Finite Group Databasebeta

← #128343

Subgroups — #128343

Order 512.

698 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13782C211yesmaximal, characteristic
2C2×#1326C212yesmaximal
3C4×#323C211yesmaximal, characteristic
4C4×#323C211yesmaximal, characteristic
5#7187C212yesmaximal
6C2×#385C411yescharacteristic
7C2×#385C2211yescharacteristic
8C2×C4×C22⋊C4C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10C22×#95C411yescharacteristic
11C4×C22⋊C8—21no—
12C4×C22⋊C8—21no—
13C2×#323C2212yes—
14C2×#323C2212yes—
15#1436—21no—
16#1436—21no—
17C4⋊#95—21no—
18#1434—21no—
19C22⋊#95C411yescharacteristic
20C22⋊#95C2211yescharacteristic
21#1326C418yescomplemented
22#1280—21no—
23#1280—21no—
24C24×C4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4C2311yescharacteristic
33C22×C42—21no—
34C22×C42D411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C4×C8—41no—
39C2×C4×C8—41no—
40#385—22no—
41#385—22no—
42#385C814yescomplemented
43#385C2×C414yes—
44C4×C22⋊C4—22no—
45C4×C22⋊C4—41no—
46C4×C22⋊C4—41no—
47C4×C22⋊C4C814yescomplemented
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—41no—
58C2×#95—41no—
59C2×#95D411yescharacteristic
60C2×#95D411yescharacteristic
61#336—41no—
62#336—41no—
63#323—22no—
64#323—22no—
65#323C2×C414yes—
66#323C2×C414yes—
67C25C22×C411yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—41no—
78C23×C4—41no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—81no—
82C23×C4—81no—
83C23×C4—81no—
84C23×C4—81no—
85C23×C4—81no—
86C23×C4—81no—
87C23×C4C4212yes—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C2×C812yes—
93C23×C4C2×C812yes—
94C23×C4M1612yes—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C22×C8—42no—
98C22×C8—42no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—81no—
102C2×C4⋊C4—81no—
103C2×C4⋊C4—81no—
104C2×C4⋊C4—81no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4C4212yes—
116C2×C22⋊C4C2×C812yes—
117C2×C22⋊C4C2×C812yes—
118C2×C22⋊C4C2×C812yes—
119C2×C22⋊C4C2×C812yes—
120C2×C22⋊C4M1612yes—
121C2×C22⋊C4C22×C412yes—
122C2×C22⋊C4#4112yes—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—24no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—81no—
129C2×C42—81no—
130C2×C42—81no—
131C2×C42—81no—
132C22⋊C8—24no—
133C22⋊C8—24no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136C4×C8—82no—
137C4×C8—82no—
138#95—24no—
139#95—24no—
140#95—24no—
141#95—41no—
142#95—41no—
143#95—42no—
144#95—41no—
145#95—41no—
146#95—81no—
147#95—81no—
148#95—81no—
149#95—81no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—41no—
155C24—41no—
156C24—41no—
157C24—41no—
158C24C2×C4211yescharacteristic
159C24C2×C22⋊C411yescharacteristic
160C24#11711yescharacteristic
161C24C22×C811yescharacteristic
162C24C22×C811yescharacteristic
163C24C2×M1611yescharacteristic
164C24C2×M1611yescharacteristic
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—84no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4C22⋊C812yes—
241C22×C4C22⋊C812yes—
242C22×C4C22⋊C812yes—
243C22×C4C22⋊C812yes—
244C22×C4#9911yescharacteristic
245C22×C4#9911yescharacteristic
246C22×C4#10112yes—
247C22×C4C4×D412yes—
248C22×C4C4×D412yes—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4C4×C814yescomplemented
267C42—42no—
268C42—42no—
269C42—81no—
270C42—81no—
271C42—84no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23C4×C22⋊C411yescharacteristic
311C23C2×C4×C811yescharacteristic
312C23C4×M1611yescharacteristic
313C23C2×C22⋊C811yescharacteristic
314C23C2×C22⋊C811yescharacteristic
315C23C2×#9911yescharacteristic
316C23#41011yescharacteristic
317C23C2×#10111yescharacteristic
318C23#41311yescharacteristic
319C23#43111yescharacteristic
320C23#43111yescharacteristic
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4#32412yes—
389C2×C4#32412yes—
390C8—84no—
391C8—84no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22C2×#32411yescharacteristic
431C22#145811yescharacteristic
432C22#145911yescharacteristic
433C22#146111yescharacteristic
434C22C4×C22⋊C811yescharacteristic
435C22C4×#9911yescharacteristic
436C22C4×#10111yescharacteristic
437C22#212111yescharacteristic
438C22#212111yescharacteristic
439C22#212211yescharacteristic
440C22#212311yescharacteristic
441C4—22no—
442C4—22no—
443C4—44no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—82no—
450C4—82no—
451C4—82no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—41no—
457C2—41no—
458C2—41no—
459C2—41no—
460C2C4×#32411yescharacteristic
461C2#1145311yescharacteristic
462C2#1145411yescharacteristic
463C2#1147311yescharacteristic
464C2#1147411yescharacteristic
465C2#1262111yescharacteristic
466C2#1262211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23C2×C4×C8
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1326C4semidirect
42. #385C8semidirect
47. C4×C22⋊C4C8semidirect
266. C22⋊C4C4×C8semidirect

Maximal subgroups

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

subgroupindex
#137822
C2×#13262
C4×#3232
#71872