Finite Group Databasebeta

← #120848

Subgroups — #120848

Order 512.

595 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21934C211yesmaximal, characteristic
2#8112C211yesmaximal, characteristic
3#8055C211yesmaximal, characteristic
4#8051C211yesmaximal, characteristic
5C2×#1297C211yesmaximal, characteristic
6#7942C211yesmaximal, characteristic
7#7929C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×C4×C22⋊C4C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1851—22no—
16#1851—22no—
17#1797—22no—
18#1306—22no—
19#1298—22no—
20#1297—22no—
21#1297C412yescomplemented
22#1297C412yescomplemented
23#1284—22no—
24#1282—22no—
25C24×C4C2311yescharacteristic
26C22×M16—21no—
27C22×M16—21no—
28C22×M16—21no—
29C22×M16Q812yes—
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—
44#407—21no—
45#407—21no—
46#407—21no—
47#407—21no—
48#407—21no—
49#407—21no—
50#407—21no—
51#407C2×C412yes—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8D412yes—
59C2×C22⋊C8D412yes—
60#381D412yes—
61#380—21no—
62#380—21no—
63#379—21no—
64C4×C22⋊C4—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⋊C4C2×C411yescharacteristic
72C4×C22⋊C4C2×C411yescharacteristic
73C4×C22⋊C4C2×C412yes—
74C4×C22⋊C4D412yes—
75C2×#95—21no—
76C2×#95Q811yescharacteristic
77C2×#95Q811yescharacteristic
78#336—24no—
79C25#4111yescharacteristic
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—41no—
83C23×C4—82no—
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×Q811yescharacteristic
90C23×C4C2×Q811yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C23×C4#4111yescharacteristic
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—82no—
107C2×M16—82no—
108C2×M16—82no—
109C2×M16—82no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—41no—
114#117—82no—
115C2×C4⋊C4—41no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42—82no—
134C2×C42—82no—
135C4⋊C8—84no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—41no—
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—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95—22no—
157#95—22no—
158#95—41no—
159#95C4⋊C412yes—
160#95C4⋊C412yes—
161C24—42no—
162C24—41no—
163C24—41no—
164C24—81no—
165C24C4×D411yescharacteristic
166C24C22⋊Q811yescharacteristic
167C24#12311yescharacteristic
168C24#12311yescharacteristic
169C24#12511yescharacteristic
170C24#12611yescharacteristic
171C24#12611yescharacteristic
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4C2×C4⋊C411yescharacteristic
201C22×C4#11711yescharacteristic
202C22×C4#11711yescharacteristic
203C22×C4C4×D411yescharacteristic
204C22×C4C4×D411yescharacteristic
205C22×C4C4×Q811yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4C22⋊Q811yescharacteristic
211C22×C4C22⋊Q811yescharacteristic
212C22×C4C22⋊Q811yescharacteristic
213C22×C4C22⋊Q811yescharacteristic
214C22×C4#12311yescharacteristic
215C22×C4#12311yescharacteristic
216C22×C4#12311yescharacteristic
217C22×C4#12411yescharacteristic
218C22×C4#12411yescharacteristic
219C22×C4#12411yescharacteristic
220C22×C4#12411yescharacteristic
221C22×C4#12511yescharacteristic
222C22×C4#12511yescharacteristic
223C22×C4#12511yescharacteristic
224C22×C4#12511yescharacteristic
225C22×C4#12611yescharacteristic
226C22×C4#12611yescharacteristic
227C22×C4C4⋊Q811yescharacteristic
228M16—82no—
229M16—82no—
230M16—82no—
231M16—82no—
232M16—82no—
233M16—82no—
234M16—82no—
235M16—82no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—82no—
243C2×C8—82no—
244C4⋊C4—82no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—44no—
252C22⋊C4—82no—
253C22⋊C4—81no—
254C22⋊C4—81no—
255C22⋊C4—82no—
256C42—44no—
257C42—82no—
258C42—82no—
259C42—82no—
260C42—81no—
261C42—81no—
262C23—22no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—21no—
268C23—21no—
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—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—84no—
287C23—84no—
288C23#38511yescharacteristic
289C23#38711yescharacteristic
290C23#38811yescharacteristic
291C23#39711yescharacteristic
292C23#39811yescharacteristic
293C23#40011yescharacteristic
294C23#40011yescharacteristic
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
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—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—84no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—84no—
337C2×C4—88no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4#38211yescharacteristic
342C2×C4#38711yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#38911yescharacteristic
345C2×C4#39311yescharacteristic
346C2×C4#39411yescharacteristic
347C2×C4#39811yescharacteristic
348C2×C4#40011yescharacteristic
349C8—82no—
350C8—82no—
351C8—82no—
352C8—82no—
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—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—44no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22#144211yescharacteristic
383C22#186111yescharacteristic
384C22#186111yescharacteristic
385C22#210212yes—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—44no—
394C4—82no—
395C4—82no—
396C4—82no—
397C4—162no—
398C4#203212yes—
399C2—21no—
400C2—21no—
401C2—22no—
402C2—41no—
403C2—41no—
404C2—41no—
405C2—41no—
406C2—41no—
407C2#1126111yescharacteristic
408C2#1127511yescharacteristic
409C2#1127911yescharacteristic
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
21. #1297C4semidirect
22. #1297C4semidirect

Maximal subgroups

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

subgroupindex
#219342
#81122
#80552
#80512
C2×#12972
#79422
#79292