Finite Group Databasebeta

← #242698

Subgroups — #242698

Order 512.

669 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21994C211yesmaximal, characteristic
2#21799C211yesmaximal, characteristic
3#17816C211yesmaximal, characteristic
4#17810C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6C2×#1546C211yesmaximal, characteristic
7#10608C214yesmaximal, complemented
8#10604C214yesmaximal, complemented
9#8003C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#1966—21no—
18#1966—21no—
19#1966C2214yes—
20#1950C2214yes—
21#1855—22no—
22#1849—21no—
23#1838—21no—
24#1784—22no—
25#1573—22no—
26#1572—22no—
27#1572—22no—
28#1571C2212yes—
29#1570C2212yes—
30#1568—22no—
31#1565C2212yes—
32#1565C2212yes—
33#1556—22no—
34#1556—22no—
35#1546—22no—
36#1546C2212yes—
37#1546C2212yes—
38C2×#332—22no—
39C2×#332C2214yes—
40#1292C2214yes—
41C23×Q8—21no—
42C22×M16—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8Q812yes—
54C2×C4⋊Q8C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4Q812yes—
57C22×C42C2311yescharacteristic
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432D412yes—
69#432D412yes—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431D412yes—
77#431C2311yescharacteristic
78#431C2311yescharacteristic
79C2×#108—24no—
80#423—21no—
81#423—21no—
82#423—21no—
83#423D411yescharacteristic
84#423D411yescharacteristic
85#423C2312yes—
86#423C2312yes—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8D412yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#101—24no—
93C2×#101—24no—
94#407—21no—
95#407D412yes—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C4×M16—21no—
100C4×M16C2312yes—
101C2×#97—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104#341—24no—
105#335—22no—
106#335—22no—
107#335—24no—
108#334—22no—
109#334—22no—
110#334—24no—
111#332—24no—
112#332—24no—
113#332—28no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—41no—
117C22×Q8—41no—
118C22×Q8—41no—
119C22×Q8—41no—
120C22×Q8—82no—
121C22×Q8—82no—
122C23×C4—22no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×Q811yescharacteristic
125C23×C4#4111yescharacteristic
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—41no—
129C2×M16—82no—
130C2×M16C2×D412yes—
131C2×M16C2×D412yes—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—41no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148C4⋊Q8—41no—
149C4⋊Q8—41no—
150C4⋊Q8—41no—
151C4⋊Q8—41no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4C2×Q814yes—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×Q811yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42C2411yescharacteristic
176#108—84no—
177C4⋊C8—22no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8C2×D412yes—
192C4⋊C8C2×D412yes—
193C4⋊C8C2×D414yes—
194#101—84no—
195#101—84no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207#97—22no—
208#97—22no—
209#97—22no—
210#97—41no—
211#97—41no—
212#97—41no—
213C4×C8—22no—
214C4×C8—21no—
215C4×C8—22no—
216C4×C8—21no—
217C4×C8—41no—
218C24C22⋊Q811yescharacteristic
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—84no—
226C2×Q8—84no—
227C2×Q8—84no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—82no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—84no—
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4#12611yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×Q811yescharacteristic
257C22×C4C2×#4111yescharacteristic
258M16—24no—
259M16—24no—
260M16—42no—
261M16—44no—
262M16—42no—
263M16—82no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—82no—
275C2×C8—82no—
276C2×C8—82no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C42#12111yescharacteristic
286C42#12111yescharacteristic
287C42#12111yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42#12611yescharacteristic
296C42#12611yescharacteristic
297C42C22×D411yescharacteristic
298C42C22×D411yescharacteristic
299C42C2×#4111yescharacteristic
300C42C2×#4111yescharacteristic
301C23—22no—
302C23—22no—
303C23—41no—
304C23#39611yescharacteristic
305C23#39611yescharacteristic
306C23C2×C22⋊Q811yescharacteristic
307Q8—84no—
308Q8—84no—
309Q8—84no—
310Q8—82no—
311Q8—82no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—48no—
332C2×C4—44no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4#39611yescharacteristic
336C2×C4#39611yescharacteristic
337C2×C4#39611yescharacteristic
338C2×C4#39611yescharacteristic
339C2×C4#39611yescharacteristic
340C2×C4#39611yescharacteristic
341C2×C4C2×#12111yescharacteristic
342C2×C4C2×#12111yescharacteristic
343C2×C4C2×#12111yescharacteristic
344C2×C4C2×C22⋊Q811yescharacteristic
345C2×C4C2×C22⋊Q811yescharacteristic
346C2×C4C2×#12611yescharacteristic
347C2×C4#57612yes—
348C2×C4#57712yes—
349C8—44no—
350C8—42no—
351C8—42no—
352C8—42no—
353C8—42no—
354C8—82no—
355C8—82no—
356C8—82no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—44no—
365C22#207012yes—
366C22C2×#39611yescharacteristic
367C22#308112yes—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—84no—
378C4#304512yes—
379C4#305711yescharacteristic
380C4#305811yescharacteristic
381C4#306912yes—
382C2—22no—
383C2—22no—
384C2—41no—
385C2#2360311yescharacteristic
386C2#2361211yescharacteristic
387C2C2×#207011yescharacteristic
subgroupG/H
center, socleC22C2×#396
commutator, Frattini, cosocleC2×C42C24
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10608C2semidirect
8. #10604C2semidirect

Maximal subgroups

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

subgroupindex
#219942
#217992
#178162
#178102
C2×#15562
C2×#15462
#106082
#106042
#80032