Finite Group Databasebeta

← #188529

Subgroups — #188529

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic
3C2×#1518C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#13955C211yesmaximal, characteristic
6#8576C214yesmaximal
7#8574C214yesmaximal
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⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1939—22no—
28#1939—22no—
29#1797C2214yes—
30#1518—24no—
31#1503—24no—
32C2×#328C2212yes—
33C2×#328C2212yes—
34C2×#328C2214yes—
35#1442—22no—
36C4⋊#95—21no—
37C4⋊#95—21no—
38#1434C2212yes—
39#1434C2212yes—
40#1294C2212yes—
41#1294C2212yes—
42#1294C2214yes—
43#533—22no—
44#533—22no—
45#533D414yes—
46#533Q814yes—
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×C42C2312yes—
63C22×C42C2311yescharacteristic
64#431—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—24no—
71C2×C22⋊C8C2312yes—
72C2×C22⋊C8C2312yes—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75#391—24no—
76#391C2×C414yes—
77#389—22no—
78#389—22no—
79#389—24no—
80#389C2×C414yes—
81#387—24no—
82#387—24no—
83#387—24no—
84#385—24no—
85#384—22no—
86#384C2×C411yescharacteristic
87#384C2×C411yescharacteristic
88#382—22no—
89#382—21no—
90#382—22no—
91#382—22no—
92#380—21no—
93#380C2312yes—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C414yes—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C22⋊C4—24no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D412yes—
115C2×#95C2311yescharacteristic
116C2×#95C2311yescharacteristic
117C2×#95C2312yes—
118#336—24no—
119#328—24no—
120#328—24no—
121#328—28no—
122C22×Q8—21no—
123C22×Q8C22×C412yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8C4⋊C414yes—
137C4⋊Q8C4⋊C414yes—
138#125—24no—
139#125—24no—
140C22⋊Q8—24no—
141C22⋊Q8—24no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8C2×D414yes—
145C22⋊Q8C2×Q814yes—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—84no—
150C4×Q8C4⋊C418yes—
151#117—24no—
152#117—24no—
153#117—42no—
154#117—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42—82no—
193C2×C42C22×C414yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D414yes—
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×Q812yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203C4⋊C8—42no—
204C4⋊C8—42no—
205C4⋊C8—84no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C4×C8—42no—
211C4×C8—42no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—22no—
216#95—24no—
217#95—22no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95—21no—
222#95—21no—
223#95—22no—
224#95—44no—
225#95—44no—
226#95—42no—
227#95—82no—
228#95—82no—
229#95C22⋊C414yes—
230#95C22⋊C414yes—
231C24C22×D411yescharacteristic
232C2×Q8—22no—
233C2×Q8—24no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—81no—
237C2×Q8—81no—
238C2×Q8C2×C4⋊C414yes—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4C2×C22⋊C414yes—
277C22×C4C2×C22⋊C414yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4#12112yes—
283C22×C4#12112yes—
284C22×C4C22⋊Q812yes—
285C22×C4C22⋊Q812yes—
286C22×C4#13512yes—
287C22×C4#13512yes—
288C22×C4#13712yes—
289C22×C4#13712yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×Q811yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—28no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—48no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—81no—
319C4⋊C4—81no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4C2×C4⋊C414yes—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C42—28no—
329C42—44no—
330C42—44no—
331C42—44no—
332C42—84no—
333C42—84no—
334C42—84no—
335C42C2×C4⋊C412yes—
336C42C2×C4⋊C412yes—
337C42#11712yes—
338C42#11712yes—
339C42#12112yes—
340C42#12112yes—
341C42C22⋊Q812yes—
342C42C22⋊Q812yes—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23C22×C22⋊C411yescharacteristic
348C23C2×#12111yescharacteristic
349C23C2×C22⋊Q811yescharacteristic
350C23C2×#13511yescharacteristic
351C23C2×#13511yescharacteristic
352C23C2×#13711yescharacteristic
353C23C2×#13711yescharacteristic
354Q8—44no—
355Q8—44no—
356Q8—84no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
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—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4#38012yes—
415C2×C4#38012yes—
416C2×C4#38012yes—
417C2×C4#38012yes—
418C2×C4#41612yes—
419C2×C4#41612yes—
420C2×C4#41912yes—
421C2×C4#41912yes—
422C2×C4#46414yes—
423C2×C4#46414yes—
424C2×C4#47914yes—
425C2×C4#47914yes—
426C2×C4C22×C4⋊C411yescharacteristic
427C2×C4C2×#11711yescharacteristic
428C2×C4C2×#12111yescharacteristic
429C2×C4C2×C22⋊Q811yescharacteristic
430C8—82no—
431C8—82no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22C2×#38011yescharacteristic
445C22#288311yescharacteristic
446C22#288411yescharacteristic
447C22C2×#41611yescharacteristic
448C22C2×#41911yescharacteristic
449C22#290111yescharacteristic
450C22#290111yescharacteristic
451C22C2×#46411yescharacteristic
452C22C2×#46411yescharacteristic
453C22C2×#47911yescharacteristic
454C22C2×#47911yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—22no—
460C4—22no—
461C4—44no—
462C4—44no—
463C4—44no—
464C4—44no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4—82no—
469C4—162no—
470C4#180514yes—
471C4#180514yes—
472C2—21no—
473C2—21no—
474C2—21no—
475C2—21no—
476C2#1761611yescharacteristic
477C2#1761711yescharacteristic
478C2#1762711yescharacteristic
479C2#1763211yescharacteristic
480C2#2063411yescharacteristic
481C2C2×#180511yescharacteristic
482C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×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

Maximal subgroups

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

subgroupindex
#342132
#219512
C2×#15182
C2×#15032
#139552
#85762
#85742
#80262
#80142