Finite Group Databasebeta

← #121121

Subgroups — #121121

Order 512.

612 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#8107C211yesmaximal, characteristic
2#8020C211yesmaximal, characteristic
3C2×#1294C211yesmaximal, characteristic
4#8008C211yesmaximal, characteristic
5#7983C211yesmaximal, characteristic
6#7925C211yesmaximal, characteristic
7#7909C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C22×#95C2211yescharacteristic
15#1305—22no—
16#1294—22no—
17#1294—21no—
18#1294—21no—
19#1293—22no—
20#1289—21no—
21#1289—21no—
22#1282—22no—
23#1280C414yescomplemented
24C24×C4—21no—
25C24×C4C2311yescharacteristic
26C22×M16—21no—
27C22×M16—21no—
28C22×M16—21no—
29C23×C8—21no—
30C2×#117—21no—
31C22×C4⋊C4—21no—
32C22×C22⋊C4—21no—
33#407—21no—
34#407—21no—
35#407—21no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407D412yes—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8C2×C412yes—
55C2×C22⋊C8C2×C412yes—
56C2×C22⋊C8D412yes—
57#381—21no—
58#380—21no—
59#380—21no—
60#380—21no—
61#380—21no—
62#380D411yescharacteristic
63#380D411yescharacteristic
64#380Q812yes—
65#379—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95C2×C412yes—
75C25#4111yescharacteristic
76C23×C4—22no—
77C23×C4—44no—
78C23×C4—41no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—82no—
82C23×C4—81no—
83C23×C4—81no—
84C23×C4—82no—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×Q811yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C2×M16—22no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—82no—
106C2×M16—82no—
107C2×M16—82no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—82no—
115#117—82no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—82no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—81no—
128C2×C22⋊C4—81no—
129C2×C42—41no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146#95—22no—
147#95—22no—
148#95—22no—
149#95—21no—
150#95—21no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155C24—21no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—42no—
160C24—42no—
161C24#11711yescharacteristic
162C24#12111yescharacteristic
163C24C22⋊Q811yescharacteristic
164C24#12611yescharacteristic
165C24#12611yescharacteristic
166C24#12611yescharacteristic
167C24#12611yescharacteristic
168C22×C4—24no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—82no—
191C22×C4—82no—
192C22×C4—82no—
193C22×C4—88no—
194C22×C4—88no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—82no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4#11711yescharacteristic
203C22×C4#11711yescharacteristic
204C22×C4C4×D411yescharacteristic
205C22×C4C4×D411yescharacteristic
206C22×C4C4×D411yescharacteristic
207C22×C4C4×Q811yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4C22⋊Q811yescharacteristic
211C22×C4C22⋊Q811yescharacteristic
212C22×C4#12311yescharacteristic
213C22×C4#12311yescharacteristic
214C22×C4#12311yescharacteristic
215C22×C4#12311yescharacteristic
216C22×C4#12311yescharacteristic
217C22×C4#12311yescharacteristic
218C22×C4#12411yescharacteristic
219C22×C4#12411yescharacteristic
220C22×C4#12411yescharacteristic
221C22×C4#12511yescharacteristic
222C22×C4#12511yescharacteristic
223C22×C4#12511yescharacteristic
224C22×C4#12611yescharacteristic
225C22×C4#12611yescharacteristic
226C22×C4#12611yescharacteristic
227C22×C4#12611yescharacteristic
228C22×C4#12611yescharacteristic
229C22×C4#12611yescharacteristic
230M16—82no—
231M16—82no—
232M16—82no—
233M16—82no—
234M16—82no—
235M16—82no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C2×C8—82no—
246C4⋊C4—82no—
247C4⋊C4—81no—
248C4⋊C4—81no—
249C4⋊C4—82no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C22⋊C4—81no—
257C22⋊C4—81no—
258C22⋊C4—82no—
259C42—82no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—44no—
287C23—44no—
288C23—44no—
289C23—44no—
290C23#38311yescharacteristic
291C23#38711yescharacteristic
292C23#38811yescharacteristic
293C23#39611yescharacteristic
294C23#39711yescharacteristic
295C23#40011yescharacteristic
296C23#40111yescharacteristic
297C23#47111yescharacteristic
298C23#47111yescharacteristic
299C23#47211yescharacteristic
300C23#47311yescharacteristic
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—82no—
331C2×C4—84no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—88no—
335C2×C4—82no—
336C2×C4—84no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—164no—
347C2×C4#38711yescharacteristic
348C2×C4#38711yescharacteristic
349C2×C4#38811yescharacteristic
350C2×C4#38811yescharacteristic
351C2×C4#39711yescharacteristic
352C2×C4#39711yescharacteristic
353C2×C4#40011yescharacteristic
354C2×C4#40011yescharacteristic
355C8—82no—
356C8—82no—
357C8—82no—
358C8—82no—
359C22—24no—
360C22—24no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22#144311yescharacteristic
392C22#184112yes—
393C22#193011yescharacteristic
394C22#193111yescharacteristic
395C22#206311yescharacteristic
396C22#206411yescharacteristic
397C22#207811yescharacteristic
398C22#207911yescharacteristic
399C22#210412yes—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—84no—
409C4—161no—
410C4—161no—
411C4—162no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—22no—
418C2—22no—
419C2—21no—
420C2—41no—
421C2—41no—
422C2#1129911yescharacteristic
423C2#1130311yescharacteristic
424C2#1131811yescharacteristic
subgroupG/H
center, socleC22#1443
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #1280C4semidirect

Maximal subgroups

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

subgroupindex
#81072
#80202
C2×#12942
#80082
#79832
#79252
#79092