Finite Group Databasebeta

← #127488

Subgroups — #127488

Order 512.

559 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21849C211yesmaximal, characteristic
2#17366C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#7249C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6#2920—21no—
7#2920—21no—
8C2×#431C411yescharacteristic
9C2×#423C411yescharacteristic
10#1984C412yes—
11#1954C2212yes—
12#1870—21no—
13#1870—21no—
14#1797—21no—
15#1797—21no—
16#1467—21no—
17#1466C412yes—
18#1465—21no—
19#1465—21no—
20#1465C2212yes—
21#1465C2212yes—
22#1464—21no—
23#1464C412yes—
24#1461—21no—
25#1460C412yes—
26#1458—21no—
27#1458C412yes—
28C22×#41Q811yescharacteristic
29C22×M16—21no—
30C4×#41—41no—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8D411yescharacteristic
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4D411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4Q811yescharacteristic
41C2×C4×D4C2311yescharacteristic
42C2×#117C2×C411yescharacteristic
43C2×#117C2×C411yescharacteristic
44C2×#117D411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48#432—22no—
49#432—22no—
50#432—41no—
51#432—41no—
52#431—22no—
53#431—22no—
54#431—41no—
55#431—41no—
56#431—41no—
57#431—41no—
58#423—41no—
59#423—41no—
60#423—41no—
61#423—41no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8C2×C412yes—
65#407—22no—
66#407—22no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C4×M16—41no—
71C4×M16—41no—
72C2×C4×C8C2×C412yes—
73#336—22no—
74#336—22no—
75#324—22no—
76#324—22no—
77#324—22no—
78#324—22no—
79#323—22no—
80#323—22no—
81#323C2×C414yes—
82#323C2×C414yes—
83C2×#41—41no—
84C2×#41—41no—
85C2×#41—42no—
86C2×#41—42no—
87C22×Q8C4⋊C411yescharacteristic
88C22×D4C4⋊C411yescharacteristic
89C22×D4C4⋊C411yescharacteristic
90C22×D4C2×Q811yescharacteristic
91C23×C4C4211yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C4⋊C411yescharacteristic
94C23×C4C4⋊C411yescharacteristic
95C23×C4C4⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C2×M16—42no—
101C2×M16—42no—
102C2×M16—42no—
103C2×M16—82no—
104C22×C8—22no—
105C22×C8—22no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×D4—41no—
111C4×D4—41no—
112C4×D4—41no—
113C4×D4—41no—
114C4×D4—41no—
115C4×D4—41no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—41no—
119C4×D4—41no—
120C4×D4—41no—
121C4×D4—41no—
122#117—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127#117—41no—
128C2×C4⋊C4C4211yescharacteristic
129C2×C4⋊C4C4211yescharacteristic
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C4⋊C411yescharacteristic
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C22⋊C4C22⋊C411yescharacteristic
135C2×C22⋊C4C22⋊C411yescharacteristic
136C2×C22⋊C4C4⋊C411yescharacteristic
137C2×C22⋊C4C4⋊C411yescharacteristic
138C2×C22⋊C4C22×C411yescharacteristic
139C2×C22⋊C4C22×C411yescharacteristic
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—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42C4211yescharacteristic
153C2×C42C22⋊C411yescharacteristic
154C2×C42C22⋊C411yescharacteristic
155C2×C42C22⋊C411yescharacteristic
156C2×C42C22⋊C411yescharacteristic
157C2×C42C22⋊C411yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C4⋊C411yescharacteristic
161C2×C42C22×C411yescharacteristic
162C4⋊C8—42no—
163C4⋊C8—42no—
164C4⋊C8—42no—
165C4⋊C8—42no—
166C4⋊C8—42no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C22⋊C8—24no—
172C22⋊C8—24no—
173C22⋊C8—42no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176C22⋊C8—82no—
177C22⋊C8—82no—
178#97—42no—
179#97—42no—
180C4×C8—42no—
181C4×C8—42no—
182C4×C8—42no—
183C24#9511yescharacteristic
184C24C2×C4⋊C411yescharacteristic
185C24C2×C4⋊C411yescharacteristic
186#41—48no—
187#41—48no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C22×C4—22no—
201C22×C4—21no—
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—21no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4#9511yescharacteristic
234C22×C4#9511yescharacteristic
235C22×C4#9511yescharacteristic
236C22×C4#9511yescharacteristic
237C22×C4#9511yescharacteristic
238C22×C4#9511yescharacteristic
239C22×C4#9511yescharacteristic
240C22×C4C2×C4211yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C4⋊C411yescharacteristic
245M16—82no—
246M16—82no—
247M16—82no—
248M16—82no—
249M16—84no—
250C2×C8—44no—
251C2×C8—44no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C42—21no—
271C42—21no—
272C42—21no—
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#9911yescharacteristic
289C42#9911yescharacteristic
290C42#9911yescharacteristic
291C42#9911yescharacteristic
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23C2×#9511yescharacteristic
308Q8—44no—
309D4—44no—
310D4—44no—
311D4—44no—
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—22no—
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—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4#33812yes—
358C2×C4#34211yescharacteristic
359C2×C4#34211yescharacteristic
360C2×C4#34211yescharacteristic
361C2×C4#34211yescharacteristic
362C2×C4C2×#9911yescharacteristic
363C2×C4C2×#9911yescharacteristic
364C2×C4#43012yes—
365C8—82no—
366C8—84no—
367C8—84no—
368C8—82no—
369C22—21no—
370C22—21no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C2×#33811yescharacteristic
387C22#173811yescharacteristic
388C22C2×#34211yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4#139212yes—
403C4#212811yescharacteristic
404C4#212911yescharacteristic
405C2—21no—
406C2—21no—
407C2—81no—
408C2—81no—
409C2—81no—
410C2#800911yescharacteristic
411C2C2×#139211yescharacteristic
412C2#1258311yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC22×#41Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7249C2semidirect

Maximal subgroups

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

subgroupindex
#218492
#173662
#173612
#72492