Finite Group Databasebeta

← #119573

Subgroups — #119573

Order 512.

679 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21768C211yesmaximal, characteristic
2#21762C211yesmaximal, characteristic
3#17397C211yesmaximal, characteristic
4#8108C211yesmaximal, characteristic
5#8057C211yesmaximal, characteristic
6#8049C211yesmaximal, characteristic
7#7925C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2280C2211yescharacteristic
14#2280C2211yescharacteristic
15#1894C414yescomplemented
16#1892—22no—
17#1873—22no—
18#1873—22no—
19#1824—21no—
20#1823—21no—
21#1787—21no—
22#1787—21no—
23#1472—22no—
24#1472—22no—
25#1463—22no—
26#1305—22no—
27#1298—22no—
28#1297—22no—
29#1282—22no—
30C23×Q8D411yescharacteristic
31C24×C4C2311yescharacteristic
32C22×M16—21no—
33C22×M16—21no—
34C22×M16C2×C412yes—
35C23×C8—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⋊Q8—21no—
46C2×C22⋊Q8C2×C412yes—
47C2×C22⋊Q8D412yes—
48C2×#117—21no—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D411yescharacteristic
52C22×C22⋊C4—21no—
53C2×#103—24no—
54C2×#103—24no—
55#407—21no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407—21no—
62#407—21no—
63#407—21no—
64#407—21no—
65#407—21no—
66#407D412yes—
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⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8D412yes—
78C2×C22⋊C8D412yes—
79#381—21no—
80#381—21no—
81#380—21no—
82#380—21no—
83#380C2×C412yes—
84#380D412yes—
85#379—21no—
86#379—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91#340—24no—
92#340—24no—
93#324—28no—
94C25#4111yescharacteristic
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—82no—
101C22×Q8C22⋊C412yes—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—82no—
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4#4111yescharacteristic
113C23×C4#4111yescharacteristic
114C23×C4#4111yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—82no—
126C2×M16—82no—
127C2×M16—82no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—41no—
132C22×C8—41no—
133C22×C8—41no—
134C22×C8—82no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151#117—82no—
152#117—82no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
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×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C42—41no—
180C2×C42—41no—
181#103—84no—
182#103—84no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199#95—22no—
200#95—22no—
201#95—41no—
202#95—41no—
203C24—42no—
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—44no—
217C2×Q8—41no—
218C2×Q8—41no—
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—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4#11711yescharacteristic
249C22×C4#11711yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4#12311yescharacteristic
268C22×C4#12311yescharacteristic
269C22×C4#12311yescharacteristic
270C22×C4#12411yescharacteristic
271C22×C4#12411yescharacteristic
272C22×C4#12411yescharacteristic
273C22×C4#12611yescharacteristic
274C22×C4#12611yescharacteristic
275M16—82no—
276M16—82no—
277M16—82no—
278M16—82no—
279M16—82no—
280M16—82no—
281C2×C8—44no—
282C2×C8—44no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—82no—
288C2×C8—82no—
289C2×C8—82no—
290C2×C8—82no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—44no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C42—82no—
312C42—82no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—88no—
335C23#38611yescharacteristic
336C23#38811yescharacteristic
337C23#38811yescharacteristic
338C23#39411yescharacteristic
339C23#39711yescharacteristic
340C23#39711yescharacteristic
341C23#39711yescharacteristic
342Q8—44no—
343Q8—44no—
344Q8—82no—
345Q8—82no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
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—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4#38111yescharacteristic
389C2×C4#38611yescharacteristic
390C2×C4#38811yescharacteristic
391C2×C4#38811yescharacteristic
392C2×C4#39411yescharacteristic
393C2×C4#39411yescharacteristic
394C2×C4#39411yescharacteristic
395C2×C4#39711yescharacteristic
396C8—82no—
397C8—82no—
398C8—82no—
399C8—82no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—44no—
420C22—44no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22#144011yescharacteristic
427C22#188112yes—
428C22#204912yes—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—82no—
437C4—82no—
438C4—162no—
439C4—162no—
440C4#191312yes—
441C2—22no—
442C2—22no—
443C2—41no—
444C2—41no—
445C2—41no—
446C2—41no—
447C2—41no—
448C2#1117211yescharacteristic
449C2#1118811yescharacteristic
450C2#1119411yescharacteristic
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
15. #1894C4semidirect

Maximal subgroups

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

subgroupindex
#217682
#217622
#173972
#81082
#80572
#80492
#79252