Finite Group Databasebeta

← #127128

Subgroups — #127128

Order 512.

609 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21847C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#7205C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6C2×#432—21no—
7C2×#431C411yescharacteristic, complemented
8C22×C4⋊C8C411yescharacteristic, complemented
9#2283—21no—
10#1984C412yescomplemented
11#1954C2212yes—
12#1869—21no—
13#1869—21no—
14#1820—21no—
15#1799—21no—
16#1799C412yes—
17#1797—21no—
18#1797—21no—
19#1782C412yes—
20C4×#99—21no—
21C4×#99C2212yes—
22#1467C412yescomplemented
23#1466—21no—
24#1464—21no—
25#1464C412yescomplemented
26#1459—21no—
27#1459C2212yes—
28C22×#41C2×C411yescharacteristic
29C23×C8—21no—
30C4×#41—41no—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4D411yescharacteristic
38C2×C4×D4D411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4Q811yescharacteristic
41C2×C4×D4C2311yescharacteristic
42C2×#117C2×C411yescharacteristic
43C2×#117D411yescharacteristic
44C2×#117Q811yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48#432—22no—
49#432—22no—
50#432—41no—
51#432—41no—
52#431—41no—
53#431—41no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—41no—
57C2×C4⋊C8—41no—
58C2×C4⋊C8—41no—
59C2×C4⋊C8C2×C412yes—
60C2×#99—22no—
61C2×#99C2×C414yes—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×#97—41no—
67C2×C4×C8C2×C412yes—
68#383—41no—
69#382—41no—
70#381—22no—
71#380—22no—
72C4×C4⋊C4—41no—
73C4×C22⋊C4—22no—
74#376—41no—
75#342—22no—
76#342—22no—
77#336—22no—
78#336—22no—
79#324—22no—
80#324C2×C414yes—
81#323—22no—
82#323—22no—
83C2×#41—24no—
84C2×#41—41no—
85C2×#41—41no—
86C2×#41—42no—
87C22×Q8C4211yescharacteristic
88C22×D4C22⋊C411yescharacteristic
89C22×D4C4⋊C411yescharacteristic
90C22×D4C22×C411yescharacteristic
91C23×C4C4211yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C4⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—42no—
103C22×C8—42no—
104C22×C8—82no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×D4—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—
121#117—41no—
122#117—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C4211yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C4⋊C411yescharacteristic
131C2×C4⋊C4C4⋊C411yescharacteristic
132C2×C4⋊C4C4⋊C411yescharacteristic
133C2×C4⋊C4C22×C411yescharacteristic
134C2×C22⋊C4—24no—
135C2×C22⋊C4C22⋊C411yescharacteristic
136C2×C22⋊C4C22⋊C411yescharacteristic
137C2×C22⋊C4C4⋊C411yescharacteristic
138C2×C22⋊C4C4⋊C411yescharacteristic
139C2×C22⋊C4C2×D411yescharacteristic
140C2×C22⋊C4C2×Q811yescharacteristic
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×C42—21no—
153C2×C42—42no—
154C2×C42C4211yescharacteristic
155C2×C42C22⋊C411yescharacteristic
156C2×C42C22⋊C411yescharacteristic
157C2×C42C22⋊C411yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C4⋊C411yescharacteristic
161C2×C42C4⋊C411yescharacteristic
162C2×C42C4⋊C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C4⋊C8—42no—
165C4⋊C8—42no—
166C4⋊C8—42no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171#99—28no—
172#99—44no—
173C22⋊C8—24no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176C22⋊C8—82no—
177C22⋊C8—82no—
178#97—42no—
179#97—42no—
180C4×C8—42no—
181#95—42no—
182#95—42no—
183#95—42no—
184#95—42no—
185C24#9511yescharacteristic
186C24C2×C22⋊C411yescharacteristic
187C24C2×C4⋊C411yescharacteristic
188#41—48no—
189#41—48no—
190C2×Q8—22no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4#13114yes—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
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—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—44no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4#9511yescharacteristic
237C22×C4#9511yescharacteristic
238C22×C4#9511yescharacteristic
239C22×C4#9511yescharacteristic
240C22×C4#9511yescharacteristic
241C22×C4#9511yescharacteristic
242C22×C4#9511yescharacteristic
243C22×C4C2×C4211yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C4⋊C411yescharacteristic
247C22×C4C2×C4⋊C411yescharacteristic
248C2×C8—44no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—82no—
256C2×C8—82no—
257C2×C8—84no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—84no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—44no—
272C22⋊C4—44no—
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—82no—
292C42—82no—
293C42#9911yescharacteristic
294C42#9911yescharacteristic
295C42#9911yescharacteristic
296C42#9911yescharacteristic
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#34412yes—
310C23C2×#9511yescharacteristic
311C23#40812yes—
312C23#42412yes—
313Q8—44no—
314D4—44no—
315D4—44no—
316D4—44no—
317C2×C4—24no—
318C2×C4—24no—
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—21no—
335C2×C4—21no—
336C2×C4—21no—
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—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4#34211yescharacteristic
367C2×C4#34211yescharacteristic
368C2×C4#34211yescharacteristic
369C2×C4#34211yescharacteristic
370C2×C4#40512yes—
371C2×C4C2×#9911yescharacteristic
372C2×C4C2×#9911yescharacteristic
373C8—82no—
374C8—84no—
375C8—82no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—81no—
392C22—81no—
393C22#172911yescharacteristic
394C22C2×#34211yescharacteristic
395C22C2×#34411yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—22no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—164no—
411C4#212811yescharacteristic
412C4#212911yescharacteristic
413C2—21no—
414C2—21no—
415C2—42no—
416C2—42no—
417C2—42no—
418C2#792711yescharacteristic
419C2#1258311yescharacteristic
420C2#1258911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #7205C2semidirect
7. C2×#431C4semidirect
8. C22×C4⋊C8C4semidirect
10. #1984C4semidirect
22. #1467C4semidirect
25. #1464C4semidirect

Maximal subgroups

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

subgroupindex
#218472
#207232
#173632
#72052