Finite Group Databasebeta

← #127206

Subgroups — #127206

Order 512.

647 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20723C211yesmaximal, characteristic
2#20375C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#7214C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6#2920—21no—
7C2×#408—21no—
8C2×C4×M16C411yescharacteristic, complemented
9C22×#97C411yescharacteristic, complemented
10#2283—21no—
11#1820C2212yes—
12#1799—21no—
13#1799C412yes—
14#1782—21no—
15C4×#99—21no—
16C4×#99C412yes—
17#1750C412yescomplemented
18#1750C2212yes—
19#1749—21no—
20#1749—21no—
21#1729—21no—
22#1465—21no—
23#1465C412yescomplemented
24#1464—21no—
25#1464C2212yes—
26#1461C412yescomplemented
27#1460—21no—
28C22×#41D411yescharacteristic
29C22×M16—21no—
30C23×C8—21no—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C4×#41—41no—
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8Q811yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×#117C2×C411yescharacteristic
44C2×#117D411yescharacteristic
45C2×#117C2311yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49#432—22no—
50#432—41no—
51#431—22no—
52#431—41no—
53#423—41no—
54C2×#99—22no—
55C2×#99—22no—
56#408—28no—
57#407—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C4×M16—41no—
61C4×M16—41no—
62C4×M16—41no—
63C2×#97—21no—
64C2×#97—21no—
65C2×#97—41no—
66C2×#97—41no—
67C2×#97C2×C412yes—
68C2×C4×C8C2×C412yes—
69#383—41no—
70#382—41no—
71#381—22no—
72#380—22no—
73C4×C4⋊C4—41no—
74C4×C22⋊C4—22no—
75#376—41no—
76#342—22no—
77#342C2×C414yes—
78#336—22no—
79#336—22no—
80#324—22no—
81#324C2×C414yes—
82#323—22no—
83#323—22no—
84C2×#41—24no—
85C2×#41—41no—
86C2×#41—41no—
87C2×#41—42no—
88C22×Q8C4⋊C411yescharacteristic
89C22×D4C22⋊C411yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C22×D4C4⋊C411yescharacteristic
92C23×C4C4211yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×M16—24no—
102C2×M16—42no—
103C2×M16—42no—
104C2×M16—42no—
105C22×C8—24no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—42no—
109C22×C8—42no—
110C4×Q8—41no—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—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—
122C4×D4—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126#117—41no—
127#117—41no—
128#117—41no—
129#117—41no—
130#117—41no—
131#117—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C4211yescharacteristic
134C2×C4⋊C4C22⋊C411yescharacteristic
135C2×C4⋊C4C4⋊C411yescharacteristic
136C2×C4⋊C4C4⋊C411yescharacteristic
137C2×C4⋊C4C22×C411yescharacteristic
138C2×C4⋊C4C2×Q811yescharacteristic
139C2×C22⋊C4—24no—
140C2×C22⋊C4C4211yescharacteristic
141C2×C22⋊C4C22⋊C411yescharacteristic
142C2×C22⋊C4C4⋊C411yescharacteristic
143C2×C22⋊C4C4⋊C411yescharacteristic
144C2×C22⋊C4C22×C411yescharacteristic
145C2×C22⋊C4C2×D411yescharacteristic
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—42no—
159C2×C42C4211yescharacteristic
160C2×C42C22⋊C411yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C4⋊C411yescharacteristic
164C2×C42C4⋊C411yescharacteristic
165C2×C42C4⋊C411yescharacteristic
166C2×C42C4⋊C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C4⋊C8—42no—
170C4⋊C8—42no—
171#99—44no—
172#99—44no—
173C22⋊C8—28no—
174C22⋊C8—28no—
175C22⋊C8—24no—
176C22⋊C8—42no—
177#97—42no—
178#97—42no—
179#97—41no—
180#97—41no—
181#97—41no—
182#97—41no—
183C4×C8—42no—
184C4×C8—42no—
185#95—42no—
186#95—42no—
187#95—42no—
188#95—42no—
189C24#9511yescharacteristic
190C24#9511yescharacteristic
191C24C2×C22⋊C411yescharacteristic
192#41—48no—
193#41—48no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4#10414yes—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
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—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—44no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—81no—
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
248C22×C4C2×C4⋊C411yescharacteristic
249M16—48no—
250M16—84no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—48no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—44no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—84no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—44no—
273C22⋊C4—44no—
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—84no—
293C42#9911yescharacteristic
294C42#9911yescharacteristic
295C42#9911yescharacteristic
296C42#9911yescharacteristic
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23#33912yes—
311C23C2×#9511yescharacteristic
312C23C2×#10412yes—
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—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4H(3,ℤ/4)12yes—
362C2×C4#33912yes—
363C2×C4#34211yescharacteristic
364C2×C4#34211yescharacteristic
365C2×C4#34211yescharacteristic
366C2×C4#34211yescharacteristic
367C2×C4C2×#9911yescharacteristic
368C2×C4C2×#9911yescharacteristic
369C8—44no—
370C8—44no—
371C8—84no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—22no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—81no—
388C22—81no—
389C22C2×H(3,ℤ/4)11yescharacteristic
390C22C2×#33911yescharacteristic
391C22C2×#34211yescharacteristic
392C4—21no—
393C4—21no—
394C4—22no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—164no—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2#794211yescharacteristic
414C2#1257711yescharacteristic
415C2#1258911yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7214C2semidirect
8. C2×C4×M16C4semidirect
9. C22×#97C4semidirect
17. #1750C4semidirect
23. #1465C4semidirect
26. #1461C4semidirect

Maximal subgroups

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

subgroupindex
#207232
#203752
#173662
#72142