Finite Group Databasebeta

← #242729

Subgroups — #242729

Order 512.

677 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#21371C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3#17822C211yesmaximal, characteristic
4#17797C211yesmaximal, characteristic
5#17742C211yesmaximal, characteristic
6C2×#1556C211yesmaximal, characteristic
7#10610C214yesmaximal, complemented
8#10606C214yesmaximal, complemented
9#8003C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#1997—21no—
18#1997—21no—
19#1997C2214yes—
20#1950C2212yes—
21#1950C2212yes—
22#1855—22no—
23#1778—21no—
24#1778—21no—
25C2×#343—22no—
26#1573—22no—
27#1571C2212yes—
28#1570C2212yes—
29#1569—22no—
30#1569—22no—
31#1568—22no—
32#1565C2214yes—
33#1556—22no—
34#1556—22no—
35#1556—22no—
36#1549C2214yes—
37#1546—22no—
38C2×#332C2212yes—
39C2×#332C2212yes—
40#1292C2214yes—
41C23×Q8—21no—
42C22×M16—21no—
43C22×M16D412yes—
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⋊Q8—21no—
54C2×C4⋊Q8Q812yes—
55C2×C4⋊Q8C2312yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4Q812yes—
58C22×C42C2311yescharacteristic
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#432D412yes—
71#431—21no—
72#431—21no—
73#431—21no—
74#431C2312yes—
75C2×#108—22no—
76C2×#108—22no—
77#423—21no—
78#423—21no—
79#423—21no—
80#423D412yes—
81#423C2312yes—
82#423C2312yes—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88#407—21no—
89#407—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C4×M16—21no—
93C4×M16—21no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16D411yescharacteristic
97C4×M16D411yescharacteristic
98C4×M16C2312yes—
99C2×#97—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103#343—24no—
104#343—24no—
105#341—22no—
106#341—22no—
107#335—24no—
108#335—24no—
109#334—24no—
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—84no—
121C23×C4—22no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×Q811yescharacteristic
124C23×C4#4111yescharacteristic
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—24no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—42no—
131C2×M16—41no—
132C2×M16—44no—
133C2×M16C2×D412yes—
134C2×M16C2×D412yes—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—41no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—41no—
149C4⋊Q8—41no—
150C4⋊Q8—41no—
151C4⋊Q8—41no—
152C4⋊Q8—41no—
153C4⋊Q8—41no—
154C4⋊Q8—41no—
155C4⋊Q8—41no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4C2×Q814yes—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×Q811yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42C2411yescharacteristic
180#108—44no—
181#108—44no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8C2×D412yes—
194C4⋊C8C2×D414yes—
195C4⋊C8C2×D412yes—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#97—21no—
205#97—21no—
206#97—22no—
207#97—22no—
208#97—22no—
209#97—22no—
210#97—41no—
211#97—41no—
212#97—41no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—41no—
218C24C22⋊Q811yescharacteristic
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—84no—
224C2×Q8—84no—
225C2×Q8—84no—
226C2×Q8—84no—
227C2×Q8—84no—
228C2×Q8—84no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—84no—
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4C22⋊Q811yescharacteristic
245C22×C4C22⋊Q811yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4#12611yescharacteristic
251C22×C4#12611yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×Q811yescharacteristic
254C22×C4C2×#4111yescharacteristic
255M16—22no—
256M16—22no—
257M16—22no—
258M16—44no—
259M16—42no—
260M16—42no—
261M16—44no—
262M16—82no—
263M16—82no—
264M16#13714yes—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—24no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—44no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—82no—
277C2×C8—82no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C42#12111yescharacteristic
287C42#12111yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42#12111yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42#12611yescharacteristic
297C42#12611yescharacteristic
298C42C22×D411yescharacteristic
299C42C22×D411yescharacteristic
300C42C2×#4111yescharacteristic
301C42C2×#4111yescharacteristic
302C23—22no—
303C23—22no—
304C23—41no—
305C23#39611yescharacteristic
306C23#39611yescharacteristic
307C23C2×C22⋊Q811yescharacteristic
308Q8—84no—
309Q8—84no—
310Q8—84no—
311Q8—84no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
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—44no—
332C2×C4—84no—
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×C4C2×#13712yes—
348C2×C4#57512yes—
349C8—24no—
350C8—42no—
351C8—42no—
352C8—42no—
353C8—44no—
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#307812yes—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—84no—
378C4#303911yescharacteristic
379C4#304011yescharacteristic
380C4#305612yes—
381C4#306812yes—
382C2—22no—
383C2—22no—
384C2—41no—
385C2#2360211yescharacteristic
386C2#2361911yescharacteristic
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. #10610C2semidirect
8. #10606C2semidirect

Maximal subgroups

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

subgroupindex
#213712
C2×#19972
#178222
#177972
#177422
C2×#15562
#106102
#106062
#80032