Finite Group Databasebeta

← #242739

Subgroups — #242739

Order 512.

695 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21370C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic
4#17796C211yesmaximal, characteristic
5#17746C211yesmaximal, characteristic
6C2×#1548C211yesmaximal, characteristic
7#10609C214yesmaximal, complemented
8#10605C214yesmaximal, complemented
9#8004C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#1997—21no—
18#1997—21no—
19#1997C2214yes—
20#1950C2212yes—
21#1950C2212yes—
22#1855—22no—
23C4×#108—21no—
24C4×#108—21no—
25C2×#343—22no—
26#1574—22no—
27#1571C2212yes—
28#1570C2212yes—
29#1569—22no—
30#1567—21no—
31#1566—21no—
32#1565—22no—
33#1565C2214yes—
34#1558—22no—
35#1548—22no—
36#1548—22no—
37#1548—21no—
38#1548—21no—
39#1548C2214yes—
40C2×#333C2212yes—
41C2×#333C2212yes—
42#1292C2214yes—
43C23×Q8Q812yes—
44C22×M16—21no—
45C22×M16—21no—
46C22×M16D412yes—
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⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8Q812yes—
58C2×C4⋊Q8C2312yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C42C2311yescharacteristic
62#432—21no—
63#432—21no—
64#432—21no—
65#432D412yes—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431D412yes—
72#431D412yes—
73#431C2312yes—
74C2×#108—22no—
75C2×#108—22no—
76#423—21no—
77#423—21no—
78#423—21no—
79#423—21no—
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⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93#407—21no—
94#407—21no—
95C2×C22⋊C8—21no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16—21no—
99C4×M16—21no—
100C4×M16D411yescharacteristic
101C4×M16D411yescharacteristic
102C4×M16C2312yes—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106#343—24no—
107#343—24no—
108#341—22no—
109#341—22no—
110#335—24no—
111#335—24no—
112#334—24no—
113#334—24no—
114#333—24no—
115#333—24no—
116#333—24no—
117#333—24no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—84no—
124C22×Q8C2×Q814yes—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×Q811yescharacteristic
128C23×C4#4111yescharacteristic
129C2×M16—22no—
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—24no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—42no—
136C2×M16—41no—
137C2×M16—41no—
138C2×M16—41no—
139C2×M16—44no—
140C2×M16—82no—
141C2×M16C2×D412yes—
142C2×M16C2×D412yes—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—41no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—41no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×Q811yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42C2411yescharacteristic
188#108—44no—
189#108—44no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8C2×D412yes—
205C4⋊C8C2×D412yes—
206C4⋊C8C2×D412yes—
207C4⋊C8C2×D412yes—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214#97—21no—
215#97—21no—
216#97—22no—
217#97—22no—
218#97—41no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—41no—
226C24C22⋊Q811yescharacteristic
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—84no—
232C2×Q8—84no—
233C2×Q8—84no—
234C2×Q8—84no—
235C2×Q8—84no—
236C2×Q8—84no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—84no—
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4#12611yescharacteristic
260C22×C4#12611yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×Q811yescharacteristic
263C22×C4C2×#4111yescharacteristic
264M16—22no—
265M16—22no—
266M16—22no—
267M16—44no—
268M16—42no—
269M16—42no—
270M16—42no—
271M16—42no—
272M16—44no—
273M16—82no—
274M16—82no—
275M16—82no—
276M16#13714yes—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—44no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—82no—
290C2×C8—82no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C42#12111yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42#12111yescharacteristic
304C42#12111yescharacteristic
305C42C22⋊Q811yescharacteristic
306C42C22⋊Q811yescharacteristic
307C42C22⋊Q811yescharacteristic
308C42C22⋊Q811yescharacteristic
309C42#12611yescharacteristic
310C42#12611yescharacteristic
311C42C22×D411yescharacteristic
312C42C22×D411yescharacteristic
313C42C2×#4111yescharacteristic
314C42C2×#4111yescharacteristic
315C23—22no—
316C23—22no—
317C23—41no—
318C23#39611yescharacteristic
319C23#39611yescharacteristic
320C23C2×C22⋊Q811yescharacteristic
321Q8—84no—
322Q8—84no—
323Q8—84no—
324Q8—84no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4#39611yescharacteristic
350C2×C4#39611yescharacteristic
351C2×C4#39611yescharacteristic
352C2×C4#39611yescharacteristic
353C2×C4#39611yescharacteristic
354C2×C4#39611yescharacteristic
355C2×C4#47112yes—
356C2×C4#47312yes—
357C2×C4C2×#12111yescharacteristic
358C2×C4C2×#12111yescharacteristic
359C2×C4C2×#12111yescharacteristic
360C2×C4C2×C22⋊Q811yescharacteristic
361C2×C4C2×C22⋊Q811yescharacteristic
362C2×C4C2×#12611yescharacteristic
363C2×C4C2×#13712yes—
364C2×C4#57512yes—
365C8—24no—
366C8—42no—
367C8—42no—
368C8—42no—
369C8—42no—
370C8—42no—
371C8—82no—
372C8—82no—
373C8—82no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—44no—
382C22#206412yes—
383C22C2×#39611yescharacteristic
384C22#307812yes—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—84no—
393C4#206412yes—
394C4#206912yes—
395C4#303911yescharacteristic
396C4#304011yescharacteristic
397C4#305612yes—
398C4C2×#47111yescharacteristic
399C4C2×#47311yescharacteristic
400C2—22no—
401C2—22no—
402C2—41no—
403C2C2×#206411yescharacteristic
404C2#2361911yescharacteristic
405C2C2×#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
#213702
C2×#19972
#178092
#177962
#177462
C2×#15482
#106092
#106052
#80042