Finite Group Databasebeta

← #242718

Subgroups — #242718

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21969C211yesmaximal, characteristic
2#21879C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic
4C2×#1571C211yesmaximal, characteristic
5#17718C211yesmaximal, characteristic
6C2×#1550C211yesmaximal, characteristic
7#10605C214yesmaximal, complemented
8#10604C214yesmaximal, complemented
9#8004C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17#1997C2212yes—
18#1966C2212yes—
19#1950—22no—
20#1950C2214yes—
21#1901—22no—
22#1810—22no—
23#1801—22no—
24#1574—22no—
25#1571—22no—
26#1571—21no—
27#1571C2214yes—
28#1570—21no—
29#1565C2214yes—
30C2×#335—22no—
31#1554—22no—
32#1550—22no—
33#1550—22no—
34#1548C2212yes—
35#1546C2212yes—
36#1542—21no—
37#1542—21no—
38C2×#333C2212yes—
39C2×#332C2212yes—
40#1292C2214yes—
41C22×M16—21no—
42C22×M16—21no—
43C23×C8—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8Q812yes—
57C2×C4×Q8—21no—
58C2×C4×Q8Q812yes—
59C2×#117—21no—
60C22×C42C2311yescharacteristic
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431D412yes—
72#431C2312yes—
73C2×#108—24no—
74#423—21no—
75#423—21no—
76#423—21no—
77#423—21no—
78#423—21no—
79#423—21no—
80#423—21no—
81#423D412yes—
82#423C2312yes—
83#423C2312yes—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8D412yes—
93C2×C4⋊C8D412yes—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96C2×#101—24no—
97#407—21no—
98#407D412yes—
99C2×C22⋊C8—21no—
100C4×M16—21no—
101C4×M16C2312yes—
102C2×C4×C8—21no—
103#343—24no—
104#341—24no—
105#335—28no—
106#335—24no—
107#334—24no—
108#333—24no—
109#333—22no—
110#333—22no—
111#332—24no—
112#332—22no—
113#332—22no—
114C22×Q8—21no—
115C22×Q8C2×Q812yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×Q811yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—82no—
125C2×M16—82no—
126C2×M16C2×D414yes—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—41no—
134C22×C8—82no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143#125—41no—
144#125—41no—
145#125—41no—
146#125—41no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C22⋊Q8—41no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—41no—
160C4×Q8—41no—
161C4×Q8—41no—
162C4×Q8—41no—
163#117—41no—
164#117—41no—
165#117—41no—
166#117—41no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C2×Q812yes—
173C2×C22⋊C4—22no—
174C2×C42—22no—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×Q811yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42C2411yescharacteristic
187#108—84no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8C2×D412yes—
206C4⋊C8C2×D414yes—
207C4⋊C8C2×D412yes—
208#101—84no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215#97—22no—
216#97—41no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—41no—
221C24C22⋊Q811yescharacteristic
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—82no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—81no—
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4C22⋊Q811yescharacteristic
245C22×C4C22⋊Q811yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C22×C4#12611yescharacteristic
250C22×C4#12611yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×Q811yescharacteristic
253C22×C4C2×#4111yescharacteristic
254M16—24no—
255M16—24no—
256M16—44no—
257M16—42no—
258M16—42no—
259M16—82no—
260M16—82no—
261M16—82no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—44no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—82no—
273C2×C8—82no—
274C2×C8—82no—
275C2×C8—82no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C42—42no—
291C42—42no—
292C42#12111yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42#12111yescharacteristic
296C42#12111yescharacteristic
297C42#12111yescharacteristic
298C42C22⋊Q811yescharacteristic
299C42C22⋊Q811yescharacteristic
300C42C22⋊Q811yescharacteristic
301C42C22⋊Q811yescharacteristic
302C42#12611yescharacteristic
303C42#12611yescharacteristic
304C42C22×D411yescharacteristic
305C42C22×D411yescharacteristic
306C42C2×#4111yescharacteristic
307C42C2×#4111yescharacteristic
308C23—22no—
309C23—22no—
310C23—41no—
311C23#39611yescharacteristic
312C23#39611yescharacteristic
313C23C2×C22⋊Q811yescharacteristic
314Q8—82no—
315Q8—82no—
316Q8—81no—
317Q8—81no—
318Q8—82no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4#39611yescharacteristic
344C2×C4#39611yescharacteristic
345C2×C4#39611yescharacteristic
346C2×C4#39611yescharacteristic
347C2×C4#39611yescharacteristic
348C2×C4#39611yescharacteristic
349C2×C4#47112yes—
350C2×C4#47312yes—
351C2×C4C2×#12111yescharacteristic
352C2×C4C2×#12111yescharacteristic
353C2×C4C2×#12111yescharacteristic
354C2×C4C2×C22⋊Q811yescharacteristic
355C2×C4C2×C22⋊Q811yescharacteristic
356C2×C4C2×#12611yescharacteristic
357C2×C4#57612yes—
358C2×C4#57812yes—
359C8—44no—
360C8—44no—
361C8—42no—
362C8—42no—
363C8—82no—
364C8—82no—
365C8—82no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—44no—
374C22#206412yes—
375C22C2×#39611yescharacteristic
376C22#308312yes—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—82no—
386C4—82no—
387C4#206412yes—
388C4#304111yescharacteristic
389C4#304411yescharacteristic
390C4#306112yes—
391C4C2×#47111yescharacteristic
392C4C2×#47311yescharacteristic
393C2—22no—
394C2—22no—
395C2—41no—
396C2C2×#206411yescharacteristic
397C2#2361111yescharacteristic
398C2#2362211yescharacteristic
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. #10605C2semidirect
8. #10604C2semidirect

Maximal subgroups

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

subgroupindex
#219692
#218792
#178092
C2×#15712
#177182
C2×#15502
#106052
#106042
#80042