Finite Group Databasebeta

← #106672

Subgroups — #106672

Order 512.

813 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
1C2×#2924C211yesmaximal, characteristic
2#17357C211yesmaximal, characteristic
3C2×#1464C211yesmaximal, characteristic
4#8120C212yesmaximal, complemented
5#7921C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7#2924—24no—
8#2924—24no—
9#2920C2212yes—
10C2×#433—21no—
11C2×#433—21no—
12C2×#432C411yescharacteristic
13C2×#432C2212yes—
14C2×#432C2211yescharacteristic
15C2×#431C411yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#424—21no—
18C2×#407C411yescharacteristic
19C22×C22⋊C8C411yescharacteristic
20C2×#405—21no—
21#1464—21no—
22#1464—21no—
23#1464—21no—
24#1464—21no—
25#1464—21no—
26#1464—21no—
27#1463C412yes—
28#1462C412yes—
29#1458—21no—
30#1458—21no—
31C2×#324C412yes—
32C2×#323C412yes—
33#1307—24no—
34#1281—24no—
35C24×C4C2×C411yescharacteristic
36C22×M16—21no—
37C23×C8—21no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117D411yescharacteristic
43C2×#117D411yescharacteristic
44C2×#117D411yescharacteristic
45C2×#117D411yescharacteristic
46C2×#117D411yescharacteristic
47C2×#117D411yescharacteristic
48C2×#117Q811yescharacteristic
49C2×#117Q811yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42C2311yescharacteristic
56#433—28no—
57#433—28no—
58#432—22no—
59#432—22no—
60#432—22no—
61#432—22no—
62#432—22no—
63#432—22no—
64#432—22no—
65#432—22no—
66#432—22no—
67#432D414yes—
68#431—22no—
69#431—22no—
70#431—22no—
71#431—22no—
72#431—22no—
73#431D414yes—
74#424—28no—
75#423—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81#407—22no—
82#407—22no—
83#407C2×C414yes—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8C2×C414yes—
89C2×C22⋊C8C2×C411yescharacteristic
90C2×C22⋊C8C2×C411yescharacteristic
91C2×C22⋊C8C2×C411yescharacteristic
92C2×C22⋊C8C2×C411yescharacteristic
93#405—28no—
94C4×M16—22no—
95C2×#97—21no—
96C2×#97—21no—
97C2×#97—22no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100#324—22no—
101#324—22no—
102#324—22no—
103#324—22no—
104#323—22no—
105#323—22no—
106#323—22no—
107#323—22no—
108C25C4211yescharacteristic
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4C4211yescharacteristic
116C23×C4C4211yescharacteristic
117C23×C4C4211yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C4⋊C411yescharacteristic
121C23×C4C4⋊C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—42no—
128C2×M16—42no—
129C22×C8—24no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—24no—
135C22×C8—42no—
136C22×C8—42no—
137#117—22no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151#117—22no—
152#117—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C22⋊C411yescharacteristic
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C4⋊C411yescharacteristic
162C2×C4⋊C4C4⋊C411yescharacteristic
163C2×C4⋊C4C4⋊C411yescharacteristic
164C2×C4⋊C4C4⋊C411yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4C22⋊C411yescharacteristic
170C2×C22⋊C4C22⋊C411yescharacteristic
171C2×C22⋊C4C22⋊C411yescharacteristic
172C2×C22⋊C4C22⋊C411yescharacteristic
173C2×C22⋊C4C4⋊C411yescharacteristic
174C2×C22⋊C4C4⋊C411yescharacteristic
175C2×C22⋊C4C4⋊C411yescharacteristic
176C2×C22⋊C4C4⋊C411yescharacteristic
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C4⋊C411yescharacteristic
184C2×C42C4⋊C411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×Q811yescharacteristic
191C2×C42#4112yes—
192C2×C42#4112yes—
193C4⋊C8—24no—
194C4⋊C8—24no—
195C4⋊C8—24no—
196C4⋊C8—24no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C4⋊C8—42no—
200C4⋊C8—42no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8—24no—
208C22⋊C8—24no—
209C22⋊C8—24no—
210C22⋊C8—24no—
211C22⋊C8C22⋊C412yes—
212C22⋊C8C22⋊C412yes—
213C22⋊C8C22⋊C412yes—
214C22⋊C8C22⋊C412yes—
215#97—24no—
216#97—24no—
217#97—42no—
218#97—42no—
219#97—42no—
220C4×C8—24no—
221C4×C8—24no—
222C4×C8—42no—
223C24—21no—
224C24—21no—
225C24—22no—
226C24—41no—
227C24—42no—
228C24—81no—
229C24#9511yescharacteristic
230C24#9511yescharacteristic
231C24C2×C4211yescharacteristic
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4#9511yescharacteristic
267C22×C4#9511yescharacteristic
268C22×C4#9511yescharacteristic
269C22×C4#9511yescharacteristic
270C22×C4#9511yescharacteristic
271C22×C4#9511yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C4⋊C411yescharacteristic
278C22×C4C2×C4⋊C411yescharacteristic
279C22×C4C2×C4⋊C411yescharacteristic
280C22×C4#11711yescharacteristic
281C22×C4#11711yescharacteristic
282C22×C4C4×D412yes—
283C22×C4C4×D412yes—
284C22×C4C4×D412yes—
285C22×C4C4×D412yes—
286M16—48no—
287M16—84no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—48no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—48no—
302C2×C8—44no—
303C2×C8—44no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42#12012yes—
329C42#12012yes—
330C42#12112yes—
331C42#12112yes—
332C42C22⋊Q812yes—
333C42C22⋊Q812yes—
334C42#12312yes—
335C42#12312yes—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—81no—
354C23—81no—
355C23—84no—
356C23—84no—
357C23#34412yes—
358C23C2×#9511yescharacteristic
359C23C4×C22⋊C411yescharacteristic
360C23C4×C22⋊C411yescharacteristic
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4#34412yes—
405C2×C4#37911yescharacteristic
406C2×C4#38011yescharacteristic
407C2×C4#38011yescharacteristic
408C2×C4#38111yescharacteristic
409C2×C4#38512yes—
410C2×C4#38512yes—
411C2×C4#38612yes—
412C2×C4#38612yes—
413C8—42no—
414C8—42no—
415C8—42no—
416C8—42no—
417C8—84no—
418C8—84no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—84no—
437C22—81no—
438C22—81no—
439C22—84no—
440C22C22⋊#9511yescharacteristic
441C22C2×#34411yescharacteristic
442C22C2×#34411yescharacteristic
443C22#175712yes—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—21no—
449C4—22no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—41no—
458C2—21no—
459C2—21no—
460C2—22no—
461C2—41no—
462C2—42no—
463C2—81no—
464C2#1073811yescharacteristic
465C2#1076411yescharacteristic
466C2#1077111yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC24C2×C42
Frattini, cosocleC22×C42C23
prime radicalC25C42
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8120C2semidirect
5. #7921C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29242
#173572
C2×#14642
#81202
#79212