Finite Group Databasebeta

← #207658

Subgroups — #207658

Order 512.

893 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#541C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20472C211yesmaximal, characteristic
4#9872C211yesmaximal, characteristic, complemented
5#9871C211yesmaximal, characteristic, complemented
6#9641C211yesmaximal, characteristic, complemented
7#9637C211yesmaximal, characteristic, complemented
8#9077C212yesmaximal
9#9075C212yesmaximal
10#8175C212yesmaximal, complemented
11#8173C212yesmaximal, complemented
12C2×#541C411yescharacteristic
13C2×#431C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2336—22no—
16#2334—22no—
17#2330C411yescharacteristic
18#2327C411yescharacteristic
19#2323C412yes—
20#2320C412yes—
21#2314C411yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×#125—22no—
24C4×#125C2211yescharacteristic
25C4×#126—22no—
26C4×C22⋊Q8C2212yes—
27C42×Q8C2212yes—
28C4×#117C2211yescharacteristic
29#1924—22no—
30#1917—22no—
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33#1797C2212yes—
34#1771—22no—
35#1767C2212yes—
36#1753—22no—
37#1744C2211yescharacteristic
38#1744C2211yescharacteristic
39C4×#97C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×Q8⋊C8C2212yes—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49C22×M16—21no—
50#541—24no—
51#541—24no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#126—22no—
54C2×#125—22no—
55C2×#125C2×C411yescharacteristic
56C2×C22⋊Q8C2×C412yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2312yes—
60C2×#117C2×C411yescharacteristic
61C2×#117C2×C411yescharacteristic
62C22×C42C2311yescharacteristic
63#431—22no—
64#431—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—22no—
71C4×M16—22no—
72C4×M16—22no—
73C2×#97—21no—
74C2×#97—22no—
75C2×#97—22no—
76C2×C4×C8—22no—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#391C2×C412yes—
82#391C2×C412yes—
83#389—22no—
84#389—22no—
85#389—22no—
86#389C2×C412yes—
87#389C2×C412yes—
88#388—22no—
89#388—22no—
90#388—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#385—22no—
101#384C2×C411yescharacteristic
102#383—22no—
103#382—22no—
104#382C2×C411yescharacteristic
105#380C2×C412yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4C2312yes—
117C4×C22⋊C4—22no—
118C4×C22⋊C4D412yes—
119C4×C22⋊C4D412yes—
120#376C2×C411yescharacteristic
121#376C2×C411yescharacteristic
122C43—22no—
123C43C2311yescharacteristic
124C43C2311yescharacteristic
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130Q8⋊C8—24no—
131Q8⋊C8—24no—
132Q8⋊C8—24no—
133Q8⋊C8—24no—
134C22×Q8C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—24no—
139C2×M16—44no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—22no—
147C4⋊Q8M1614yescomplemented
148#126—216no—
149#125—216no—
150#125—22no—
151#125M1614yescomplemented
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—42no—
161C4×Q8—42no—
162#117—22no—
163#117—22no—
164#117—41no—
165#117—41no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C22⋊C412yes—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2411yescharacteristic
206C4⋊C8—24no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—82no—
212C22⋊C8—24no—
213C22⋊C8—82no—
214#97—24no—
215#97—44no—
216#97—82no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—44no—
222C4×C8#4112yes—
223C4×C8#4112yes—
224C4×C8#4112yes—
225C4×C8#4112yes—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241C24C2×C22⋊C411yescharacteristic
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8C2×M1614yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275M16—48no—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—28no—
291C4⋊C4—28no—
292C4⋊C4—22no—
293C4⋊C4—28no—
294C4⋊C4—28no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4C2×M1614yes—
308C22⋊C4—28no—
309C22⋊C4—24no—
310C22⋊C4—42no—
311C22⋊C4—82no—
312C42—28no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—42no—
330C42—42no—
331C42—44no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—82no—
337C42—82no—
338C42C2×M1612yes—
339C42C2×M1612yes—
340C42C2×#4111yescharacteristic
341C42C2×#4111yescharacteristic
342C42C2×#4111yescharacteristic
343C42C2×#4111yescharacteristic
344C23—21no—
345C23—21no—
346C23—22no—
347C23—22no—
348C23—21no—
349C23C22×C22⋊C411yescharacteristic
350Q8—24no—
351Q8—24no—
352Q8—42no—
353Q8—42no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4#40714yes—
395C2×C4C4×#4111yescharacteristic
396C2×C4C4×#4111yescharacteristic
397C2×C4#52511yescharacteristic
398C2×C4#52511yescharacteristic
399C2×C4#53211yescharacteristic
400C2×C4#53211yescharacteristic
401C2×C4C22×M1611yescharacteristic
402C2×C4C22×M1611yescharacteristic
403C2×C4#56811yescharacteristic
404C2×C4#56811yescharacteristic
405C2×C4#57712yes—
406C2×C4#57812yes—
407C8—24no—
408C8—42no—
409C8—42no—
410C8—42no—
411C8—82no—
412C8—82no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22#229711yescharacteristic
419C22C2×#40711yescharacteristic
420C22#287911yescharacteristic
421C22#289812yes—
422C22#290112yes—
423C4—24no—
424C4—21no—
425C4—21no—
426C4—42no—
427C4—44no—
428C4—42no—
429C4—42no—
430C4—41no—
431C4—44no—
432C4—41no—
433C4—41no—
434C4—82no—
435C4—82no—
436C4#297011yescharacteristic
437C4#297011yescharacteristic
438C4#297111yescharacteristic
439C4#297111yescharacteristic
440C4#317411yescharacteristic
441C4#317711yescharacteristic
442C4#317812yes—
443C2—81no—
444C2#1745412yes—
445C2#1750612yes—
446C2#2138011yescharacteristic
447C2#2141311yescharacteristic
448C2#2141911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9872C2semidirect
5. #9871C2semidirect
6. #9641C2semidirect
7. #9637C2semidirect
10. #8175C2semidirect
11. #8173C2semidirect
147. C4⋊Q8M16semidirect
151. #125M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5412
#219262
#204722
#98722
#98712
#96412
#96372
#90772
#90752
#81752
#81732