Finite Group Databasebeta

← #119250

Subgroups — #119250

Order 512.

715 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#21762C211yesmaximal, characteristic
2C2×#1508C211yesmaximal, characteristic
3C2×#1472C211yesmaximal, characteristic
4#8057C211yesmaximal, characteristic
5#7942C211yesmaximal, characteristic
6#7925C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1892—22no—
16#1873—22no—
17#1823C412yescomplemented
18#1787C412yescomplemented
19#1508—22no—
20#1508—22no—
21C2×#328—22no—
22#1472—22no—
23#1472—22no—
24C2×#324—22no—
25#1298—22no—
26#1284—22no—
27#1282—22no—
28#1282—22no—
29C23×Q8C2×C411yescharacteristic
30C24×C4C2311yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C23×C8—21no—
34C23×C8—21no—
35C2×C22⋊Q8—21no—
36C2×C22⋊Q8—21no—
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8D412yes—
46C2×C22⋊Q8D412yes—
47C2×#117—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C42—21no—
53C2×#103—24no—
54#407—21no—
55#407—21no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407D412yes—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8C2×C412yes—
76C2×C22⋊C8D412yes—
77C2×C22⋊C8D412yes—
78#381—21no—
79#380—21no—
80#380C2×C412yes—
81#379—21no—
82C4×C22⋊C4—21no—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4D412yes—
88C2×#95—21no—
89#340—24no—
90#328—28no—
91#324—28no—
92C25#4111yescharacteristic
93C22×Q8—21no—
94C22×Q8—21no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—82no—
99C22×Q8C22⋊C412yes—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—41no—
103C23×C4—82no—
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C23×C4#4111yescharacteristic
114C23×C4#4111yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C2×M16—22no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—82no—
123C2×M16—82no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8—82no—
133C22×C8—82no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150#117—82no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4C22⋊C412yes—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42—22no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42—82no—
181#103—84no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198#95—22no—
199#95—22no—
200#95—41no—
201#95—41no—
202C24—22no—
203C24—22no—
204C24—42no—
205C24—81no—
206C24C4×D411yescharacteristic
207C24#12111yescharacteristic
208C24#12311yescharacteristic
209C24#12311yescharacteristic
210C24#12411yescharacteristic
211C24#12611yescharacteristic
212C24#12611yescharacteristic
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—44no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—82no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4#11711yescharacteristic
253C22×C4#11711yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4#12311yescharacteristic
268C22×C4#12311yescharacteristic
269C22×C4#12311yescharacteristic
270C22×C4#12311yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4#12311yescharacteristic
273C22×C4#12311yescharacteristic
274C22×C4#12411yescharacteristic
275C22×C4#12411yescharacteristic
276C22×C4#12411yescharacteristic
277C22×C4#12611yescharacteristic
278C22×C4#12611yescharacteristic
279M16—82no—
280M16—82no—
281M16—82no—
282M16—82no—
283C2×C8—44no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—82no—
289C2×C8—82no—
290C2×C8—82no—
291C2×C8—82no—
292C2×C8—82no—
293C2×C8—82no—
294C2×C8—82no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—44no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C22⋊C4—24no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C42—82no—
316C42—82no—
317C42—82no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—21no—
324C23—21no—
325C23—22no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—84no—
341C23—84no—
342C23#38611yescharacteristic
343C23#38811yescharacteristic
344C23#38811yescharacteristic
345C23#39411yescharacteristic
346C23#39711yescharacteristic
347C23#39711yescharacteristic
348C23#39711yescharacteristic
349C23#44312yes—
350C23#44412yes—
351Q8—44no—
352Q8—82no—
353Q8—82no—
354Q8—82no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—88no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4#38111yescharacteristic
400C2×C4#38611yescharacteristic
401C2×C4#38811yescharacteristic
402C2×C4#38811yescharacteristic
403C2×C4#39411yescharacteristic
404C2×C4#39411yescharacteristic
405C2×C4#39411yescharacteristic
406C2×C4#39711yescharacteristic
407C8—82no—
408C8—82no—
409C8—82no—
410C8—82no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—42no—
429C22—42no—
430C22—44no—
431C22—44no—
432C22—42no—
433C22—42no—
434C22—82no—
435C22—82no—
436C22—81no—
437C22—81no—
438C22#144011yescharacteristic
439C22#187612yes—
440C22#192912yes—
441C22#193711yescharacteristic
442C22#193811yescharacteristic
443C22#208212yes—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—84no—
452C4—82no—
453C4—82no—
454C4—162no—
455C4#182812yes—
456C2—22no—
457C2—22no—
458C2—22no—
459C2—21no—
460C2—21no—
461C2—41no—
462C2—41no—
463C2—41no—
464C2#1116911yescharacteristic
465C2#1118211yescharacteristic
466C2#1119111yescharacteristic
subgroupG/H
center, socleC22#1440
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
17. #1823C4semidirect
18. #1787C4semidirect

Maximal subgroups

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

subgroupindex
#217622
C2×#15082
C2×#14722
#80572
#79422
#79252
C2×#12822