Finite Group Databasebeta

← #195301

Subgroups — #195301

Order 512.

1242 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3226C211yesmaximal, characteristic
2#34388C211yesmaximal, characteristic
3#21840C211yesmaximal, characteristic, complemented
4C2×#1514C211yesmaximal, characteristic
5#14663C211yesmaximal, characteristic, complemented
6#9043C218yesmaximal, complemented
7#8025C211yesmaximal, characteristic, complemented
8#7916C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3226—24no—
12#3226C418yescomplemented
13C2×#500—21no—
14C2×#486—21no—
15C2×#480C2214yes—
16C2×#474—22no—
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2381—21no—
20#2380—21no—
21#2364—22no—
22#2353—22no—
23#2339—22no—
24#2337—22no—
25#2325C2214yes—
26#2318—22no—
27#2283C2211yescharacteristic
28C2×#383C2211yescharacteristic
29C2×#383C2211yescharacteristic
30#1993—22no—
31#1987—22no—
32#1823C2214yes—
33#1514—24no—
34C2×#327C2214yes—
35#1446—22no—
36#1445—22no—
37#1445C2214yes—
38#1294C2214yes—
39#1280C2214yes—
40C22×#41—21no—
41C23×Q8—21no—
42#532—22no—
43#532—22no—
44#532C2×C414yes—
45#532D414yes—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yescomplemented
54C2×C4⋊Q8D411yescharacteristic
55C2×#127D411yescharacteristic
56C2×#124—21no—
57C2×#121—21no—
58C2×#121C2312yes—
59C2×C4×D4—24no—
60C2×C4×D4—21no—
61C2×C4×D4D412yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2312yes—
64C22×C42C2311yescharacteristic
65#500—28no—
66#486—28no—
67#480—28no—
68#480C2×C4116yescomplemented
69#474—216no—
70#432—22no—
71#432—22no—
72#431—22no—
73#431C2×C414yescomplemented
74C2×#107—24no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×#102—24no—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2312yes—
80C2×#97—21no—
81C2×C4×C8—21no—
82#391—24no—
83#388—24no—
84#388—24no—
85#387—24no—
86#387—24no—
87#386—24no—
88#386—24no—
89#386—24no—
90#386—24no—
91#385—24no—
92#383—21no—
93#383—21no—
94#383—21no—
95#383—21no—
96#383—21no—
97#383—22no—
98#383—22no—
99#382D412yes—
100#381—22no—
101#381—21no—
102#380—22no—
103#380C2312yes—
104C4×C4⋊C4D412yes—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—21no—
108#376—21no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—28no—
113C2×#95D412yes—
114C2×#95D412yes—
115C2×#95C2312yes—
116C2×#95C2312yes—
117#340—28no—
118#327—28no—
119C2×#41—24no—
120C2×#41—24no—
121C2×#41—84no—
122C22×Q8—24no—
123C22×Q8—21no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—42no—
127C22×Q8—84no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4C2×D412yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D414yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—22no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8C22⋊C414yes—
152#127—22no—
153#127C22⋊C414yes—
154#124—24no—
155#124—24no—
156#121—24no—
157#121—24no—
158#121—22no—
159#121—22no—
160#121C22×C414yes—
161#121C2×D414yes—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—88no—
166C4×D4C22⋊C418yes—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4C22×C414yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D414yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C42—24no—
188C2×C42—22no—
189C2×C42C22×C412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42#4112yes—
196C2×C42#4112yes—
197#107—216no—
198C4⋊C8—24no—
199C4⋊C8—42no—
200#102—216no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8C22×C414yes—
205#97—42no—
206C4×C8—24no—
207#95—24no—
208#95—24no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—42no—
222#95—42no—
223#95—88no—
224C24—22no—
225C24C22×D411yescharacteristic
226#41—88no—
227#41—88no—
228C2×Q8—24no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—88no—
232C2×Q8—82no—
233C2×Q8—84no—
234C2×Q8—84no—
235C2×Q8—88no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—22no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—48no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4C2×C22⋊C414yes—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—28no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—48no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4C4×D412yes—
270C22×C4C4×D412yes—
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12112yes—
276C22×C4#12112yes—
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4111yescharacteristic
283C2×C8—24no—
284C2×C8—44no—
285C2×C8—44no—
286C2×C8—42no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—84no—
297C4⋊C4C2×C22⋊C414yes—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—28no—
301C22⋊C4—44no—
302C22⋊C4—84no—
303C22⋊C4—84no—
304C22⋊C4—84no—
305C42—82no—
306C42—84no—
307C42C2×C22⋊C412yes—
308C42C2×C22⋊C412yes—
309C42C4×D412yes—
310C42C4×D412yes—
311C42#12112yes—
312C42#12112yes—
313C42#12312yes—
314C42#12312yes—
315C23—22no—
316C23—22no—
317C23—24no—
318C23—44no—
319C23—44no—
320C23—42no—
321C23—42no—
322C23C2×C4×D411yescharacteristic
323C23C2×#12011yescharacteristic
324C23C2×#12111yescharacteristic
325C23#57614yes—
326Q8—84no—
327Q8—88no—
328Q8—88no—
329D4—44no—
330D4—48no—
331D4—48no—
332D4—82no—
333D4—84no—
334D4—84no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—88no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—88no—
363C2×C4—88no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—168no—
368C2×C4#38612yes—
369C2×C4#38612yes—
370C2×C4#38612yes—
371C2×C4#38612yes—
372C2×C4C22×C22⋊C411yescharacteristic
373C2×C4C2×C4×D411yescharacteristic
374C2×C4C2×#12111yescharacteristic
375C2×C4C2×#12311yescharacteristic
376C8—44no—
377C8—82no—
378C22—28no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—44no—
384C22—44no—
385C22—44no—
386C22—84no—
387C22—84no—
388C22C2×#38611yescharacteristic
389C22#294812yes—
390C22#300812yes—
391C22#302212yes—
392C22#302812yes—
393C22#304112yes—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—44no—
403C4—84no—
404C4—84no—
405C4—84no—
406C4—162no—
407C2—22no—
408C2—22no—
409C2—84no—
410C2#1847014yes—
411C2#2093411yescharacteristic
412C2#2094411yescharacteristic
413C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
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
3. #21840C2semidirect
5. #14663C2semidirect
6. #9043C2semidirect
7. #8025C2semidirect
8. #7916C2semidirect
12. #3226C4semidirect
53. C2×C4⋊Q8C2×C4semidirect
68. #480C2×C4semidirect
73. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32262
#343882
#218402
C2×#15142
#146632
#90432
#80252
#79162