Finite Group Databasebeta

← #242749

Subgroups — #242749

Order 512.

676 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21969C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic
3#17807C211yesmaximal, characteristic
4#17807C211yesmaximal, characteristic
5#17742C211yesmaximal, characteristic
6C22×#332C211yesmaximal, characteristic
7#10610C214yesmaximal, complemented
8#10606C214yesmaximal, complemented
9#8002C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17#1997—22no—
18#1997C2212yes—
19#1997C2212yes—
20#1950—22no—
21#1950C2214yes—
22#1821—22no—
23#1810—22no—
24#1573—22no—
25#1573—22no—
26#1571—21no—
27#1571C2212yes—
28#1570—21no—
29#1570C2212yes—
30#1565—22no—
31#1565C2214yes—
32#1556—22no—
33#1549C2214yes—
34#1546—21no—
35#1546—21no—
36C2×#332—22no—
37C2×#332—22no—
38C2×#332C2214yes—
39#1292C2212yes—
40#1292C2212yes—
41C23×Q8—21no—
42C23×C8—21no—
43C23×C8—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⋊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#432D412yes—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431C2312yes—
75C2×#108—24no—
76#423D412yes—
77#423D411yescharacteristic
78#423D411yescharacteristic
79#423C2312yes—
80#423C2311yescharacteristic
81#423C2311yescharacteristic
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
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⋊C8D412yes—
91C2×C4⋊C8D412yes—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94#407—21no—
95#407—21no—
96#407—21no—
97#407D412yes—
98C4×M16—21no—
99C4×M16—21no—
100C4×M16—21no—
101C4×M16C2312yes—
102#343—24no—
103#343—24no—
104#341—24no—
105#335—24no—
106#335—24no—
107#334—24no—
108#334—24no—
109#332—28no—
110#332—24no—
111#332—24no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—41no—
115C22×Q8—41no—
116C22×Q8—41no—
117C22×Q8—41no—
118C22×Q8—82no—
119C22×Q8—82no—
120C23×C4—22no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×Q811yescharacteristic
123C23×C4#4111yescharacteristic
124C2×M16—21no—
125C2×M16—21no—
126C2×M16—22no—
127C2×M16—21no—
128C2×M16—21no—
129C2×M16C2×D414yes—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—82no—
137C22×C8—82no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148C4⋊Q8—41no—
149C4⋊Q8—41no—
150C4⋊Q8—41no—
151C4⋊Q8—41no—
152C4⋊Q8—41no—
153C4⋊Q8—41no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C2×Q814yes—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×Q811yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42C2411yescharacteristic
178#108—84no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—21no—
188C4⋊C8—22no—
189C4⋊C8—21no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8C2×D412yes—
197C4⋊C8C2×D414yes—
198C4⋊C8C2×D412yes—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207#97—22no—
208#97—22no—
209#97—41no—
210#97—41no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—41no—
214C4×C8—41no—
215C24C22⋊Q811yescharacteristic
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—84no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—84no—
224C2×Q8—84no—
225C2×Q8—82no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—82no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—84no—
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4C22⋊Q811yescharacteristic
245C22×C4C22⋊Q811yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4#12611yescharacteristic
252C22×C4#12611yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×Q811yescharacteristic
255C22×C4C2×#4111yescharacteristic
256M16—24no—
257M16—24no—
258M16—42no—
259M16—42no—
260M16—42no—
261M16—42no—
262M16—42no—
263M16—42no—
264C2×C8—24no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—24no—
268C2×C8—44no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—82no—
274C2×C8—82no—
275C2×C8—82no—
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—22no—
305C23#39611yescharacteristic
306C23#39611yescharacteristic
307C23C2×C22⋊Q811yescharacteristic
308Q8—84no—
309Q8—84no—
310Q8—84no—
311Q8—82no—
312Q8—82no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4#39611yescharacteristic
338C2×C4#39611yescharacteristic
339C2×C4#39611yescharacteristic
340C2×C4#39611yescharacteristic
341C2×C4#39611yescharacteristic
342C2×C4#39611yescharacteristic
343C2×C4#47112yes—
344C2×C4#47212yes—
345C2×C4C2×#12111yescharacteristic
346C2×C4C2×#12111yescharacteristic
347C2×C4C2×#12111yescharacteristic
348C2×C4C2×C22⋊Q811yescharacteristic
349C2×C4C2×C22⋊Q811yescharacteristic
350C2×C4C2×#12611yescharacteristic
351C2×C4#57712yes—
352C2×C4#57812yes—
353C8—44no—
354C8—44no—
355C8—42no—
356C8—42no—
357C8—81no—
358C8—81no—
359C8—82no—
360C8—82no—
361C22—28no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22#206312yes—
369C22#207012yes—
370C22C2×#39611yescharacteristic
371C22#308212yes—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—84no—
381C4#206312yes—
382C4#304211yescharacteristic
383C4#304411yescharacteristic
384C4#306312yes—
385C4C2×#47111yescharacteristic
386C4C2×#47211yescharacteristic
387C2—22no—
388C2—22no—
389C2—22no—
390C2C2×#206311yescharacteristic
391C2#2361511yescharacteristic
392C2C2×#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
#219692
#218292
#178072
#177422
C22×#3322
#106102
#106062
#80022