Finite Group Databasebeta

← #192438

Subgroups — #192438

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic
6#8871C218yesmaximal
7#8014C211yesmaximal, characteristic
8#7908C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2370—22no—
13#2369—21no—
14#2365—21no—
15#2355—21no—
16#2355—22no—
17Q8×C4⋊C4—21no—
18#2329C2214yes—
19#2317—21no—
20C4×C22⋊Q8—22no—
21C2×#389—22no—
22#2285C2211yescharacteristic
23C2×#384—21no—
24C2×#384C2211yescharacteristic
25#2274—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990—21no—
29#1956—21no—
30#1948—22no—
31#1872C2214yes—
32#1853C2214yes—
33#1844—21no—
34#1834—21no—
35#1803—22no—
36#1503C418yescomplemented
37C2×#328C2214yes—
38#1444C2214yes—
39C4⋊#95—21no—
40#1294C2214yes—
41#1280C2214yes—
42#533—22no—
43#533—22no—
44#533C2×C414yes—
45#533Q814yes—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8C2312yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—21no—
53C2×C4×Q8D412yes—
54C2×#117—21no—
55C2×#117—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C42—21no—
64C22×C42D412yes—
65C22×C42C2311yescharacteristic
66#431—22no—
67#431—22no—
68#431—22no—
69#431C2×C414yes—
70C2×#106—24no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×#103—24no—
74C2×C22⋊C8C2312yes—
75C2×C22⋊C8C2312yes—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#391—22no—
79#391—24no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—24no—
84#389—48no—
85#387—24no—
86#387—24no—
87#387—24no—
88#385—24no—
89#384—22no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384—44no—
95#384D411yescharacteristic
96#384D411yescharacteristic
97#382—22no—
98#382—21no—
99#380—21no—
100#380C2312yes—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4C2×C414yes—
114C4×C4⋊C4D412yes—
115C4×C22⋊C4—24no—
116C2×#95—24no—
117C2×#95C2312yes—
118C2×#95C2312yes—
119#340—24no—
120#340—24no—
121#328C2×C4116yes—
122C22×Q8—21no—
123C22×Q8C2×D412yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8C4⋊C414yescomplemented
136C4⋊Q8C4⋊C414yescomplemented
137#125—24no—
138#125—24no—
139C22⋊Q8—24no—
140C22⋊Q8—24no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8C22×C414yes—
144C22⋊Q8C2×Q814yes—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—84no—
149C4×Q8C4⋊C418yescomplemented
150#117—24no—
151#117—24no—
152#117—42no—
153#117—42no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—84no—
188C2×C4⋊C4SD1614yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—22no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—44no—
204C2×C42C22×C412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q812yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212#106—84no—
213C4⋊C8—24no—
214C4⋊C8—42no—
215#103—84no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8C22×C414yes—
220C4×C8—24no—
221C4×C8—42no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—48no—
232#95C22×C414yes—
233C24C22×D411yescharacteristic
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8C2×C4⋊C414yes—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4C4×D412yes—
271C22×C4C4×D412yes—
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4C22⋊Q812yes—
277C22×C4C22⋊Q812yes—
278C22×C4C2×SD1614yes—
279C22×C4C2×SD1612yes—
280C22×C4#13512yes—
281C22×C4#13712yes—
282C22×C4#13712yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×Q811yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C2×C8—24no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—216no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—28no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C4⋊C4—816no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4C2×C4⋊C414yes—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C42—28no—
328C42—48no—
329C42—44no—
330C42—44no—
331C42—82no—
332C42—84no—
333C42—84no—
334C42C2×C4⋊C412yes—
335C42C2×C4⋊C412yes—
336C42C4×D412yes—
337C42C4×D412yes—
338C42C22⋊Q812yes—
339C42C22⋊Q812yes—
340C42#12312yes—
341C42#12312yes—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23C2×C4×D411yescharacteristic
347C23C2×#12011yescharacteristic
348C23C2×C22⋊Q811yescharacteristic
349C23C22×SD1611yescharacteristic
350C23C2×#13511yescharacteristic
351C23C2×#13711yescharacteristic
352C23C2×#13711yescharacteristic
353Q8—44no—
354Q8—48no—
355Q8—82no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—28no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
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—84no—
398C2×C4—84no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#38512yes—
403C2×C4#38512yes—
404C2×C4#38512yes—
405C2×C4#38512yes—
406C2×C4C4×SD1614yes—
407C2×C4#44114yes—
408C2×C4#44812yes—
409C2×C4#44812yes—
410C2×C4#45212yes—
411C2×C4#45212yes—
412C2×C4#45312yes—
413C2×C4#45612yes—
414C2×C4#47514yes—
415C2×C4#47914yes—
416C2×C4C22×C4⋊C411yescharacteristic
417C2×C4C2×C4×D411yescharacteristic
418C2×C4C2×C22⋊Q811yescharacteristic
419C2×C4C2×#12311yescharacteristic
420C8—44no—
421C8—82no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22C2×#38511yescharacteristic
435C22C2×C4×SD1611yescharacteristic
436C22C2×#44111yescharacteristic
437C22C2×#44811yescharacteristic
438C22C2×#45211yescharacteristic
439C22C2×#45311yescharacteristic
440C22C2×#45611yescharacteristic
441C22#302711yescharacteristic
442C22#302811yescharacteristic
443C22C2×#47511yescharacteristic
444C22C2×#47911yescharacteristic
445C4—22no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—48no—
451C4—44no—
452C4—44no—
453C4—44no—
454C4—82no—
455C4—82no—
456C4—82no—
457C4—162no—
458C4#185914yes—
459C4#186514yes—
460C4#186514yes—
461C2—21no—
462C2—21no—
463C2—21no—
464C2—21no—
465C2#1838911yescharacteristic
466C2#1839211yescharacteristic
467C2#1841011yescharacteristic
468C2#1841311yescharacteristic
469C2C2×#185911yescharacteristic
470C2C2×#186511yescharacteristic
471C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
36. #1503C4semidirect
135. C4⋊Q8C4⋊C4semidirect
136. C4⋊Q8C4⋊C4semidirect
149. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#219812
#218602
C2×#15032
#139102
#88712
#80142
#79082