Finite Group Databasebeta

← #128108

Subgroups — #128108

Order 512.

607 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21374C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#7320C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6C2×#432—21no—
7C2×#431C411yescharacteristic, complemented
8C2×C4×M16C411yescharacteristic, complemented
9#2283—21no—
10#1985—21no—
11#1955—21no—
12#1870C412yescomplemented
13#1870C2212yes—
14#1820—21no—
15#1799—21no—
16#1799C2212yes—
17#1782C412yes—
18C4×#99—21no—
19C4×#99C412yes—
20C4×C22⋊C8—21no—
21C4×C22⋊C8—21no—
22#1467—21no—
23#1466C2212yes—
24#1464—21no—
25#1464C412yescomplemented
26#1459—21no—
27#1459C412yescomplemented
28C22×#41C2×C411yescharacteristic
29C22×M16C2×C412yes—
30C4×#41—41no—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C2×C4×Q8D411yescharacteristic
35C2×C4×Q8Q811yescharacteristic
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4D411yescharacteristic
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—41no—
56C2×#99—22no—
57C2×#99C2×C414yes—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8C2×C412yes—
62C4×M16—41no—
63C4×M16—41no—
64C2×#97—21no—
65C2×#97—41no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
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#324—22no—
81#323—22no—
82#323C2×C414yes—
83C2×#41—41no—
84C2×#41—41no—
85C2×#41—42no—
86C2×#41—42no—
87C22×Q8C4⋊C411yescharacteristic
88C22×D4C4211yescharacteristic
89C22×D4C22⋊C411yescharacteristic
90C22×D4C22×C411yescharacteristic
91C23×C4C4211yescharacteristic
92C23×C4C4211yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C2×M16—44no—
101C2×M16—42no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—42no—
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⋊C4C22⋊C411yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C4⋊C411yescharacteristic
131C2×C4⋊C4C4⋊C411yescharacteristic
132C2×C4⋊C4C4⋊C411yescharacteristic
133C2×C4⋊C4C2×Q811yescharacteristic
134C2×C22⋊C4—24no—
135C2×C22⋊C4C4211yescharacteristic
136C2×C22⋊C4C22⋊C411yescharacteristic
137C2×C22⋊C4C4⋊C411yescharacteristic
138C2×C22⋊C4C4⋊C411yescharacteristic
139C2×C22⋊C4C22×C411yescharacteristic
140C2×C22⋊C4C2×D411yescharacteristic
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×C42C22⋊C411yescharacteristic
155C2×C42C22⋊C411yescharacteristic
156C2×C42C22⋊C411yescharacteristic
157C2×C42C22⋊C411yescharacteristic
158C2×C42C4⋊C411yescharacteristic
159C2×C42C4⋊C411yescharacteristic
160C2×C42C4⋊C411yescharacteristic
161C2×C42C4⋊C411yescharacteristic
162C2×C42C4⋊C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C4⋊C8—42no—
165C4⋊C8—42no—
166C4⋊C8—42no—
167#99—44no—
168#99—44no—
169C22⋊C8—24no—
170C22⋊C8—44no—
171C22⋊C8—44no—
172C22⋊C8—42no—
173C22⋊C8—42no—
174#97—42no—
175#97—42no—
176#97—42no—
177C4×C8—41no—
178C4×C8—41no—
179C4×C8—41no—
180C4×C8—41no—
181#95—42no—
182#95—42no—
183#95—42no—
184#95—42no—
185C24#9511yescharacteristic
186C24C2×C4211yescharacteristic
187C24C2×C22⋊C411yescharacteristic
188#41—48no—
189#41—48no—
190C2×Q8—41no—
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—41no—
202C22×C4—22no—
203C22×C4—22no—
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—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4#9511yescharacteristic
234C22×C4#9511yescharacteristic
235C22×C4#9511yescharacteristic
236C22×C4#9511yescharacteristic
237C22×C4#9511yescharacteristic
238C22×C4#9511yescharacteristic
239C22×C4#9511yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C4⋊C411yescharacteristic
243C22×C4C2×C4⋊C411yescharacteristic
244C22×C4C2×C4⋊C411yescharacteristic
245M16—44no—
246M16—44no—
247C2×C8—24no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—84no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—84no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—44no—
269C22⋊C4—44no—
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—84no—
289C42#9911yescharacteristic
290C42#9911yescharacteristic
291C42#9911yescharacteristic
292C42#9911yescharacteristic
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23#34412yes—
307C23C2×#9511yescharacteristic
308C23#41312yes—
309Q8—44no—
310D4—44no—
311D4—44no—
312D4—44no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
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—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
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—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4#34211yescharacteristic
360C2×C4#34211yescharacteristic
361C2×C4#34211yescharacteristic
362C2×C4#34211yescharacteristic
363C2×C4C2×#9911yescharacteristic
364C2×C4C2×#9911yescharacteristic
365C8—44no—
366C8—82no—
367C8—84no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—22no—
372C22—42no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C2×#34211yescharacteristic
386C22#174211yescharacteristic
387C22C2×#34411yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—164no—
402C4#139112yes—
403C4#212012yes—
404C2—21no—
405C2—21no—
406C2—42no—
407C2—42no—
408C2—81no—
409C2#810411yescharacteristic
410C2#1257311yescharacteristic
411C2C2×#139111yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC24C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C23 ▷ 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. #7320C2semidirect
7. C2×#431C4semidirect
8. C2×C4×M16C4semidirect
12. #1870C4semidirect
25. #1464C4semidirect
27. #1459C4semidirect

Maximal subgroups

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

subgroupindex
#213742
#207232
#173632
#73202