Finite Group Databasebeta

← #242700

Subgroups — #242700

Order 512.

653 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21994C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic
3#17816C211yesmaximal, characteristic
4#17810C211yesmaximal, characteristic
5C2×#1558C211yesmaximal, characteristic
6C2×#1548C211yesmaximal, characteristic
7#10609C214yesmaximal, complemented
8#10605C214yesmaximal, complemented
9#8003C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#1997—21no—
18#1997—21no—
19#1997C2214yes—
20#1950C2214yes—
21#1855—22no—
22#1849—21no—
23#1838—21no—
24#1821—22no—
25#1573—22no—
26#1572—22no—
27#1572—22no—
28#1571C2212yes—
29#1570C2212yes—
30#1568—22no—
31#1565C2212yes—
32#1565C2212yes—
33#1558—22no—
34#1558—22no—
35#1548—22no—
36#1548C2212yes—
37#1548C2212yes—
38C2×#333—22no—
39C2×#333C2214yes—
40#1292C2214yes—
41C23×Q8Q812yes—
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⋊C4—21no—
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—
92#407—21no—
93#407D412yes—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C4×M16—21no—
98C4×M16C2312yes—
99C2×#97—21no—
100C2×#97—21no—
101C2×C4×C8—21no—
102#343—24no—
103#343—24no—
104#341—24no—
105#335—22no—
106#335—22no—
107#335—24no—
108#334—22no—
109#334—22no—
110#334—24no—
111#333—24no—
112#333—24no—
113#333—28no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—84no—
120C22×Q8C2×Q814yes—
121C23×C4—22no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×Q811yescharacteristic
124C23×C4#4111yescharacteristic
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—41no—
128C2×M16—82no—
129C2×M16C2×D412yes—
130C2×M16C2×D412yes—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—41no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148C4⋊Q8—41no—
149C4⋊Q8—41no—
150C4⋊Q8—41no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×Q811yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42C2411yescharacteristic
175#108—84no—
176C4⋊C8—22no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8C2×D412yes—
191C4⋊C8C2×D412yes—
192C4⋊C8C2×D414yes—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—21no—
199C22⋊C8—21no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#97—22no—
205#97—22no—
206#97—22no—
207#97—41no—
208#97—41no—
209#97—41no—
210C4×C8—22no—
211C4×C8—21no—
212C4×C8—22no—
213C4×C8—21no—
214C4×C8—41no—
215C24C22⋊Q811yescharacteristic
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—84no—
221C2×Q8—84no—
222C2×Q8—84no—
223C2×Q8—84no—
224C2×Q8—84no—
225C2×Q8—84no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—84no—
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4C22⋊Q811yescharacteristic
245C22×C4C22⋊Q811yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4#12611yescharacteristic
248C22×C4#12611yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×Q811yescharacteristic
251C22×C4C2×#4111yescharacteristic
252M16—24no—
253M16—24no—
254M16—44no—
255M16—42no—
256M16—42no—
257M16—82no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—82no—
269C2×C8—82no—
270C2×C8—82no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42#12111yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42C22⋊Q811yescharacteristic
286C42C22⋊Q811yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42#12611yescharacteristic
290C42#12611yescharacteristic
291C42C22×D411yescharacteristic
292C42C22×D411yescharacteristic
293C42C2×#4111yescharacteristic
294C42C2×#4111yescharacteristic
295C23—22no—
296C23—22no—
297C23—41no—
298C23#39611yescharacteristic
299C23#39611yescharacteristic
300C23C2×C22⋊Q811yescharacteristic
301Q8—84no—
302Q8—84no—
303Q8—84no—
304Q8—84no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4#39611yescharacteristic
329C2×C4#39611yescharacteristic
330C2×C4#39611yescharacteristic
331C2×C4#39611yescharacteristic
332C2×C4#39611yescharacteristic
333C2×C4#39611yescharacteristic
334C2×C4C2×#12111yescharacteristic
335C2×C4C2×#12111yescharacteristic
336C2×C4C2×#12111yescharacteristic
337C2×C4C2×C22⋊Q811yescharacteristic
338C2×C4C2×C22⋊Q811yescharacteristic
339C2×C4C2×#12611yescharacteristic
340C2×C4#57712yes—
341C2×C4#57812yes—
342C8—44no—
343C8—42no—
344C8—42no—
345C8—42no—
346C8—42no—
347C8—82no—
348C8—82no—
349C8—82no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—44no—
358C22#206912yes—
359C22C2×#39611yescharacteristic
360C22#308212yes—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—84no—
371C4#304612yes—
372C4#305711yescharacteristic
373C4#305911yescharacteristic
374C4#306912yes—
375C2—22no—
376C2—22no—
377C2—41no—
378C2#2360311yescharacteristic
379C2#2361311yescharacteristic
380C2C2×#206911yescharacteristic
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. #10609C2semidirect
8. #10605C2semidirect

Maximal subgroups

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

subgroupindex
#219942
#218292
#178162
#178102
C2×#15582
C2×#15482
#106092
#106052
#80032