Finite Group Databasebeta

← #121024

Subgroups — #121024

Order 512.

617 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21932C211yesmaximal, characteristic
2#8057C211yesmaximal, characteristic
3C2×#1297C211yesmaximal, characteristic
4#8012C211yesmaximal, characteristic
5#7942C211yesmaximal, characteristic
6#7925C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×C4×C22⋊C4C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1851—22no—
16#1850—22no—
17#1797—22no—
18#1298—22no—
19#1297—22no—
20#1294C412yescomplemented
21#1294C412yescomplemented
22#1284—22no—
23#1282—22no—
24#1282—22no—
25C24×C4C2311yescharacteristic
26C22×M16—21no—
27C22×M16Q812yes—
28C23×C8—21no—
29C23×C8—21no—
30C2×#117—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C42—21no—
34C22×C42—21no—
35C2×C4⋊C8—24no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44C2×C22⋊C8—21no—
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⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8C2×C412yes—
58C2×C22⋊C8D412yes—
59C2×C22⋊C8D412yes—
60#381—21no—
61#380—21no—
62#380C2×C411yescharacteristic
63#380C2×C411yescharacteristic
64#379—21no—
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4D412yes—
73C4×C22⋊C4D412yes—
74C4×C22⋊C4Q811yescharacteristic
75C4×C22⋊C4Q811yescharacteristic
76C2×#95—21no—
77C2×#95C2×C412yes—
78#336—24no—
79C25#4111yescharacteristic
80C23×C4—22no—
81C23×C4—41no—
82C23×C4—82no—
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×Q811yescharacteristic
89C23×C4C2×Q811yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—82no—
102C2×M16—82no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—82no—
112C22×C8—82no—
113#117—82no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C42—22no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—22no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42—82no—
135C2×C42—82no—
136C4⋊C8—84no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153#95—22no—
154#95—21no—
155#95—21no—
156#95—22no—
157#95—41no—
158#95—41no—
159#95C4⋊C412yes—
160#95C4⋊C412yes—
161C24—22no—
162C24—22no—
163C24—42no—
164C24—81no—
165C24C4×D411yescharacteristic
166C24C22⋊Q811yescharacteristic
167C24#12311yescharacteristic
168C24#12311yescharacteristic
169C24#12511yescharacteristic
170C24#12611yescharacteristic
171C24#12611yescharacteristic
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4C2×C4⋊C411yescharacteristic
199C22×C4#11711yescharacteristic
200C22×C4#11711yescharacteristic
201C22×C4C4×D411yescharacteristic
202C22×C4C4×D411yescharacteristic
203C22×C4C4×Q811yescharacteristic
204C22×C4#12011yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4C22⋊Q811yescharacteristic
209C22×C4C22⋊Q811yescharacteristic
210C22×C4C22⋊Q811yescharacteristic
211C22×C4C22⋊Q811yescharacteristic
212C22×C4#12311yescharacteristic
213C22×C4#12311yescharacteristic
214C22×C4#12311yescharacteristic
215C22×C4#12411yescharacteristic
216C22×C4#12411yescharacteristic
217C22×C4#12411yescharacteristic
218C22×C4#12411yescharacteristic
219C22×C4#12511yescharacteristic
220C22×C4#12511yescharacteristic
221C22×C4#12511yescharacteristic
222C22×C4#12511yescharacteristic
223C22×C4#12611yescharacteristic
224C22×C4#12611yescharacteristic
225C22×C4C4⋊Q811yescharacteristic
226M16—82no—
227M16—82no—
228M16—82no—
229M16—82no—
230C2×C8—44no—
231C2×C8—44no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—82no—
237C2×C8—82no—
238C2×C8—82no—
239C2×C8—82no—
240C2×C8—82no—
241C2×C8—82no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C42—44no—
255C42—82no—
256C42—82no—
257C42—81no—
258C42—81no—
259C42—82no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—21no—
266C23—21no—
267C23—22no—
268C23—22no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—84no—
283C23—84no—
284C23#38511yescharacteristic
285C23#38711yescharacteristic
286C23#38811yescharacteristic
287C23#39711yescharacteristic
288C23#39811yescharacteristic
289C23#40011yescharacteristic
290C23#40011yescharacteristic
291C23#44312yes—
292C23#44412yes—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
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—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—88no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4#38211yescharacteristic
340C2×C4#38711yescharacteristic
341C2×C4#38811yescharacteristic
342C2×C4#38911yescharacteristic
343C2×C4#39311yescharacteristic
344C2×C4#39411yescharacteristic
345C2×C4#39811yescharacteristic
346C2×C4#40011yescharacteristic
347C8—82no—
348C8—82no—
349C8—82no—
350C8—82no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—42no—
369C22—42no—
370C22—44no—
371C22—44no—
372C22—42no—
373C22—42no—
374C22—82no—
375C22—81no—
376C22—81no—
377C22—82no—
378C22#144211yescharacteristic
379C22#186612yes—
380C22#192212yes—
381C22#193711yescharacteristic
382C22#193811yescharacteristic
383C22#210212yes—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—44no—
392C4—82no—
393C4—84no—
394C4—161no—
395C4—161no—
396C4#203212yes—
397C2—22no—
398C2—21no—
399C2—21no—
400C2—22no—
401C2—22no—
402C2—41no—
403C2—41no—
404C2—41no—
405C2#1126111yescharacteristic
406C2#1127911yescharacteristic
407C2#1129111yescharacteristic
subgroupG/H
centerC2×C4#393
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC22#1442
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1294C4semidirect
21. #1294C4semidirect

Maximal subgroups

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

subgroupindex
#219322
#80572
C2×#12972
#80122
#79422
#79252
C2×#12822