Finite Group Databasebeta

← #109708

Subgroups — #109708

Order 512.

736 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21941C211yesmaximal, characteristic
2#20528C211yesmaximal, characteristic
3C2×#1298C211yesmaximal, characteristic
4#8020C211yesmaximal, characteristic
5C2×#1294C211yesmaximal, characteristic
6#7915C211yesmaximal, characteristic
7C2×#1280C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C22×#95C2211yescharacteristic
15#1854—22no—
16#1853C412yescomplemented
17#1852C412yescomplemented
18#1808—21no—
19#1807—21no—
20#1752—21no—
21#1752—21no—
22C8×C22⋊C4—21no—
23C8×C22⋊C4—21no—
24#1750—22no—
25#1298C414yescomplemented
26#1294—21no—
27#1294—21no—
28#1294—21no—
29#1294—21no—
30#1280—21no—
31#1280—21no—
32#1280C414yescomplemented
33C24×C4—21no—
34C24×C4C2311yescharacteristic
35C22×M16—21no—
36C23×C8—21no—
37C23×C8—21no—
38C23×C8—21no—
39C2×#117D412yes—
40C22×C4⋊C4C2×C412yes—
41C22×C22⋊C4—21no—
42C2×#107—22no—
43C2×#106—22no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407C2×C412yes—
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⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8C2×C412yes—
60C2×C22⋊C8C2×C412yes—
61C2×C22⋊C8C2×C411yescharacteristic
62C2×C22⋊C8C2×C411yescharacteristic
63C2×C22⋊C8C2×C412yes—
64C2×C22⋊C8C2×C412yes—
65C2×C22⋊C8D411yescharacteristic
66C2×C22⋊C8D411yescharacteristic
67C2×C22⋊C8D411yescharacteristic
68C2×C22⋊C8D411yescharacteristic
69C2×C22⋊C8Q812yes—
70C2×C22⋊C8Q812yes—
71C2×#97—22no—
72C2×C4×C8—22no—
73#381—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380C2×C412yes—
79#380D411yescharacteristic
80#380D411yescharacteristic
81#379—21no—
82C4×C22⋊C4—21no—
83C4×C22⋊C4—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95C2×C412yes—
91#340—22no—
92#340—22no—
93#336—22no—
94#336—22no—
95C25#4111yescharacteristic
96C23×C4—24no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—42no—
100C23×C4—41no—
101C23×C4—41no—
102C23×C4—82no—
103C23×C4—81no—
104C23×C4—81no—
105C23×C4—82no—
106C23×C4C22×C411yescharacteristic
107C23×C4C22×C411yescharacteristic
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×Q811yescharacteristic
114C23×C4C2×Q811yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C2×M16—22no—
121C2×M16—41no—
122C2×M16—82no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—41no—
136C22×C8—82no—
137C22×C8—82no—
138C22×C8—82no—
139#117—82no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—82no—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—81no—
152C2×C22⋊C4—81no—
153C2×C42—22no—
154#107—82no—
155#106—82no—
156C22⋊C8—21no—
157C22⋊C8—21no—
158C22⋊C8—22no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—22no—
163C22⋊C8—22no—
164C22⋊C8—21no—
165C22⋊C8—21no—
166C22⋊C8—21no—
167C22⋊C8—21no—
168C22⋊C8—22no—
169C22⋊C8—22no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172C22⋊C8C4212yes—
173C22⋊C8C4212yes—
174C22⋊C8C4⋊C412yes—
175C22⋊C8C4⋊C412yes—
176C22⋊C8C4⋊C412yes—
177C22⋊C8C4⋊C412yes—
178#97—82no—
179C4×C8—82no—
180#95—22no—
181#95—21no—
182#95—21no—
183#95—22no—
184#95—22no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189C24—22no—
190C24—22no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—21no—
195C24—42no—
196C24#11711yescharacteristic
197C24C4×D411yescharacteristic
198C24C4×Q811yescharacteristic
199C24#12111yescharacteristic
200C24#12311yescharacteristic
201C24#12511yescharacteristic
202C24#12611yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—84no—
231C22×C4—84no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—84no—
237C22×C4—84no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4C2×C4211yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C4⋊C411yescharacteristic
243C22×C4C2×C4⋊C411yescharacteristic
244C22×C4#11711yescharacteristic
245C22×C4#11711yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×Q811yescharacteristic
254C22×C4C4×Q811yescharacteristic
255C22×C4C4×Q811yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12411yescharacteristic
265C22×C4#12511yescharacteristic
266C22×C4#12611yescharacteristic
267C22×C4C4⋊Q811yescharacteristic
268M16—82no—
269M16—82no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—82no—
284C2×C8—82no—
285C2×C8—82no—
286C2×C8—82no—
287C2×C8—82no—
288C2×C8—82no—
289C2×C8—82no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—82no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—81no—
304C22⋊C4—81no—
305C42—42no—
306C42—42no—
307C23—22no—
308C23—21no—
309C23—21no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—21no—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23C4×C4⋊C411yescharacteristic
343C23#38611yescharacteristic
344C23#38711yescharacteristic
345C23#38711yescharacteristic
346C23#38711yescharacteristic
347C23#38811yescharacteristic
348C23#38911yescharacteristic
349C23#44312yes—
350C23#44412yes—
351C23#47111yescharacteristic
352C23#47111yescharacteristic
353C23#47211yescharacteristic
354C23#47311yescharacteristic
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
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—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—82no—
393C2×C4—88no—
394C2×C4—84no—
395C2×C4—82no—
396C2×C4—84no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—162no—
401C2×C4—162no—
402C2×C4C4×C22⋊C411yescharacteristic
403C2×C4C4×C4⋊C411yescharacteristic
404C2×C4#38511yescharacteristic
405C2×C4#38511yescharacteristic
406C2×C4#38711yescharacteristic
407C2×C4#38811yescharacteristic
408C2×C4#38911yescharacteristic
409C2×C4#39111yescharacteristic
410C8—42no—
411C8—42no—
412C8—42no—
413C8—42no—
414C8—42no—
415C8—42no—
416C8—82no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—42no—
440C22—41no—
441C22—41no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—42no—
452C22—42no—
453C22#143611yescharacteristic
454C22#177211yescharacteristic
455C22#177311yescharacteristic
456C22#187612yes—
457C22#190711yescharacteristic
458C22#190811yescharacteristic
459C22#191312yes—
460C22#192212yes—
461C22#193011yescharacteristic
462C22#193111yescharacteristic
463C22#195111yescharacteristic
464C22#195211yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—84no—
474C4—82no—
475C4—82no—
476C4—162no—
477C2—21no—
478C2—22no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—21no—
484C2—41no—
485C2—41no—
486C2#926312yes—
487C2#926412yes—
488C2#1089611yescharacteristic
489C2#1090911yescharacteristic
490C2#1091511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1853C4semidirect
17. #1852C4semidirect
25. #1298C4semidirect
32. #1280C4semidirect

Maximal subgroups

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

subgroupindex
#219412
#205282
C2×#12982
#80202
C2×#12942
#79152
C2×#12802