
21C4×C22⋊Q8C2214yes—
22#2285C2211yescharacteristic
23C2×#382C2211yescharacteristic
24#2274—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C8×C4⋊C4—22no—
29C8×C4⋊C4—22no—
30C4×C22⋊C8C2214yes—
31#1518C418yescomplemented
32#1503C418yescomplemented
33C2×#328C2212yes—
34C2×#328C2214yes—
35C2×#328C2212yes—
36#1436—22no—
37C4⋊#95—21no—
38#1434C2212yes—
39C4×#95C2212yes—
40#1294C2212yes—
41#1294C2214yes—
42#1294C2212yes—
43#533—22no—
44#533—22no—
45#533C2×C414yes—
46#533C2×C414yes—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8C2312yes—
52C2×C4×Q8—21no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C414yes—
55C2×#117—21no—
56C2×#117—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C42—21no—
62C22×C42—22no—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65#431—22no—
66#431—22no—
67#431C2×C414yes—
68#431C2×C414yes—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C22⋊C8C2312yes—
72C2×C22⋊C8C2312yes—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—24no—
76#391—24no—
77#391C2×C414yes—
78#389—22no—
79#389—22no—
80#389—24no—
81#389C2×C414yes—
82#387—24no—
83#387—24no—
84#387—24no—
85#385—24no—
86#384—22no—
87#384C2×C411yescharacteristic
88#384C2×C411yescharacteristic
89#382—22no—
90#382—21no—
91#382—22no—
92#382C2×C414yes—
93#380—21no—
94#380C2312yes—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—44no—
103C4×C4⋊C4C2×C414yes—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4C2×C414yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—24no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D412yes—
116C2×#95C2311yescharacteristic
117C2×#95C2312yes—
118C2×#95C2311yescharacteristic
119#336—24no—
120#328C2×C418yes—
121#328C2×C418yes—
122#328C2×C4116yes—
123C22×Q8—21no—
124C22×Q8C22×C412yes—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8C4214yescomplemented
138C4⋊Q8C4214yescomplemented
139#125—24no—
140#125—24no—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8C22×C414yes—
146C22⋊Q8C22×C414yes—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—84no—
151C4×Q8C4218yescomplemented
152#117—24no—
153#117—24no—
154#117—42no—
155#117—42no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4SD1614yes—
180C2×C4⋊C4Q1614yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42—44no—
194C2×C42—44no—
195C2×C42—82no—
196C2×C42—82no—
197C2×C42C22×C414yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C412yes—
201C2×C42C22×C412yes—
202C2×C42C2×D414yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C4⋊C8—24no—
208C4⋊C8—24no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8C22×C414yes—
212C22⋊C8C22×C414yes—
213C4×C8—24no—
214C4×C8—24no—
215C4×C8—84no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—22no—
220#95—24no—
221#95—24no—
222#95—22no—
223#95—21no—
224#95—21no—
225#95—22no—
226#95—44no—
227#95—44no—
228#95—82no—
229#95C22⋊C414yes—
230#95C22⋊C414yes—
231#95C22×C412yes—
232#95C22×C412yes—
233#95C22×C414yes—
234C24C22×D411yescharacteristic
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—81no—
240C2×Q8—81no—
241C2×Q8C2×C4214yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4C2×C22⋊C414yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C414yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C4×D412yes—
284C22×C4C4×D412yes—
285C22×C4C4×D412yes—
286C22×C4C4×D412yes—
287C22×C4C2×SD1614yes—
288C22×C4C2×SD1612yes—
289C22×C4C2×Q1614yes—
290C22×C4C2×Q1612yes—
291C22×C4#13512yes—
292C22×C4#13512yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C23×C411yescharacteristic
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—28no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—48no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—48no—
320C4⋊C4—81no—
321C4⋊C4—81no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C4⋊C4#10318yes—
325C4⋊C4#10318yes—
326C4⋊C4C2×C4214yes—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C42—28no—
332C42—44no—
333C42—44no—
334C42—44no—
335C42—48no—
336C42—84no—
337C42—84no—
338C42—88no—
339C42C2×C4212yes—
340C42C2×C4212yes—
341C42#11712yes—
342C42#11712yes—
343C42C4×D412yes—
344C42C4×D412yes—
345C42C4×D412yes—
346C42C4×D412yes—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23C22×C22⋊C411yescharacteristic
352C23C2×C4×D411yescharacteristic
353C23C2×C4×D411yescharacteristic
354C23C22×SD1611yescharacteristic
355C23C22×Q1611yescharacteristic
356C23C2×#13511yescharacteristic
357C23C2×#13511yescharacteristic
358Q8—44no—
359Q8—44no—
360Q8—84no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—24no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4C4×C22⋊C412yes—
418C2×C4C4×C22⋊C412yes—
419C2×C4C4×C22⋊C412yes—
420C2×C4C4×C22⋊C412yes—
421C2×C4C2×#10318yes—
422C2×C4C2×#10312yes—
423C2×C4C2×#10312yes—
424C2×C4#41612yes—
425C2×C4#41612yes—
426C2×C4C4×SD1614yes—
427C2×C4C4×SD1614yes—
428C2×C4C4×Q1614yes—
429C2×C4C4×Q1614yes—
430C2×C4C22×C4211yescharacteristic
431C2×C4C2×#11711yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×C4×D411yescharacteristic
434C8—44no—
435C8—44no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22C2×C4×C22⋊C411yescharacteristic
449C22#288311yescharacteristic
450C22#288411yescharacteristic
451C22C22×#10311yescharacteristic
452C22C2×#41611yescharacteristic
453C22#290111yescharacteristic
454C22#290111yescharacteristic
455C22C2×C4×SD1611yescharacteristic
456C22C2×C4×SD1611yescharacteristic
457C22C2×C4×Q1611yescharacteristic
458C22C2×C4×Q1611yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—22no—
464C4—22no—
465C4—28no—
466C4—44no—
467C4—44no—
468C4—44no—
469C4—44no—
470C4—82no—
471C4—84no—
472C4—82no—
473C4—82no—
474C4C4×#10314yes—
475C4C4×#10314yes—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2#1761411yescharacteristic
481C2#1761611yescharacteristic
482C2#1761711yescharacteristic
483C2#1761911yescharacteristic
484C2#2039311yescharacteristic
485C2#2040511yescharacteristic
486C2C2×C4×#10311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2369C4semidirect
15. #2365C4semidirect
17. #2355C4semidirect
18. Q8×C4⋊C4C4semidirect
31. #1518C4semidirect
32. #1503C4semidirect
137. C4⋊Q8C42semidirect
138. C4⋊Q8C42semidirect
151. C4×Q8C42semidirect

Maximal subgroups

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

subgroupindex
#342132
#205332
C2×#15182
C2×#15032
#138042
#82212
C4×#3282
#80262
#80142