Finite Group Databasebeta

← #127572

Subgroups — #127572

Order 512.

625 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20694C211yesmaximal, characteristic
2#20627C211yesmaximal, characteristic
3#17357C211yesmaximal, characteristic
4#7253C214yesmaximal, complemented
5C22×#117C2211yescharacteristic
6#2920—21no—
7C2×#423C411yescharacteristic
8C2×#423C411yescharacteristic
9C2×#407—21no—
10C2×#407—21no—
11#1822—21no—
12#1822C412yes—
13#1821—21no—
14#1821C412yes—
15#1798—21no—
16#1798—21no—
17#1798C412yes—
18#1798C2212yes—
19#1742—21no—
20#1742C2212yes—
21#1728—21no—
22#1464—21no—
23#1464C412yes—
24#1463—21no—
25#1462C412yes—
26#1458—21no—
27#1458C2212yes—
28C24×C4D411yescharacteristic
29C22×M16—21no—
30C22×M16—21no—
31C22×M16—22no—
32C2×#117—41no—
33C2×#117—41no—
34C2×#117—41no—
35C2×#117—41no—
36C2×#117C2×C411yescharacteristic
37C2×#117C2×C411yescharacteristic
38C2×#117C2×C411yescharacteristic
39C2×#117D411yescharacteristic
40C2×#117D411yescharacteristic
41C2×#117D411yescharacteristic
42C2×#117Q811yescharacteristic
43C2×#117C2311yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C22⋊C4C2×C411yescharacteristic
47C22×C22⋊C4Q811yescharacteristic
48C22×C42C2×C411yescharacteristic
49C22×C42D411yescharacteristic
50#432—22no—
51#432—41no—
52#431—22no—
53#431—41no—
54#423—41no—
55#423—41no—
56#423—41no—
57#423—41no—
58#423—41no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2×C412yes—
62C2×C4⋊C8C2×C412yes—
63#407—22no—
64#407—22no—
65#407—22no—
66#407—44no—
67#407—41no—
68#407—41no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—41no—
72C2×C22⋊C8—41no—
73C4×M16—41no—
74#343—22no—
75#343—22no—
76#343—22no—
77#343C2×C414yes—
78#336—22no—
79#336—22no—
80#324—22no—
81#324—22no—
82#323—22no—
83#323C2×C414yes—
84C25C4⋊C411yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—41no—
92C23×C4—41no—
93C23×C4C4211yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C4⋊C411yescharacteristic
101C23×C4C4⋊C411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C2×M16—24no—
104C2×M16—42no—
105C2×M16—42no—
106C2×M16—42no—
107C2×M16—42no—
108C2×M16—42no—
109C2×M16—42no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—42no—
115C2×M16—42no—
116C2×M16—42no—
117C2×M16—42no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—42no—
121#117—41no—
122#117—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127#117—41no—
128#117—41no—
129#117—41no—
130#117—41no—
131#117—41no—
132#117—41no—
133#117—41no—
134#117—41no—
135#117—41no—
136#117—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4C4211yescharacteristic
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C4⋊C411yescharacteristic
146C2×C4⋊C4C4⋊C411yescharacteristic
147C2×C4⋊C4C22×C411yescharacteristic
148C2×C4⋊C4C2×D411yescharacteristic
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4C4211yescharacteristic
154C2×C22⋊C4C22⋊C411yescharacteristic
155C2×C22⋊C4C4⋊C411yescharacteristic
156C2×C22⋊C4C4⋊C411yescharacteristic
157C2×C22⋊C4C4⋊C411yescharacteristic
158C2×C22⋊C4C4⋊C411yescharacteristic
159C2×C22⋊C4C22×C411yescharacteristic
160C2×C22⋊C4C2×Q811yescharacteristic
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42C4211yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C4⋊C411yescharacteristic
174C2×C42C4⋊C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C4⋊C8—42no—
178C4⋊C8—42no—
179C4⋊C8—42no—
180C4⋊C8—42no—
181C4⋊C8—42no—
182C4⋊C8—42no—
183C22⋊C8—24no—
184C22⋊C8—44no—
185C22⋊C8—42no—
186C22⋊C8—42no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189C22⋊C8—42no—
190C22⋊C8—44no—
191#97—42no—
192C4×C8—42no—
193C24—21no—
194C24—21no—
195C24—22no—
196C24—41no—
197C24—81no—
198C24—81no—
199C24#9511yescharacteristic
200C24#9511yescharacteristic
201C24C2×C4⋊C411yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4#9511yescharacteristic
249C22×C4#9511yescharacteristic
250C22×C4#9511yescharacteristic
251C22×C4#9511yescharacteristic
252C22×C4#9511yescharacteristic
253C22×C4#9511yescharacteristic
254C22×C4#9911yescharacteristic
255C22×C4#9911yescharacteristic
256C22×C4#9911yescharacteristic
257C22×C4#9911yescharacteristic
258C22×C4C2×C4211yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C4⋊C411yescharacteristic
263C22×C4C2×C4⋊C411yescharacteristic
264C22×C4AGL(1,ℤ/8)12yes—
265C22×C4#13712yes—
266M16—44no—
267M16—44no—
268M16—82no—
269M16—82no—
270M16—82no—
271M16—82no—
272M16—82no—
273M16—82no—
274M16—84no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—82no—
284C2×C8—82no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—81no—
335C23—81no—
336C23—84no—
337C23—84no—
338C23#34211yescharacteristic
339C23#34211yescharacteristic
340C23C2×#9511yescharacteristic
341C23#42812yes—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
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—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4#34211yescharacteristic
396C2×C4#34211yescharacteristic
397C2×C4C2×#9911yescharacteristic
398C2×C4C2×#9911yescharacteristic
399C2×C4#41712yes—
400C2×C4#41811yescharacteristic
401C2×C4#41911yescharacteristic
402C2×C4#42111yescharacteristic
403C2×C4#42111yescharacteristic
404C8—84no—
405C8—84no—
406C8—82no—
407C8—82no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—81no—
427C22—81no—
428C22—84no—
429C22—84no—
430C22#173311yescharacteristic
431C22#173611yescharacteristic
432C22C2×#34211yescharacteristic
433C4—22no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4#151011yescharacteristic
446C4#151011yescharacteristic
447C4#151111yescharacteristic
448C4#151111yescharacteristic
449C2—21no—
450C2—21no—
451C2—41no—
452C2—41no—
453C2—42no—
454C2—81no—
455C2#801511yescharacteristic
456C2#1257911yescharacteristic
457C2#1258911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
socleC22C2×#342
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7253C2semidirect

Maximal subgroups

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

subgroupindex
#206942
#206272
#173572
#72532