Finite Group Databasebeta

← #201382

Subgroups — #201382

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#34971C211yesmaximal, characteristic
3#20735C212yesmaximal, complemented
4#20723C211yesmaximal, characteristic
5#9474C214yesmaximal, complemented
6#9471C214yesmaximal, complemented
7#7980C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2989—24no—
11#2983—21no—
12#2975—21no—
13#2973C2214yes—
14#2965—24no—
15#2920—22no—
16C2×#431C2212yes—
17#2599C414yescomplemented
18#2581—21no—
19#2536—21no—
20#2486C2212yes—
21#2484—21no—
22#2409C2212yes—
23#2406—21no—
24#2399C412yescomplemented
25#2398C412yescomplemented
26C4×C22⋊Q8—22no—
27C4×#123—22no—
28#2283C2212yes—
29#2283C2211yescharacteristic
30#1823C2214yes—
31#1820—21no—
32#1805—24no—
33#1799C2214yes—
34#1787C2214yes—
35#1782—21no—
36C4×#103—24no—
37C4×#99—22no—
38#1289C2218yes—
39C22×#41D411yescharacteristic
40C23×Q8C2311yescharacteristic
41C2×#123—22no—
42C2×C22⋊Q8—22no—
43C4×#41—22no—
44C4×#41—22no—
45C4×#41—22no—
46C4×#41—22no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D412yes—
59C2×C4×D4D411yescharacteristic
60C2×C4×D4C2312yes—
61C2×#117C2×C412yes—
62C2×#117D411yescharacteristic
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42D412yes—
67C22×C42C2311yescharacteristic
68#446—28no—
69#436—28no—
70#435—216no—
71#432—24no—
72#432—24no—
73#431—24no—
74#431—22no—
75#431—22no—
76#431—22no—
77#431—22no—
78#423—24no—
79C2×C4⋊C8D414yes—
80C2×#103—28no—
81C2×#99—24no—
82#407—24no—
83C2×C22⋊C8C2314yes—
84C4×M16—24no—
85C2×C4×C8D414yes—
86#400—22no—
87#398—22no—
88#398C2×C414yes—
89#397C2×C414yes—
90#395—22no—
91#395C2×C414yes—
92#394—22no—
93#393C2×C414yes—
94#391—22no—
95#389—22no—
96#388—24no—
97#387—24no—
98#387—24no—
99#386—22no—
100#385—24no—
101#385—22no—
102#383—22no—
103#383C2×C412yes—
104#382—21no—
105#382—22no—
106#381—21no—
107#381—22no—
108#381—22no—
109#380—22no—
110#380C2314yes—
111#380C2312yes—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—22no—
118#376—22no—
119#376C2×C412yes—
120#342—24no—
121#340—28no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—44no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—44no—
129C22×Q8C22⋊C411yescharacteristic
130C22×Q8C22×C412yes—
131C22×D4C22⋊C411yescharacteristic
132C22×D4C2×D412yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C2×M16—28no—
142C22×C8C2×D418yes—
143#123—48no—
144C22⋊Q8—48no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—42no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—42no—
158C4×Q8—41no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167#117—22no—
168#117—22no—
169#117—24no—
170#117—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C22×C412yes—
193C2×C22⋊C4C2×D412yes—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4112yes—
214C2×C42#4112yes—
215C4⋊C8—28no—
216C4⋊C8—24no—
217C4⋊C8—24no—
218#103—416no—
219#99—48no—
220C22⋊C8—28no—
221C22⋊C8—24no—
222C22⋊C8—24no—
223#97—28no—
224C4×C8—24no—
225C4×C8—24no—
226#95—22no—
227#95—24no—
228#95—22no—
229#95—24no—
230#95—24no—
231#95—22no—
232#95—24no—
233#95—24no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95C22×C414yes—
238C24C2×C22⋊C412yes—
239C24C22×D411yescharacteristic
240#41—48no—
241#41—48no—
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—28no—
245C2×Q8—24no—
246C2×Q8—42no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—42no—
250C2×Q8—48no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—42no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4#12112yes—
289C22×C4#12112yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D412yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4112yes—
294M16—88no—
295C2×C8—28no—
296C2×C8—48no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—48no—
315C22⋊C4—48no—
316C22⋊C4—44no—
317C22⋊C4—48no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—44no—
325C42—42no—
326C42—44no—
327C42—42no—
328C42—84no—
329C42C4×D412yes—
330C42C4×D412yes—
331C42C4×D412yes—
332C42C4×D412yes—
333C42#12012yes—
334C42#12011yescharacteristic
335C42#12011yescharacteristic
336C42#12311yescharacteristic
337C42#12312yes—
338C42#12311yescharacteristic
339C23—22no—
340C23—22no—
341C23—22no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—42no—
346C23C22×C22⋊C411yescharacteristic
347C23C2×#12112yes—
348C23#57514yes—
349Q8—28no—
350Q8—44no—
351Q8—44no—
352Q8—44no—
353Q8—44no—
354D4—44no—
355D4—48no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4#38611yescharacteristic
395C2×C4#38611yescharacteristic
396C2×C4#38611yescharacteristic
397C2×C4#38612yes—
398C2×C4#38612yes—
399C2×C4#38611yescharacteristic
400C2×C4C2×C4×D412yes—
401C2×C4C2×#12011yescharacteristic
402C2×C4C2×#12311yescharacteristic
403C8—48no—
404C22—21no—
405C22—22no—
406C22—21no—
407C22—22no—
408C22—44no—
409C22—44no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—82no—
414C22C2×#38611yescharacteristic
415C22#289612yes—
416C22#305414yes—
417C4—21no—
418C4—22no—
419C4—21no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—42no—
425C4—44no—
426C4—42no—
427C4—42no—
428C4—84no—
429C4—82no—
430C4—164no—
431C4#190912yes—
432C4#302512yes—
433C2—22no—
434C2—24no—
435C2—82no—
436C2#2095911yescharacteristic
437C2#2097011yescharacteristic
438C2C2×#190911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20735C2semidirect
5. #9474C2semidirect
6. #9471C2semidirect
7. #7980C2semidirect
17. #2599C4semidirect
24. #2399C4semidirect
25. #2398C4semidirect

Maximal subgroups

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

subgroupindex
#602772
#349712
#207352
#207232
#94742
#94712
#79802