Finite Group Databasebeta

← #192830

Subgroups — #192830

Order 512.

1008 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6#8895C214yesmaximal, complemented
7#8884C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×#107—21no—
14C22×C4⋊C8C2211yescharacteristic
15#2370—22no—
16#2369—21no—
17#2365—21no—
18#2355—22no—
19#2355C2212yes—
20Q8×C4⋊C4C2212yes—
21#2329—22no—
22#2317—21no—
23C4×C22⋊Q8—22no—
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#2274—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990—21no—
29#1956—21no—
30#1947—22no—
31#1947—22no—
32#1947C2214yes—
33#1872—22no—
34#1852—22no—
35#1845C2212yes—
36#1834C2212yes—
37#1803C2214yes—
38C2×#340—21no—
39#1481C418yescomplemented
40C2×Q8⋊C8C2214yes—
41#1296C2212yes—
42#1296C2212yes—
43#1279C2214yes—
44C23×C8—21no—
45#533—22no—
46#533—22no—
47#533C2×C414yes—
48#533Q814yes—
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8D411yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8D412yes—
54C2×C4×Q8—22no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2312yes—
57C2×#117—21no—
58C2×#117—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66#431—22no—
67#431—22no—
68#431—22no—
69#431C2×C414yes—
70C2×#107—24no—
71C2×#107—24no—
72C2×#107—44no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8C2×C414yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#103—24no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8C2312yes—
84#391—24no—
85#391—22no—
86#389—22no—
87#389—22no—
88#389—24no—
89#389—22no—
90#387—24no—
91#387—24no—
92#387—24no—
93#385—24no—
94#384—22no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384C2312yes—
99#384C2311yescharacteristic
100#384C2311yescharacteristic
101#382—22no—
102#382—21no—
103#380—21no—
104#380D412yes—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4C2312yes—
117C4×C22⋊C4—24no—
118C2×#95—21no—
119#340—24no—
120#340—24no—
121#340—24no—
122Q8⋊C8C2×C4116yescomplemented
123C22×Q8—21no—
124C22×Q8C2×D412yes—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—41no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C22×C8—24no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—42no—
136C22×C8—42no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8C4⋊C414yes—
140C4⋊Q8C4⋊C414yes—
141#125—24no—
142#125—24no—
143C22⋊Q8—24no—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8C4⋊C418yes—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—84no—
152C4×Q8C22×C414yes—
153C4×Q8C2×Q814yes—
154#117—24no—
155#117—24no—
156#117—42no—
157#117—42no—
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—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4Q1614yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—82no—
199C2×C42C22×C412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×Q812yes—
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42C2411yescharacteristic
207#107—48no—
208#107—84no—
209C4⋊C8—24no—
210C4⋊C8—24no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8C22×C414yes—
215#103—84no—
216C22⋊C8—24no—
217C22⋊C8—42no—
218C4×C8—22no—
219C4×C8—42no—
220C4×C8C22×C414yes—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229C24#12011yescharacteristic
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—81no—
235C2×Q8—81no—
236C2×Q8C2×C4⋊C414yes—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4C2×C4⋊C412yes—
258C22×C4C4×D412yes—
259C22×C4C4×D412yes—
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4C22⋊Q812yes—
264C22×C4C22⋊Q812yes—
265C22×C4#12312yes—
266C22×C4C2×Q1612yes—
267C22×C4#13512yes—
268C22×C4AGL(1,ℤ/8)12yes—
269C22×C4#13712yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—48no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C4⋊C4—82no—
312C4⋊C4C2×C4⋊C414yes—
313C4⋊C4C2×Q1614yes—
314C22⋊C4—28no—
315C22⋊C4—44no—
316C42—24no—
317C42—24no—
318C42—44no—
319C42—44no—
320C42—82no—
321C42—84no—
322C42—84no—
323C42C2×C4⋊C412yes—
324C42C4×D412yes—
325C42C4×D412yes—
326C42C22⋊Q812yes—
327C42C22⋊Q812yes—
328C42#12312yes—
329C42C23×C411yescharacteristic
330C42C22×Q811yescharacteristic
331C42C2×#4111yescharacteristic
332C42C2×#4111yescharacteristic
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23#38512yes—
338C23#44912yes—
339C23#45112yes—
340C23C2×#12011yescharacteristic
341Q8—28no—
342Q8—44no—
343Q8—84no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—164no—
378C2×C4#38512yes—
379C2×C4#38512yes—
380C2×C4#38512yes—
381C2×C4C4×Q1614yes—
382C2×C4#44014yes—
383C2×C4#44912yes—
384C2×C4#45112yes—
385C2×C4#47714yes—
386C2×C4#47914yes—
387C2×C4C22×C4⋊C411yescharacteristic
388C2×C4C2×C4×D411yescharacteristic
389C2×C4C2×C4×D411yescharacteristic
390C2×C4C2×C22⋊Q811yescharacteristic
391C2×C4C2×C22⋊Q811yescharacteristic
392C2×C4C2×#12311yescharacteristic
393C2×C4C22×Q1611yescharacteristic
394C2×C4C2×#13511yescharacteristic
395C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
396C2×C4C2×#13711yescharacteristic
397C2×C4#57612yes—
398C2×C4#57812yes—
399C8—44no—
400C8—44no—
401C8—82no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22#186314yes—
409C22C2×#38511yescharacteristic
410C22C2×#45111yescharacteristic
411C22C2×#44911yescharacteristic
412C22#300811yescharacteristic
413C22#301111yescharacteristic
414C22#302111yescharacteristic
415C22#302211yescharacteristic
416C4—22no—
417C4—21no—
418C4—21no—
419C4—22no—
420C4—44no—
421C4—44no—
422C4—44no—
423C4—82no—
424C4—82no—
425C4—82no—
426C4—162no—
427C4#186314yes—
428C4#291312yes—
429C4C2×C4×Q1611yescharacteristic
430C4C2×#44011yescharacteristic
431C4C2×#47711yescharacteristic
432C4C2×#47911yescharacteristic
433C4#309812yes—
434C2—24no—
435C2#1890511yescharacteristic
436C2#1890611yescharacteristic
437C2#1891511yescharacteristic
438C2#1891611yescharacteristic
439C2#2084711yescharacteristic
440C2#2086511yescharacteristic
441C2C2×#186311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8895C2semidirect
7. #8884C2semidirect
39. #1481C4semidirect
122. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#219822
#218602
C2×#19472
C2×#14812
#88952
#88842
#80392
#78922