Finite Group Databasebeta

← #99111

Subgroups — #99111

Order 512.

898 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14685C211yesmaximal, characteristic
2#10712C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4#7025C212yesmaximal, complemented
5#7025C212yesmaximal, complemented
6C23×C42C2211yescharacteristic
7C22×C22⋊C8—21no—
8C22×C22⋊C8—21no—
9C2×#381—21no—
10C2×#380C411yescharacteristic
11C2×#380C411yescharacteristic
12C2×#336—21no—
13C2×#336—21no—
14C2×#336C2212yes—
15C2×#336C2212yes—
16C4⋊#95C411yescharacteristic
17C4⋊#95C411yescharacteristic
18C4⋊#95C2211yescharacteristic
19C4⋊#95C2211yescharacteristic
20#1434—21no—
21#1434—21no—
22C22⋊#95—21no—
23C24×C4C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C24×C4D411yescharacteristic
26C23×C8—22no—
27C23×C8—22no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C42—24no—
35C22×C42—42no—
36C22×C42—42no—
37C22×C42C2×C411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42C2311yescharacteristic
41C2×C22⋊C8—24no—
42C2×C22⋊C8—24no—
43C2×C22⋊C8—42no—
44C2×C22⋊C8—42no—
45C2×C22⋊C8—42no—
46C2×C22⋊C8—42no—
47#381—44no—
48#380—42no—
49#380—42no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60#336—24no—
61#336—24no—
62#336—42no—
63#336—42no—
64C25C22⋊C411yescharacteristic
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4—41no—
87C23×C4—41no—
88C23×C4—41no—
89C23×C4—41no—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4SD1612yes—
94C23×C4Q1612yes—
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C22×C8—24no—
99C22×C8—24no—
100C22×C8—42no—
101C22×C8—42no—
102C22×C8—42no—
103C22×C8—42no—
104C22×C8—42no—
105C22×C8—42no—
106C22×C8—42no—
107C22×C8—42no—
108C22×C8—42no—
109C22×C8—42no—
110C22×C8—42no—
111C22×C8—42no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42—416no—
142C2×C42#4114yes—
143C22⋊C8—48no—
144C22⋊C8—48no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—42no—
149#95—42no—
150#95—42no—
151#95—42no—
152#95—41no—
153#95—42no—
154#95—42no—
155#95—41no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—21no—
161C24—21no—
162C24—21no—
163C24—21no—
164C24—21no—
165C24—21no—
166C24—21no—
167C24—21no—
168C24#9912yes—
169C24#10312yes—
170C24#10412yes—
171C24C2×C22⋊C411yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—24no—
185C22×C4—28no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—42no—
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—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4#9912yes—
258C22×C4#10312yes—
259C22×C4#10412yes—
260C22×C4#11712yes—
261C22×C4#12312yes—
262C22×C4#12312yes—
263C22×C4#13112yes—
264C22×C4#13112yes—
265C22×C4C2×SD1611yescharacteristic
266C22×C4C2×Q1611yescharacteristic
267C22×C4#13512yes—
268C22×C4AGL(1,ℤ/8)12yes—
269C22×C4AGL(1,ℤ/8)11yescharacteristic
270C22×C4#13711yescharacteristic
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—82no—
278C2×C8—82no—
279C2×C8—82no—
280C2×C8—82no—
281C2×C8—82no—
282C2×C8—82no—
283C2×C8—82no—
284C2×C8—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C22⋊C4—84no—
290C22⋊C4—84no—
291C42—44no—
292C42—44no—
293C42—44no—
294C42—44no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23#32814yes—
330C23#38111yescharacteristic
331C23#40811yescharacteristic
332C23#40811yescharacteristic
333C23C2×#9911yescharacteristic
334C23C2×#9911yescharacteristic
335C23#41311yescharacteristic
336C23#41311yescharacteristic
337C23C2×#10311yescharacteristic
338C23#41611yescharacteristic
339C23#41711yescharacteristic
340C23#41811yescharacteristic
341C23C2×#10411yescharacteristic
342C23C2×#10411yescharacteristic
343C23#42111yescharacteristic
344C23#42111yescharacteristic
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—48no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—416no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
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—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4#43314yes—
422C2×C4#48112yes—
423C2×C4#48212yes—
424C2×C4#48412yes—
425C2×C4#48512yes—
426C8—84no—
427C8—84no—
428C22—24no—
429C22—24no—
430C22—24no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22#146811yescharacteristic
444C22#146811yescharacteristic
445C22#146811yescharacteristic
446C22#146811yescharacteristic
447C22#148111yescharacteristic
448C22#148211yescharacteristic
449C22#148311yescharacteristic
450C22#148411yescharacteristic
451C22#149311yescharacteristic
452C22#149411yescharacteristic
453C22#149511yescharacteristic
454C22#149611yescharacteristic
455C22C2×#32811yescharacteristic
456C22#150111yescharacteristic
457C22#150411yescharacteristic
458C22#150711yescharacteristic
459C22#151011yescharacteristic
460C22#151311yescharacteristic
461C22#151611yescharacteristic
462C22#152011yescharacteristic
463C22#152411yescharacteristic
464C22#152911yescharacteristic
465C22#153211yescharacteristic
466C22#153511yescharacteristic
467C22#154211yescharacteristic
468C22#155011yescharacteristic
469C22#155311yescharacteristic
470C22#155711yescharacteristic
471C22#181811yescharacteristic
472C22#181911yescharacteristic
473C22#182211yescharacteristic
474C22#182411yescharacteristic
475C22#182511yescharacteristic
476C22#182611yescharacteristic
477C22#182611yescharacteristic
478C4—22no—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—24no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—161no—
487C4—161no—
488C4—161no—
489C4—161no—
490C2—44no—
491C2#765511yescharacteristic
492C2#766111yescharacteristic
493C2#767411yescharacteristic
494C2#768111yescharacteristic
495C2#769411yescharacteristic
496C2#769811yescharacteristic
497C2#771511yescharacteristic
498C2#771811yescharacteristic
499C2#870211yescharacteristic
500C2#871011yescharacteristic
501C2#871111yescharacteristic
502C2#871311yescharacteristic
503C2#871511yescharacteristic
504C2#872011yescharacteristic
505C2#872711yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7025C2semidirect
5. #7025C2semidirect

Maximal subgroups

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

subgroupindex
#146852
#107122
#70252