Finite Group Databasebeta

← #108632

Subgroups — #108632

Order 512.

768 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13778C211yesmaximal, characteristic
2C2×#1305C212yesmaximal
3C2×#1305C212yesmaximal
4#7909C211yesmaximal, characteristic
5#7909C211yesmaximal, characteristic
6C2×#407C2212yes—
7C2×#407C2212yes—
8C2×#380C2211yescharacteristic
9C2×#380C2211yescharacteristic
10C2×C4×C22⋊C4C411yescharacteristic
11C2×C4×C22⋊C4C411yescharacteristic
12C22×#95C2211yescharacteristic
13#1436—21no—
14#1436—21no—
15#1436—21no—
16#1436—21no—
17C22⋊#95C411yescharacteristic
18C22⋊#95C411yescharacteristic
19#1305—24no—
20#1305C418yescomplemented
21#1280—21no—
22#1280—21no—
23#1280—21no—
24#1280—21no—
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2311yescharacteristic
28C22×M16—22no—
29C22×M16—22no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4C2×C411yescharacteristic
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C42—21no—
37C22×C42—21no—
38#407—22no—
39#407—22no—
40#407—22no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407C2×C414yes—
45#407C2×C414yes—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50#380—22no—
51#380—21no—
52#380—21no—
53#380C2×C414yes—
54#380D412yes—
55#380Q812yes—
56C4×C22⋊C4—42no—
57C4×C22⋊C4—42no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95D411yescharacteristic
69C2×#95D411yescharacteristic
70C2×#95D411yescharacteristic
71C2×#95D411yescharacteristic
72C25C22×C411yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—41no—
84C23×C4—41no—
85C23×C4—81no—
86C23×C4—81no—
87C23×C4—81no—
88C23×C4—81no—
89C23×C4—81no—
90C23×C4—81no—
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22×C412yes—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×Q811yescharacteristic
100C23×C4#4112yes—
101C23×C4#4112yes—
102C23×C4#4111yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4#4112yes—
105C2×M16—24no—
106C2×M16—24no—
107C2×M16—42no—
108C2×M16—42no—
109C2×M16—82no—
110C2×M16—82no—
111C2×M16—82no—
112C2×M16—82no—
113C22×C8—42no—
114C22×C8—42no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—81no—
119C2×C4⋊C4—81no—
120C2×C4⋊C4—81no—
121C2×C4⋊C4—81no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4C4212yes—
135C2×C22⋊C4C4212yes—
136C2×C22⋊C4C4⋊C412yes—
137C2×C22⋊C4C4⋊C412yes—
138C2×C42—42no—
139C2×C42—42no—
140C2×C42—81no—
141C2×C42—81no—
142C2×C42—81no—
143C2×C42—81no—
144C22⋊C8—24no—
145C22⋊C8—24no—
146C22⋊C8—24no—
147C22⋊C8—24no—
148C22⋊C8—42no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151C22⋊C8—42no—
152#95—24no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—42no—
160#95—42no—
161#95—81no—
162#95—81no—
163#95—81no—
164#95—81no—
165C24—21no—
166C24—21no—
167C24—21no—
168C24—21no—
169C24—41no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24C2×C4211yescharacteristic
174C24C2×C22⋊C411yescharacteristic
175C24C2×C4⋊C411yescharacteristic
176C24#11711yescharacteristic
177C24#11711yescharacteristic
178C24#11711yescharacteristic
179C24#11711yescharacteristic
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
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—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—82no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—82no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—84no—
253C22×C4—84no—
254C22×C4—84no—
255C22×C4—84no—
256C22×C4#10012yes—
257C22×C4#10112yes—
258C22×C4#11712yes—
259C22×C4#11712yes—
260C22×C4C4×D412yes—
261C22×C4C4×D412yes—
262C22×C4C4×D412yes—
263C22×C4C4×Q812yes—
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4#12312yes—
269C22×C4#12312yes—
270C22×C4#12412yes—
271C22×C4#12512yes—
272C22×C4#12612yes—
273C22×C4#12612yes—
274M16—84no—
275M16—84no—
276M16—84no—
277M16—84no—
278C2×C8—44no—
279C2×C8—44no—
280C2×C8—82no—
281C2×C8—82no—
282C2×C8—82no—
283C2×C8—82no—
284C4⋊C4—84no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C42—84no—
296C42—84no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23#37611yescharacteristic
338C23C4×C22⋊C411yescharacteristic
339C23C4×C4⋊C411yescharacteristic
340C23#38011yescharacteristic
341C23#38111yescharacteristic
342C23#38211yescharacteristic
343C23#38311yescharacteristic
344C23C2×#10011yescharacteristic
345C23C2×#10111yescharacteristic
346C23#41311yescharacteristic
347C23#41311yescharacteristic
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4#38712yes—
419C2×C4#38712yes—
420C2×C4#38812yes—
421C2×C4#38812yes—
422C8—84no—
423C8—84no—
424C22—21no—
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—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22#143411yescharacteristic
465C22C4×#10011yescharacteristic
466C22C4×#10111yescharacteristic
467C22#180011yescharacteristic
468C22#180111yescharacteristic
469C22#182111yescharacteristic
470C22#182211yescharacteristic
471C22#182811yescharacteristic
472C22#182811yescharacteristic
473C22#182911yescharacteristic
474C22#183011yescharacteristic
475C4—22no—
476C4—22no—
477C4—22no—
478C4—22no—
479C4—42no—
480C4—42no—
481C4—82no—
482C4—82no—
483C4—82no—
484C4—82no—
485C4—81no—
486C4—81no—
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—41no—
492C2—41no—
493C2—41no—
494C2—41no—
495C2#875611yescharacteristic
496C2#875711yescharacteristic
497C2#875811yescharacteristic
498C2#875911yescharacteristic
499C2#1080211yescharacteristic
500C2#1082911yescharacteristic
501C2#1083011yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ 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
20. #1305C4semidirect

Maximal subgroups

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

subgroupindex
#137782
C2×#13052
#79092