Finite Group Databasebeta

← #192181

Subgroups — #192181

Order 512.

1098 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34380C211yesmaximal, characteristic
2#21944C211yesmaximal, characteristic
3#21940C211yesmaximal, characteristic
4C2×#1472C211yesmaximal, characteristic
5#14660C211yesmaximal, characteristic
6#8856C214yesmaximal, complemented
7#8855C214yesmaximal, complemented
8#8024C211yesmaximal, characteristic
9#8015C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2378—22no—
14#2360—22no—
15#2344—21no—
16#2342—22no—
17Q8×C22⋊C4—21no—
18#2329—22no—
19#2323—22no—
20#2316C2212yes—
21#2313C2212yes—
22#2280C2211yescharacteristic
23C2×#381C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26#1854—22no—
27#1853C2212yes—
28#1852—22no—
29#1852C2214yes—
30#1852C2212yes—
31#1809—22no—
32#1808—22no—
33#1472C418yescomplemented
34C2×#324C2214yes—
35#1446—21no—
36#1445—21no—
37#1441C2214yes—
38#1440—22no—
39#1294C2214yes—
40#1294C2212yes—
41#1294C2212yes—
42C23×Q8D411yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—21no—
45C23×C8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8C2×C414yes—
53C2×C22⋊Q8Q814yes—
54C2×C4×Q8—22no—
55C2×#117—22no—
56C2×#117D412yes—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D412yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4D412yes—
65C2×#107—24no—
66C2×#107—22no—
67C2×#106—22no—
68#407—22no—
69#407C2×C414yes—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8C2×C414yes—
74C2×C22⋊C8C2312yes—
75C2×C22⋊C8C2312yes—
76#391—22no—
77#391—24no—
78#389—22no—
79#389—24no—
80#389—24no—
81#388—24no—
82#388—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#385—24no—
87#381—22no—
88#381—21no—
89#381—22no—
90#381—22no—
91#380—22no—
92#380—21no—
93#380—21no—
94#380—21no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380D412yes—
99#380D412yes—
100#380C2311yescharacteristic
101#380C2312yes—
102#380C2311yescharacteristic
103#379—21no—
104C4×C4⋊C4—22no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—24no—
113C2×#95—24no—
114C2×#95C2312yes—
115#340—24no—
116#340—22no—
117#340—22no—
118#324C2×C4116yescomplemented
119C25C2×D411yescharacteristic
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—82no—
124C22×Q8C4⋊C414yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—82no—
129C23×C4C22×C412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×Q812yes—
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4C2411yescharacteristic
141C2×M16—24no—
142C2×M16—82no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—24no—
146C22×C8—82no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C22⋊Q8—24no—
151C22⋊Q8—24no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8—24no—
155C4×Q8—84no—
156#117—82no—
157#117—84no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4C4⋊C414yes—
173C2×C4⋊C4C22×C414yes—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×Q814yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4C4⋊C418yes—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—24no—
197#107—84no—
198#107—82no—
199#106—82no—
200C22⋊C8—24no—
201C22⋊C8—24no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8C22×C414yes—
205C22⋊C8C22×C414yes—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95—22no—
216#95—22no—
217#95—41no—
218#95—41no—
219#95—42no—
220#95—84no—
221#95—84no—
222C24—22no—
223C24—22no—
224C24—42no—
225C24—42no—
226C24C2×C4⋊C412yes—
227C24#12011yescharacteristic
228C24#12011yescharacteristic
229C24#12312yes—
230C24C22×D411yescharacteristic
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—44no—
234C2×Q8—82no—
235C2×Q8—84no—
236C2×Q8—84no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4C2×C4⋊C414yes—
270C22×C4C2×C4⋊C414yes—
271C22×C4C2×C4⋊C412yes—
272C22×C4C4×D412yes—
273C22×C4C4×D412yes—
274C22×C4C4×D412yes—
275C22×C4C4×D412yes—
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4C22⋊Q812yes—
279C22×C4C22⋊Q812yes—
280C22×C4C22⋊Q812yes—
281C22×C4C22⋊Q812yes—
282C22×C4#12312yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×Q811yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289M16—84no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—44no—
294C2×C8—42no—
295C2×C8—84no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C22⋊C4—28no—
312C22⋊C4—28no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—84no—
322C42—44no—
323C42—84no—
324C42—84no—
325C23—22no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—24no—
336C23—24no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—84no—
342C23—84no—
343C23#38512yes—
344C23#38512yes—
345C23C22×C4⋊C411yescharacteristic
346C23C2×#12011yescharacteristic
347C23C2×#12311yescharacteristic
348C23#57612yes—
349C23#57812yes—
350Q8—48no—
351Q8—84no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—28no—
357C2×C4—24no—
358C2×C4—24no—
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—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—82no—
392C2×C4—88no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—164no—
398C2×C4#38512yes—
399C2×C4#38512yes—
400C2×C4C2×C4×D411yescharacteristic
401C2×C4C2×C4×D411yescharacteristic
402C2×C4C2×C22⋊Q811yescharacteristic
403C2×C4C2×C22⋊Q811yescharacteristic
404C8—44no—
405C8—44no—
406C22—24no—
407C22—24no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—84no—
423C22—84no—
424C22C2×#38511yescharacteristic
425C22#291312yes—
426C22#300811yescharacteristic
427C22#301111yescharacteristic
428C22#302012yes—
429C22#302612yes—
430C22#309812yes—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—22no—
436C4—22no—
437C4—44no—
438C4—44no—
439C4—82no—
440C4—82no—
441C4—84no—
442C4—162no—
443C4—162no—
444C2—22no—
445C2—22no—
446C2—42no—
447C2—42no—
448C2#1845512yes—
449C2#1845612yes—
450C2#2084711yescharacteristic
451C2#2085711yescharacteristic
452C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8856C2semidirect
7. #8855C2semidirect
33. #1472C4semidirect
118. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343802
#219442
#219402
C2×#14722
#146602
#88562
#88552
#80242
#80152