Finite Group Databasebeta

← #201539

Subgroups — #201539

Order 512.

1185 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic, complemented
2#34971C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic, complemented
4#17443C212yesmaximal
5#9490C214yesmaximal, complemented
6#9488C214yesmaximal, complemented
7#7956C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2983—24no—
11#2983—22no—
12#2983C2214yes—
13C2×#446—24no—
14#2920C2212yes—
15C2×#432—22no—
16C2×#431C2212yes—
17#2599—22no—
18#2581C2212yes—
19#2536C2212yes—
20#2486—21no—
21#2484C412yes—
22#2409—21no—
23#2406C412yes—
24#2399C412yes—
25#2398C412yes—
26C4×C22⋊Q8—22no—
27C4×#123—22no—
28#2283C2211yescharacteristic
29#1821—21no—
30#1801C2214yes—
31#1784—21no—
32C4×#101—22no—
33#1488—24no—
34#1487C2214yes—
35#1486C2214yes—
36#1476—24no—
37#1286C2218yes—
38C22×#41C2×C411yescharacteristic
39C23×Q8D411yescharacteristic
40C2×#123—22no—
41C2×C22⋊Q8—22no—
42C4×#41—22no—
43C4×#41—22no—
44C4×#41—22no—
45C4×#41—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8D412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8C2311yescharacteristic
55C2×C4×Q8C2312yes—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2311yescharacteristic
60C2×#117—21no—
61C2×#117C2×C411yescharacteristic
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67#446—216no—
68#446—216no—
69#432—28no—
70#432—28no—
71#432—22no—
72#432—22no—
73#431—22no—
74#431—22no—
75#431—22no—
76#431—22no—
77#431D414yes—
78#431C2314yes—
79#423C2314yes—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82C2×#101—24no—
83#407—22no—
84C2×C22⋊C8—22no—
85C4×M16—22no—
86C2×#97—24no—
87C2×C4×C8D414yes—
88#400C2×C414yes—
89#398—22no—
90#398C2×C414yes—
91#397—22no—
92#395—22no—
93#395C2×C414yes—
94#394C2×C414yes—
95#393—22no—
96#391—22no—
97#389—22no—
98#388—24no—
99#387—24no—
100#387—24no—
101#386—22no—
102#385—22no—
103#385—24no—
104#383—21no—
105#382C2312yes—
106#381—22no—
107#381C2×C412yes—
108#380—21no—
109#380—22no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—24no—
114#376C2×C412yes—
115#343—24no—
116Q8⋊C8—28no—
117D4⋊C8—28no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—44no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—44no—
124C22×Q8C22⋊C412yes—
125C22×Q8C22×C411yescharacteristic
126C22×Q8C2×D412yes—
127C22×D4—21no—
128C22×D4C22×C411yescharacteristic
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×M16—24no—
138C22×C8—24no—
139#123—48no—
140C22⋊Q8—48no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—41no—
158C4×D4—41no—
159C4×D4—42no—
160C4×D4—42no—
161C4×D4—41no—
162C4×D4—41no—
163#117—22no—
164#117—22no—
165#117—42no—
166#117—42no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C414yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C414yes—
181C2×C4⋊C4C2×D412yes—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42C22⋊C414yes—
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D414yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42C2411yescharacteristic
208C4⋊C8—216no—
209C4⋊C8—24no—
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8—24no—
213#101—48no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—42no—
217C22⋊C8—42no—
218#97—216no—
219#97—42no—
220C4×C8—24no—
221C4×C8—42no—
222C4×C8C2×D418yes—
223#95—22no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—22no—
228#95—24no—
229#95—22no—
230#95C22×C414yes—
231C24—21no—
232C24C2×C22⋊C411yescharacteristic
233#41—28no—
234#41—44no—
235#41—44no—
236#41—44no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—22no—
242C2×Q8—44no—
243C2×Q8—42no—
244C2×Q8—48no—
245C2×Q8—48no—
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4C2×C22⋊C414yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C414yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C4×D412yes—
279C22×C4C4×D412yes—
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12112yes—
283C22×C4#12112yes—
284C22×C4#12311yescharacteristic
285C22×C4#12311yescharacteristic
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289M16—48no—
290C2×C8—28no—
291C2×C8—48no—
292C2×C8—44no—
293C4⋊C4—28no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—28no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—48no—
300C4⋊C4—48no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C22⋊C4—48no—
306C22⋊C4—48no—
307C22⋊C4—44no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—42no—
313C42—24no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—44no—
318C42—42no—
319C42—42no—
320C42—41no—
321C42—41no—
322C42C4×D412yes—
323C42C4×D412yes—
324C42#12012yes—
325C42#12112yes—
326C42#12112yes—
327C42#12312yes—
328C42C22×D412yes—
329C42C2×#4112yes—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23#38611yescharacteristic
339C23#38611yescharacteristic
340C23C22×C22⋊C411yescharacteristic
341Q8—24no—
342Q8—48no—
343Q8—42no—
344Q8—48no—
345D4—24no—
346D4—42no—
347D4—44no—
348D4—44no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—48no—
368C2×C4—41no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—41no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—84no—
385C2×C4#38611yescharacteristic
386C2×C4#38612yes—
387C2×C4#38612yes—
388C2×C4#38611yescharacteristic
389C2×C4C2×C4×D412yes—
390C2×C4C2×#12011yescharacteristic
391C2×C4C2×#12112yes—
392C2×C4C2×#12311yescharacteristic
393C2×C4#57514yes—
394C8—48no—
395C8—84no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—44no—
404C22—44no—
405C22—42no—
406C22—82no—
407C22—81no—
408C22—81no—
409C22#189112yes—
410C22C2×#38611yescharacteristic
411C22#289612yes—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—22no—
416C4—22no—
417C4—22no—
418C4—42no—
419C4—44no—
420C4—44no—
421C4—84no—
422C4—81no—
423C4—81no—
424C4#191212yes—
425C4#305414yes—
426C2—22no—
427C2—22no—
428C2—41no—
429C2—82no—
430C2C2×#189111yescharacteristic
431C2#2097011yescharacteristic
432C2C2×#191211yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2983C2semidirect
3. #20727C2semidirect
5. #9490C2semidirect
6. #9488C2semidirect
7. #7956C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#349712
#207272
#174432
#94902
#94882
#79562