Finite Group Databasebeta

← #242937

Subgroups — #242937

Order 512.

625 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21329C211yesmaximal, characteristic
2#20755C211yesmaximal, characteristic
3#17708C211yesmaximal, characteristic
4C2×#1542C211yesmaximal, characteristic
5#10630C218yesmaximal, complemented
6#8079C211yesmaximal, characteristic
7C2×#1301C211yesmaximal, characteristic
8#7854C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C4×#117C2211yescharacteristic
16#1995C2214yes—
17#1994—22no—
18#1832C2214yes—
19#1831—22no—
20#1733—21no—
21#1733—21no—
22#1729—21no—
23#1729—21no—
24#1550—22no—
25#1549—21no—
26#1548—21no—
27#1542—22no—
28#1542C2214yes—
29#1542C2214yes—
30C2×#333—21no—
31C2×#332—21no—
32#1301C2214yes—
33#1301C2214yes—
34#1274C2214yes—
35C22×M16—21no—
36C22×M16—21no—
37C23×C8—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533D412yes—
45#533C2312yes—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×#117—21no—
54C2×#117D411yescharacteristic
55C2×#117D411yescharacteristic
56C22×C42C2311yescharacteristic
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2312yes—
64#431C2312yes—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423C2312yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93#376C2312yes—
94C43D412yes—
95#339—24no—
96#339—24no—
97#336—24no—
98#336—24no—
99#333—24no—
100#333—24no—
101#332—24no—
102#332—24no—
103C22×Q8—21no—
104C22×Q8—21no—
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C23×C4#4111yescharacteristic
108C2×M16—22no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—42no—
113C2×M16—42no—
114C2×M16—42no—
115C2×M16—42no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—42no—
124C22×C8—42no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8C2×D412yes—
132C4⋊Q8C2×D412yes—
133#125—22no—
134#125—22no—
135#125—41no—
136#125—41no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153#117—22no—
154#117—22no—
155#117—22no—
156#117—22no—
157#117—22no—
158#117—22no—
159#117—41no—
160#117—41no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—41no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—41no—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42C2411yescharacteristic
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—41no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—41no—
212C4×C8—41no—
213#95—22no—
214#95—22no—
215C24#12311yescharacteristic
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—82no—
221C2×Q8—82no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—81no—
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12411yescharacteristic
238C22×C4#12411yescharacteristic
239C22×C4#12611yescharacteristic
240C22×C4#12611yescharacteristic
241C22×C4#12611yescharacteristic
242C22×C4#12611yescharacteristic
243C22×C4#13512yes—
244C22×C4#13512yes—
245C22×C4#13712yes—
246C22×C4#13712yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C2×#4111yescharacteristic
249C22×C4C2×#4111yescharacteristic
250M16—42no—
251M16—42no—
252M16—82no—
253M16—82no—
254M16—82no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—82no—
265C2×C8—82no—
266C2×C8—82no—
267C2×C8—82no—
268C2×C8—82no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—44no—
292C42—42no—
293C42—42no—
294C42—82no—
295C42#12111yescharacteristic
296C42#12111yescharacteristic
297C42#12311yescharacteristic
298C42#12311yescharacteristic
299C42#12311yescharacteristic
300C42#12311yescharacteristic
301C42#12411yescharacteristic
302C42#12411yescharacteristic
303C42#12611yescharacteristic
304C42#12611yescharacteristic
305C42#12611yescharacteristic
306C42#12611yescharacteristic
307C42C22×D411yescharacteristic
308C42C2×#4111yescharacteristic
309C42C2×#4111yescharacteristic
310C42C2×#4111yescharacteristic
311C23—41no—
312C23—41no—
313C23#39711yescharacteristic
314C23#39711yescharacteristic
315C23#48512yes—
316C23#48512yes—
317C23C2×#12311yescharacteristic
318Q8—82no—
319Q8—82no—
320Q8—82no—
321Q8—82no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
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—82no—
343C2×C4—82no—
344C2×C4#39711yescharacteristic
345C2×C4#39711yescharacteristic
346C2×C4#39711yescharacteristic
347C2×C4#39711yescharacteristic
348C2×C4#39711yescharacteristic
349C2×C4#39711yescharacteristic
350C2×C4#46412yes—
351C2×C4#46412yes—
352C2×C4#46412yes—
353C2×C4#46412yes—
354C2×C4#48512yes—
355C2×C4#48512yes—
356C2×C4#48812yes—
357C2×C4#48812yes—
358C2×C4#49112yes—
359C2×C4#49112yes—
360C2×C4C2×#12111yescharacteristic
361C2×C4C2×#12311yescharacteristic
362C2×C4C2×#12311yescharacteristic
363C2×C4C2×#12411yescharacteristic
364C2×C4C2×#12611yescharacteristic
365C2×C4C2×#12611yescharacteristic
366C2×C4C2×#13511yescharacteristic
367C2×C4C2×#13511yescharacteristic
368C2×C4C2×#13711yescharacteristic
369C2×C4C2×#13711yescharacteristic
370C8—82no—
371C8—82no—
372C8—82no—
373C8—82no—
374C8—82no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—42no—
382C22—42no—
383C22#207412yes—
384C22#207412yes—
385C22C2×#39711yescharacteristic
386C22C2×#48511yescharacteristic
387C22C2×#48511yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—82no—
396C4—82no—
397C4—82no—
398C4—82no—
399C4#207412yes—
400C4#207412yes—
401C4#208012yes—
402C4C2×#46411yescharacteristic
403C4C2×#46411yescharacteristic
404C4C2×#48811yescharacteristic
405C4C2×#49111yescharacteristic
406C2—21no—
407C2—21no—
408C2—22no—
409C2—22no—
410C2C2×#207411yescharacteristic
411C2C2×#208011yescharacteristic
412C2#2364911yescharacteristic
subgroupG/H
centerC2×C4C2×#124
commutator, Frattini, cosocleC2×C42C24
socleC22C2×#397
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10630C2semidirect

Maximal subgroups

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

subgroupindex
#213292
#207552
#177082
C2×#15422
#106302
#80792
C2×#13012
#78542