Finite Group Databasebeta

← #99918

Subgroups — #99918

Order 512.

454 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23237C211yesmaximal, characteristic
2#22079C211yesmaximal, characteristic
3#9879C211yesmaximal, characteristic
4#7589C211yesmaximal, characteristic
5C2×#1271C211yesmaximal, characteristic
6#7029C211yesmaximal, characteristic
7#7016C211yesmaximal, characteristic
8C2×#484C411yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10C2×#389C2211yescharacteristic
11#1984—21no—
12#1942—21no—
13#1933—21no—
14#1921—21no—
15#1916—21no—
16#1916—21no—
17#1872C411yescharacteristic
18#1863—21no—
19#1852C411yescharacteristic
20#1850—21no—
21#1824C411yescharacteristic
22#1797C2211yescharacteristic
23C8×C4⋊C4—21no—
24C8×C22⋊C4—21no—
25C4×C22⋊C8C2211yescharacteristic
26C4×C22⋊C8C2211yescharacteristic
27C2×#336C2211yescharacteristic
28C2×#336C2211yescharacteristic
29#1280—21no—
30#1280—21no—
31#1272—21no—
32#1271—21no—
33#1271—21no—
34#1271—21no—
35C23×C8—21no—
36C23×C8—21no—
37C2×C22⋊Q8C2×C411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C42C2311yescharacteristic
43#484—22no—
44#484C814yescomplemented
45C2×#107—21no—
46C2×#107—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×#103—21no—
51C2×#103—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8C2×C411yescharacteristic
54C2×C22⋊C8C2×C411yescharacteristic
55C2×C22⋊C8D411yescharacteristic
56C2×C22⋊C8D411yescharacteristic
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61#391D411yescharacteristic
62#389—21no—
63#389—21no—
64#389—21no—
65#389—21no—
66#389D411yescharacteristic
67#387—21no—
68#385—21no—
69#380C2×C411yescharacteristic
70C4×C4⋊C4—21no—
71C4×C4⋊C4D411yescharacteristic
72C4×C22⋊C4—21no—
73C2×#95—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#340—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336D411yescharacteristic
89#336D411yescharacteristic
90C22×Q8C22⋊C411yescharacteristic
91C23×C4—21no—
92C23×C4—21no—
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4#4111yescharacteristic
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—42no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—81no—
110C22×C8—81no—
111C22×C8—81no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8C2×C812yes—
115C22⋊Q8M1612yes—
116C4×Q8—81no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—81no—
130C2×C4⋊C4—81no—
131C2×C4⋊C4—81no—
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C4⋊C4C22⋊C411yescharacteristic
134C2×C4⋊C4C22⋊C411yescharacteristic
135C2×C4⋊C4C2×C812yes—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—81no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42#4111yescharacteristic
145#107—24no—
146#107—81no—
147C4⋊C8—42no—
148C4⋊C8—81no—
149C4⋊C8—81no—
150#103—24no—
151#103—81no—
152C22⋊C8—21no—
153C22⋊C8—21no—
154C22⋊C8—21no—
155C22⋊C8—21no—
156C22⋊C8—21no—
157C22⋊C8—21no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160C22⋊C8C2×C812yes—
161C22⋊C8Q1612yes—
162C4×C8—42no—
163C4×C8—81no—
164C4×C8—81no—
165C4×C8—81no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—41no—
172#95—41no—
173C24C4×D411yescharacteristic
174C2×Q8—21no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8C22⋊C812yes—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4C2×C22⋊C411yescharacteristic
202C22×C4C4×D411yescharacteristic
203C22×C4#12011yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12311yescharacteristic
207C22×C4C22×C811yescharacteristic
208C22×C4C2×M1611yescharacteristic
209C22×C4#13111yescharacteristic
210C22×C4#13111yescharacteristic
211C22×C4C2×Q1611yescharacteristic
212C22×C4#13511yescharacteristic
213C22×C4#13711yescharacteristic
214C22×C4#13711yescharacteristic
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—42no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—82no—
235C2×C8—82no—
236C2×C8—81no—
237C2×C8—82no—
238C2×C8—81no—
239C2×C8—82no—
240C2×C8—81no—
241C2×C8—81no—
242C2×C8—81no—
243C4⋊C4—21no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C4⋊C4—82no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C4⋊C4C22⋊C812yes—
262C22⋊C4—22no—
263C22⋊C4—22no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C42—41no—
267C42—41no—
268C42—41no—
269C42—41no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23#38611yescharacteristic
279C23C8×D411yescharacteristic
280C23#43511yescharacteristic
281C23C4×Q1611yescharacteristic
282C23#44111yescharacteristic
283C23#44311yescharacteristic
284C23#44411yescharacteristic
285Q8—42no—
286Q8—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—161no—
333C2×C4—161no—
334C2×C4C2×C22⋊C811yescharacteristic
335C2×C4#40811yescharacteristic
336C2×C4C8×D411yescharacteristic
337C2×C4#43511yescharacteristic
338C2×C4#45111yescharacteristic
339C2×C4#45211yescharacteristic
340C2×C4#46211yescharacteristic
341C2×C4#46411yescharacteristic
342C2×C4#47111yescharacteristic
343C2×C4#47211yescharacteristic
344C8—42no—
345C8—42no—
346C8—42no—
347C8—81no—
348C8—81no—
349C8—81no—
350C8—82no—
351C8—161no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22C8×Q1611yescharacteristic
365C22#157911yescharacteristic
366C22#158411yescharacteristic
367C22#158711yescharacteristic
368C22#186911yescharacteristic
369C22#187611yescharacteristic
370C22#187811yescharacteristic
371C22#190611yescharacteristic
372C22#190711yescharacteristic
373C22#191311yescharacteristic
374C22#191511yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—41no—
386C4—42no—
387C4—42no—
388C4—81no—
389C4—81no—
390C4—82no—
391C4—161no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2—21no—
396C2#894611yescharacteristic
397C2#909511yescharacteristic
398C2#910811yescharacteristic
399C2#926411yescharacteristic
400C2#926611yescharacteristic
401C2#927011yescharacteristic
402C2#927611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
44. #484C8semidirect

Maximal subgroups

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

subgroupindex
#232372
#220792
#98792
#75892
C2×#12712
#70292
#70162