Finite Group Databasebeta

← #127239

Subgroups — #127239

Order 512.

630 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20727C211yesmaximal, characteristic
2#17365C211yesmaximal, characteristic
3#13961C211yesmaximal, characteristic
4#7215C214yesmaximal
5C22×C4×Q8C2211yescharacteristic
6#2920—21no—
7#2920—21no—
8C2×#383C411yescharacteristic
9C2×#376C411yescharacteristic
10#1821C2212yes—
11#1801—21no—
12#1801C412yescomplemented
13#1784—21no—
14C4×#101—21no—
15C4×#101C412yescomplemented
16#1467—21no—
17#1467C412yescomplemented
18#1463—21no—
19#1463C2212yes—
20#1461—21no—
21#1461C412yescomplemented
22#1446—21no—
23#1445—21no—
24#1436C412yes—
25#1436C2212yes—
26#1434—21no—
27#1434—21no—
28C23×Q8D411yescharacteristic
29C2×C4×Q8—41no—
30C2×C4×Q8—41no—
31C2×C4×Q8—41no—
32C2×C4×Q8—41no—
33C2×C4×Q8C2×C411yescharacteristic
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C2×C4×Q8Q811yescharacteristic
40C2×C4×Q8Q811yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4C2311yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42C2×C412yes—
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48#432—22no—
49#432—41no—
50#432—41no—
51#432—41no—
52#431—22no—
53#431—41no—
54#431—41no—
55#431—41no—
56#423—22no—
57#423—41no—
58C2×#101—22no—
59C2×#101—22no—
60#407—22no—
61#407—22no—
62C4×M16—22no—
63C4×M16—41no—
64#383—41no—
65#383—41no—
66#376—41no—
67#376—41no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95C2×C412yes—
77#343—22no—
78#343C2×C414yes—
79#324—22no—
80#324—22no—
81#324—22no—
82#324C2×C414yes—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8C22⋊C411yescharacteristic
88C22×Q8C22⋊C411yescharacteristic
89C22×Q8C4⋊C411yescharacteristic
90C22×Q8C4⋊C411yescharacteristic
91C23×C4—22no—
92C23×C4—22no—
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C2×M16—24no—
101C2×M16—42no—
102C4×Q8—41no—
103C4×Q8—41no—
104C4×Q8—41no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×Q8—41no—
116C4×Q8—41no—
117C4×Q8—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×C4⋊C4C4211yescharacteristic
125C2×C4⋊C4C4211yescharacteristic
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C4⋊C4C4⋊C411yescharacteristic
129C2×C4⋊C4C4⋊C411yescharacteristic
130C2×C4⋊C4C4⋊C411yescharacteristic
131C2×C4⋊C4C4⋊C411yescharacteristic
132C2×C4⋊C4C22×C411yescharacteristic
133C2×C4⋊C4C22×C411yescharacteristic
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×Q811yescharacteristic
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—42no—
149C2×C42—44no—
150C2×C42C4211yescharacteristic
151C2×C42C4211yescharacteristic
152C2×C42C22⋊C411yescharacteristic
153C2×C42C22⋊C411yescharacteristic
154C2×C42C22⋊C411yescharacteristic
155C2×C42C22⋊C411yescharacteristic
156C2×C42C4⋊C411yescharacteristic
157C2×C42C4⋊C411yescharacteristic
158C2×C42C4⋊C411yescharacteristic
159C2×C42C4⋊C411yescharacteristic
160C2×C42C4⋊C411yescharacteristic
161C2×C42C4⋊C411yescharacteristic
162C4⋊C8—42no—
163C4⋊C8—42no—
164C4⋊C8—42no—
165C4⋊C8—42no—
166#101—44no—
167#101—44no—
168C22⋊C8—24no—
169C22⋊C8—42no—
170C22⋊C8—42no—
171C22⋊C8—42no—
172#97—42no—
173#97—42no—
174C4×C8—42no—
175C4×C8—42no—
176#95—42no—
177#95—44no—
178#95—44no—
179#95—44no—
180#95—44no—
181#95—42no—
182#95—42no—
183#95—41no—
184#95—41no—
185#95—41no—
186#95—41no—
187C24C2×C22⋊C411yescharacteristic
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—48no—
201C2×Q8—48no—
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—22no—
216C22×C4—24no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4#9511yescharacteristic
239C22×C4#9511yescharacteristic
240C22×C4#9511yescharacteristic
241C22×C4#9511yescharacteristic
242C22×C4#9511yescharacteristic
243C22×C4#9511yescharacteristic
244C22×C4#9511yescharacteristic
245C22×C4#9511yescharacteristic
246C22×C4#9911yescharacteristic
247C22×C4#9911yescharacteristic
248C22×C4#9911yescharacteristic
249C22×C4#9911yescharacteristic
250C22×C4C2×C4211yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C4⋊C411yescharacteristic
254C22×C4C2×C4⋊C411yescharacteristic
255C22×C4C2×C4⋊C411yescharacteristic
256M16—44no—
257M16—44no—
258M16—84no—
259C2×C8—44no—
260C2×C8—44no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—44no—
298C42—44no—
299C23—22no—
300C23—22no—
301C23—21no—
302C23—21no—
303C23C2×#9511yescharacteristic
304C23C2×#9911yescharacteristic
305C23C2×#9911yescharacteristic
306Q8—44no—
307Q8—44no—
308Q8—44no—
309Q8—44no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4H(3,ℤ/4)12yes—
371C2×C4#34211yescharacteristic
372C2×C4#34211yescharacteristic
373C2×C4#34211yescharacteristic
374C2×C4#34211yescharacteristic
375C2×C4#42112yes—
376C8—84no—
377C8—84no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22C2×H(3,ℤ/4)11yescharacteristic
389C22#173311yescharacteristic
390C22C2×#34211yescharacteristic
391C22#212012yes—
392C22#213312yes—
393C4—22no—
394C4—22no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—84no—
410C4—162no—
411C2—22no—
412C2—22no—
413C2—21no—
414C2—21no—
415C2#794611yescharacteristic
416C2#1257311yescharacteristic
417C2#1258611yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1801C4semidirect
15. C4×#101C4semidirect
17. #1467C4semidirect
21. #1461C4semidirect

Maximal subgroups

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

subgroupindex
#207272
#173652
#139612
#72152