Finite Group Databasebeta

← #128102

Subgroups — #128102

Order 512.

621 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21853C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#7322C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6#2920—21no—
7C2×#432C411yescharacteristic, complemented
8C2×#423C411yescharacteristic, complemented
9C2×#410C412yes—
10#2285—21no—
11#1985C412yescomplemented
12#1955C2212yes—
13#1870—21no—
14#1870—21no—
15#1799—21no—
16#1799—21no—
17#1799C412yes—
18#1799C2212yes—
19#1797—21no—
20#1797—21no—
21#1740—21no—
22#1467C412yescomplemented
23#1466—21no—
24#1465—21no—
25#1465C2212yes—
26#1458—21no—
27#1458C412yescomplemented
28C22×#41C2×C411yescharacteristic
29C22×M16—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×#117—22no—
43C2×#117C2×C411yescharacteristic
44C2×#117D411yescharacteristic
45C2×#117Q811yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49#432—22no—
50#432—41no—
51#432—41no—
52#432—41no—
53#431—22no—
54#431—41no—
55#423—41no—
56#423—41no—
57#423—41no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2×C412yes—
61#410—44no—
62C2×#99—22no—
63C2×#99C2×C414yes—
64#407—22no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C4×M16—41no—
69C2×#97C2×C412yes—
70#384—41no—
71#382—41no—
72#380—22no—
73#380—22no—
74C4×C4⋊C4—41no—
75C4×C4⋊C4—41no—
76#342—22no—
77#342—22no—
78#336—22no—
79#336—22no—
80#324—22no—
81#324C2×C414yes—
82#323—22no—
83#323—22no—
84C2×#41—22no—
85C2×#41—22no—
86C2×#41—41no—
87C2×#41—41no—
88C2×#41—42no—
89C22×Q8C4211yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C22×D4C4⋊C411yescharacteristic
92C22×D4C22×C411yescharacteristic
93C23×C4C4211yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C2×M16—42no—
103C2×M16—42no—
104C2×M16—82no—
105C22×C8—22no—
106C22×C8—22no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×Q8—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—
122C4×D4—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127#117—41no—
128#117—41no—
129#117—44no—
130#117—44no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C4211yescharacteristic
135C2×C4⋊C4C22⋊C411yescharacteristic
136C2×C4⋊C4C4⋊C411yescharacteristic
137C2×C4⋊C4C4⋊C411yescharacteristic
138C2×C4⋊C4C4⋊C411yescharacteristic
139C2×C4⋊C4C22×C411yescharacteristic
140C2×C22⋊C4—24no—
141C2×C22⋊C4C22⋊C411yescharacteristic
142C2×C22⋊C4C22⋊C411yescharacteristic
143C2×C22⋊C4C4⋊C411yescharacteristic
144C2×C22⋊C4C4⋊C411yescharacteristic
145C2×C22⋊C4C2×D411yescharacteristic
146C2×C22⋊C4C2×Q811yescharacteristic
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—21no—
159C2×C42—42no—
160C2×C42C4211yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C22⋊C411yescharacteristic
166C2×C42C4⋊C411yescharacteristic
167C2×C42C4⋊C411yescharacteristic
168C2×C42C4⋊C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C4⋊C8—42no—
171C4⋊C8—42no—
172C4⋊C8—42no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177#99—44no—
178#99—44no—
179C22⋊C8—24no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182C22⋊C8—82no—
183C22⋊C8—82no—
184#97—42no—
185#97—42no—
186C4×C8—42no—
187#95—42no—
188#95—42no—
189C24#9511yescharacteristic
190C24C2×C22⋊C411yescharacteristic
191C24C2×C4⋊C411yescharacteristic
192#41—44no—
193#41—44no—
194#41—44no—
195#41—44no—
196C2×Q8—22no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—44no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4#9511yescharacteristic
244C22×C4#9511yescharacteristic
245C22×C4#9511yescharacteristic
246C22×C4#9511yescharacteristic
247C22×C4#9511yescharacteristic
248C22×C4#9511yescharacteristic
249C22×C4#9511yescharacteristic
250C22×C4#10012yes—
251C22×C4#10112yes—
252C22×C4C2×C4211yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C2×C4⋊C411yescharacteristic
257M16—82no—
258M16—82no—
259M16—84no—
260C2×C8—44no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—84no—
274C4⋊C4—84no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—44no—
284C22⋊C4—44no—
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—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—84no—
304C42#9911yescharacteristic
305C42#9911yescharacteristic
306C42#9911yescharacteristic
307C42#9911yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23C2×#9511yescharacteristic
323C23C2×#10011yescharacteristic
324C23C2×#10111yescharacteristic
325Q8—44no—
326D4—44no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4#34211yescharacteristic
381C2×C4#34211yescharacteristic
382C2×C4#34211yescharacteristic
383C2×C4#34211yescharacteristic
384C2×C4#34312yes—
385C2×C4#34312yes—
386C2×C4#34412yes—
387C2×C4C2×#9911yescharacteristic
388C2×C4C2×#9911yescharacteristic
389C8—82no—
390C8—84no—
391C8—82no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—42no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22C2×#34211yescharacteristic
412C22C2×#34311yescharacteristic
413C22C2×#34411yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—164no—
427C4#130512yes—
428C4#130512yes—
429C4#212811yescharacteristic
430C4#212911yescharacteristic
431C2—21no—
432C2—21no—
433C2—41no—
434C2—41no—
435C2—81no—
436C2—81no—
437C2C2×#130511yescharacteristic
438C2#1258311yescharacteristic
439C2#1258911yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ 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. #7322C2semidirect
7. C2×#432C4semidirect
8. C2×#423C4semidirect
11. #1985C4semidirect
22. #1467C4semidirect
27. #1458C4semidirect

Maximal subgroups

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

subgroupindex
#218532
#206322
#173612
#73222