Finite Group Databasebeta

← #205790

Subgroups — #205790

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3250C211yesmaximal, characteristic
2#21959C211yesmaximal, characteristic, complemented
3#21875C212yesmaximal, complemented
4#20544C211yesmaximal, characteristic, complemented
5#17581C212yesmaximal, complemented
6#9785C212yesmaximal
7#9784C212yesmaximal, complemented
8#9781C214yesmaximal, complemented
9C2×#529C2212yes—
10#3250—24no—
11#3250—22no—
12#3250—22no—
13C2×#479C412yescomplemented
14C2×#478C412yescomplemented
15C2×#467C2211yescharacteristic
16C2×#466C2211yescharacteristic
17C2×#465C2212yes—
18C2×#464C412yescomplemented
19C2×#463C412yescomplemented
20C2×#432C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22#2285C2212yes—
23#2283C2211yescharacteristic
24#1996C2214yes—
25#1964C2214yes—
26#1948—24no—
27#1899—24no—
28#1807C2212yes—
29#1807C2212yes—
30#1799—24no—
31#1771—22no—
32#1768—22no—
33#1752C2212yes—
34C8×C22⋊C4C2212yes—
35#1520C2214yes—
36#1519C2214yes—
37#1512—24no—
38#1509—24no—
39C22×Q16C2×C411yescharacteristic
40C22×SD16C2×C412yes—
41C22×D8C2×C411yescharacteristic
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44#529—22no—
45#529—22no—
46#529—22no—
47#529—22no—
48#529—22no—
49#529—22no—
50#529—22no—
51#529—22no—
52C2×#126—22no—
53C2×#125C2×C412yes—
54C2×#124C2×C412yes—
55C2×#123—22no—
56C2×C22⋊Q8C2312yes—
57C2×#121C2312yes—
58C2×C4×Q8C2×C412yes—
59C2×C4×D4—22no—
60C2×C4×D4C2×C412yes—
61C2×#117—22no—
62C2×#117—22no—
63C22×C42C2311yescharacteristic
64#479—24no—
65#479—24no—
66#478—24no—
67#478—24no—
68#467—22no—
69#467—22no—
70#466—22no—
71#466—22no—
72#465—24no—
73#465—24no—
74#464—24no—
75#464—24no—
76#463—24no—
77#463—24no—
78#432—22no—
79#432D414yes—
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#106—22no—
83C2×#106—22no—
84C2×#106C2×C412yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2×C412yes—
87C2×C4⋊C8C2×C412yes—
88C2×C4⋊C8D414yes—
89C2×C4⋊C8D412yes—
90C2×C4⋊C8D412yes—
91C2×#103C2×C412yes—
92C2×#103C2×C412yes—
93C2×#102C2×C412yes—
94C2×#102C2×C412yes—
95C2×#99—28no—
96C2×C22⋊C8C2312yes—
97C2×#97—21no—
98C2×#97—22no—
99C2×C4×C8—22no—
100#384—22no—
101#383—21no—
102#382—21no—
103#382—22no—
104#381D412yes—
105#380—22no—
106#380D412yes—
107#380C2314yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C22⋊C4C2312yes—
112#376—21no—
113#342—28no—
114#340—22no—
115#340—22no—
116#336—22no—
117#336—22no—
118#328—28no—
119#327—28no—
120C22×Q8C22×C412yes—
121C22×D4—22no—
122C22×D4C22×C412yes—
123C23×C4—22no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
127C2×Q16—22no—
128C2×Q16—81no—
129C2×SD16—24no—
130C2×SD16—82no—
131C2×D8—22no—
132C2×D8—81no—
133C22×C8—22no—
134C22×C8—81no—
135C22×C8—81no—
136C22×C8C22×C411yescharacteristic
137C22×C8C22×C412yes—
138C22×C8C22×C411yescharacteristic
139C22×C8C2×D412yes—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C22×C8C2×D412yes—
143#126—42no—
144#126—42no—
145#126—42no—
146#126—42no—
147#125—22no—
148#125—22no—
149#125—22no—
150#125—22no—
151#124—22no—
152#124—22no—
153#124—22no—
154#124—22no—
155#123—42no—
156#123—42no—
157#123—42no—
158#123—42no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163#121—22no—
164#121—22no—
165#121—22no—
166#121—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—42no—
179#117—42no—
180#117—42no—
181#117—42no—
182#117—42no—
183#117—84no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4C2×D414yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—42no—
204C2×C42C22×C412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42#4111yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4112yes—
212#107—22no—
213#107—22no—
214#106—24no—
215#106—82no—
216#106—82no—
217C4⋊C8—24no—
218C4⋊C8C22⋊C414yes—
219C4⋊C8C22⋊C414yes—
220C4⋊C8C22⋊C414yes—
221C4⋊C8C22⋊C414yes—
222#103—24no—
223#103—24no—
224#102—24no—
225#102—24no—
226#99—88no—
227C22⋊C8—22no—
228C22⋊C8C2×D414yes—
229#97—24no—
230#97—82no—
231C4×C8—82no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—24no—
237#95—42no—
238C24—22no—
239C24C22×D411yescharacteristic
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—44no—
251C2×D4—44no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—81no—
263C22×C4—84no—
264C22×C4C4×D412yes—
265C22×C4C4×D412yes—
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12712yes—
271C22×C4#12712yes—
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4111yescharacteristic
278Q16—82no—
279SD16—84no—
280D8—82no—
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8C2×C22⋊C412yes—
292C2×C8C2×C22⋊C414yes—
293C2×C8C2×C22⋊C412yes—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—84no—
309C4⋊C4—82no—
310C4⋊C4—84no—
311C22⋊C4—28no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—48no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—84no—
328C42C2×C22⋊C412yes—
329C42C2×C22⋊C412yes—
330C42C4×D412yes—
331C42C4×D412yes—
332C42#12112yes—
333C42#12112yes—
334C42#12412yes—
335C42#12412yes—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—44no—
340C23—82no—
341C23—82no—
342C23C2×C4×D411yescharacteristic
343C23C2×#12111yescharacteristic
344C23C2×#12711yescharacteristic
345C23#57714yes—
346Q8—42no—
347Q8—42no—
348Q8—82no—
349D4—42no—
350D4—42no—
351D4—82no—
352D4—82no—
353D4—82no—
354D4—84no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4#39012yes—
388C2×C4#39012yes—
389C2×C4#39012yes—
390C2×C4#39012yes—
391C2×C4C22×C22⋊C411yescharacteristic
392C2×C4C2×C4×D411yescharacteristic
393C2×C4C2×#12111yescharacteristic
394C2×C4C2×#12411yescharacteristic
395C8—22no—
396C8—24no—
397C8—22no—
398C8—44no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—44no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—164no—
410C22C2×#39011yescharacteristic
411C22#295212yes—
412C22#306012yes—
413C22#315412yes—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—22no—
421C4—84no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—164no—
426C4#288311yescharacteristic
427C4#288411yescharacteristic
428C2—22no—
429C2—22no—
430C2—22no—
431C2—162no—
432C2#2121411yescharacteristic
433C2#2122911yescharacteristic
434C2#2124411yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#466C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21959C2semidirect
3. #21875C2semidirect
4. #20544C2semidirect
5. #17581C2semidirect
7. #9784C2semidirect
8. #9781C2semidirect
13. C2×#479C4semidirect
14. C2×#478C4semidirect
18. C2×#464C4semidirect
19. C2×#463C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32502
#219592
#218752
#205442
#175812
#97852
#97842
#97812