Finite Group Databasebeta

← #127822

Subgroups — #127822

Order 512.

673 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21849C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3C4×#413C211yesmaximal, characteristic
4#7282C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6C2×#431C411yescharacteristic, complemented
7C2×#423C411yescharacteristic, complemented
8C2×#413—21no—
9C2×C4×M16—21no—
10#2283—21no—
11#1984—21no—
12#1954—21no—
13#1870C412yescomplemented
14#1870C2212yes—
15#1820C412yes—
16#1799—21no—
17#1799C412yes—
18#1797—21no—
19#1797—21no—
20#1782—21no—
21C4×#101—21no—
22C4×#101C412yescomplemented
23C4×#100—21no—
24C4×#100C2212yes—
25C4×#99—21no—
26C4×#99C2212yes—
27#1742C412yescomplemented
28C22×#41D411yescharacteristic
29C22×M16—22no—
30C22×M16C2×C412yes—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C4×#41—41no—
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4Q811yescharacteristic
42C2×C4×D4C2311yescharacteristic
43C2×#117C2×C411yescharacteristic
44C2×#117C2×C411yescharacteristic
45C2×#117Q811yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49#431—41no—
50#431—41no—
51#423—41no—
52#423—41no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56#413—28no—
57C2×#101—22no—
58C2×#100—22no—
59C2×#99—22no—
60C2×#99—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8C2×C412yes—
64C4×M16—22no—
65C4×M16—22no—
66C4×M16—41no—
67C4×M16—41no—
68C2×#97—41no—
69C2×C4×C8—21no—
70C2×C4×C8—41no—
71#383—41no—
72#382—41no—
73#381—22no—
74#380—22no—
75C4×C4⋊C4—41no—
76C4×C22⋊C4—22no—
77#376—41no—
78#343—22no—
79#343C2×C414yes—
80#342—22no—
81#342C2×C414yes—
82#336—22no—
83#336—22no—
84C2×#41—24no—
85C2×#41—41no—
86C2×#41—41no—
87C2×#41—42no—
88C22×Q8C22⋊C411yescharacteristic
89C22×D4C22⋊C411yescharacteristic
90C22×D4C4⋊C411yescharacteristic
91C22×D4C2×D411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×M16—28no—
102C2×M16—24no—
103C2×M16—44no—
104C2×M16—42no—
105C2×M16—42no—
106C2×M16—82no—
107C22×C8—22no—
108C22×C8—22no—
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⋊C4C4⋊C411yescharacteristic
138C2×C4⋊C4C22×C411yescharacteristic
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×Q811yescharacteristic
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×C42C4211yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C4⋊C411yescharacteristic
166C2×C42C4⋊C411yescharacteristic
167C2×C42C4⋊C411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C4⋊C8—42no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174#101—28no—
175#100—28no—
176#99—44no—
177#99—44no—
178C22⋊C8—42no—
179C22⋊C8—82no—
180C22⋊C8—82no—
181#97—42no—
182#97—42no—
183C4×C8—42no—
184C4×C8—42no—
185C4×C8—42no—
186#95—42no—
187#95—42no—
188#95—42no—
189#95—42no—
190C24#9511yescharacteristic
191C24C2×C22⋊C411yescharacteristic
192C24C2×C4⋊C411yescharacteristic
193#41—48no—
194#41—48no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—41no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4#10414yes—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
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—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—44no—
241C22×C4#9511yescharacteristic
242C22×C4#9511yescharacteristic
243C22×C4#9511yescharacteristic
244C22×C4#9511yescharacteristic
245C22×C4#9511yescharacteristic
246C22×C4#9511yescharacteristic
247C22×C4#9511yescharacteristic
248C22×C4C2×C4211yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C2×C4⋊C411yescharacteristic
253M16—28no—
254M16—48no—
255M16—44no—
256M16—84no—
257C2×C8—44no—
258C2×C8—48no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—84no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—44no—
279C22⋊C4—44no—
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—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—82no—
299C42—82no—
300C42#9911yescharacteristic
301C42#9911yescharacteristic
302C42#9911yescharacteristic
303C42#9911yescharacteristic
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23#33912yes—
318C23C2×#9511yescharacteristic
319C23C2×#10412yes—
320Q8—44no—
321D4—44no—
322D4—44no—
323D4—44no—
324C2×C4—24no—
325C2×C4—24no—
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—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
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—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4#33912yes—
373C2×C4#34211yescharacteristic
374C2×C4#34211yescharacteristic
375C2×C4#34211yescharacteristic
376C2×C4#34211yescharacteristic
377C2×C4C2×#9911yescharacteristic
378C2×C4C2×#9911yescharacteristic
379C2×C4#42112yes—
380C8—48no—
381C8—82no—
382C8—82no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22C2×#33911yescharacteristic
401C22#173311yescharacteristic
402C22C2×#34211yescharacteristic
403C4—21no—
404C4—21no—
405C4—22no—
406C4—21no—
407C4—21no—
408C4—22no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—164no—
417C4#139512yes—
418C4#212211yescharacteristic
419C4#212311yescharacteristic
420C2—21no—
421C2—21no—
422C2—42no—
423C2—42no—
424C2—81no—
425C2#805111yescharacteristic
426C2#1257511yescharacteristic
427C2C2×#139511yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ 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. #7282C2semidirect
6. C2×#431C4semidirect
7. C2×#423C4semidirect
13. #1870C4semidirect
22. C4×#101C4semidirect
27. #1742C4semidirect

Maximal subgroups

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

subgroupindex
#218492
#207232
C4×#4132
#72822