Finite Group Databasebeta

← #72234326

Subgroups — #72234326

Order 1024.

559 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#513082C211yesmaximal, characteristic
2#484128C211yesmaximal, characteristic, complemented
3#484118C211yesmaximal, characteristic, complemented
4#151129C212yesmaximal, complemented
5#151115C212yesmaximal, complemented
6#151115C212yesmaximal, complemented
7#151085C212yesmaximal, complemented
8#151038C211yesmaximal, characteristic, complemented
9#150126C211yesmaximal, characteristic, complemented
10#150124C211yesmaximal, characteristic, complemented
11#149509C211yesmaximal, characteristic, complemented
12#33733C2211yescharacteristic
13#33733C2211yescharacteristic
14#18084—21no—
15#18066—21no—
16#13733C411yescharacteristic
17#13725C411yescharacteristic
18#13719C411yescharacteristic
19#13719C411yescharacteristic
20#13697C412yes—
21#13697C412yes—
22#13654C2212yes—
23#13654C2212yes—
24#13640C2211yescharacteristic
25C4×M64C2211yescharacteristic
26#12086C2212yes—
27#12086C2212yes—
28#12073—22no—
29#12069—22no—
30#12045C2212yes—
31#12045C2212yes—
32#12008C2211yescharacteristic
33C2×#1372C2211yescharacteristic
34#11649—21no—
35#11645—21no—
36#11408C2211yescharacteristic
37#7463C2211yescharacteristic
38#7463C2211yescharacteristic
39#7463C2211yescharacteristic
40#7463C2211yescharacteristic
41#7385C2212yes—
42#7385C2212yes—
43#7385C2212yes—
44#7385C2212yes—
45#7332C2212yes—
46#7330C2212yes—
47#3169—21no—
48#3169—21no—
49#2979—41no—
50#2935—41no—
51C2×D32C2×C412yes—
52C2×D32C2×C412yes—
53C2×M64C2×C411yescharacteristic
54C2×M64C2×C411yescharacteristic
55#2252C2×C411yescharacteristic
56#2252C2×C411yescharacteristic
57#2246C2×C411yescharacteristic
58#2246C2×C411yescharacteristic
59#2240—21no—
60#2240—21no—
61#2215—21no—
62#2215—21no—
63#2215C2×C412yes—
64#2215C2×C412yes—
65C4×D16—21no—
66C4×D16—21no—
67C4×D16C2312yes—
68C4×D16C2312yes—
69#2162D411yescharacteristic
70C2×C4×C16C2311yescharacteristic
71#1589—22no—
72#1589—22no—
73#1586—42no—
74#1581—42no—
75C8×D8—22no—
76C8×D8—22no—
77#1564—21no—
78C2×#335—21no—
79#1484—41no—
80#1480—41no—
81C8×M16D411yescharacteristic
82#1424C2×C412yes—
83#1424C2×C412yes—
84C4⋊C32—21no—
85#1415C2×C412yes—
86#1415C2×C412yes—
87#1415C2×C412yes—
88#1415C2×C412yes—
89C22⋊C32—21no—
90#1397C2312yes—
91#1397C2311yescharacteristic
92#1397C2311yescharacteristic
93C4×C32C2×C411yescharacteristic
94C4×C32C2×C411yescharacteristic
95#1372—21no—
96#1372C2311yescharacteristic
97#1372C2311yescharacteristic
98#1372C2311yescharacteristic
99#1372C2311yescharacteristic
100#1371—21no—
101#1331—22no—
102#1331—22no—
103#1331—22no—
104#1331—22no—
105#1312D412yes—
106#1311C2312yes—
107C2×#131—81no—
108#532—41no—
109#532—41no—
110C2×D16—21no—
111C2×D16—21no—
112C2×D16C22×C412yes—
113C2×D16C22×C412yes—
114C22×C16C22×C411yescharacteristic
115C22×C16#4111yescharacteristic
116#500—21no—
117#500—21no—
118#493—21no—
119#493—21no—
120#486—41no—
121#486—41no—
122#466—22no—
123#466—22no—
124#466—41no—
125#466—41no—
126C4×D8—22no—
127C4×D8—22no—
128C4×D8—41no—
129C4×D8—41no—
130#436—42no—
131#436—42no—
132#436—81no—
133#435—82no—
134C8×D4—42no—
135C8×D4—42no—
136C8×D4—81no—
137#431—41no—
138#431D811yescharacteristic
139C2×C4⋊C8—41no—
140#408—82no—
141#408—81no—
142C4×M16D811yescharacteristic
143C2×C4×C8C2×D411yescharacteristic
144D32—24no—
145D32—42no—
146M64—22no—
147M64—41no—
148C2×C32—22no—
149C2×C32C22×C411yescharacteristic
150C2×C32C22×C411yescharacteristic
151C2×C32C22×C411yescharacteristic
152C2×C32C22×C411yescharacteristic
153#366—21no—
154#366—21no—
155#366C22×C412yes—
156#366C22×C412yes—
157C4⋊C16—21no—
158#357—22no—
159#357—22no—
160#357—22no—
161#357—22no—
162C22⋊C16—21no—
163C4×C16C22×C411yescharacteristic
164C4×C16C2×D412yes—
165C4×C16C2×D411yescharacteristic
166C4×C16#4111yescharacteristic
167C4×C16C2411yescharacteristic
168#335—21no—
169#335—21no—
170#335—21no—
171#335—21no—
172#335—41no—
173#334—41no—
174D4⋊C8—42no—
175D4⋊C8—42no—
176D4⋊C8—42no—
177D4⋊C8—42no—
178#322D812yes—
179C82C2×D412yes—
180C2×#41—81no—
181C2×#41—81no—
182C2×D8—22no—
183C2×D8—22no—
184C2×D8—41no—
185C2×D8—41no—
186#131—164no—
187C2×M16—21no—
188C2×M16—82no—
189C2×M16—82no—
190C22×C8—82no—
191C22×C8—82no—
192C22×C8—81no—
193C22×C8C4×D411yescharacteristic
194C22×C8#13511yescharacteristic
195C4⋊Q8—41no—
196C4⋊Q8—41no—
197#127—41no—
198#127—41no—
199#124—81no—
200#124—81no—
201#121—42no—
202#121—42no—
203#121—81no—
204#121—81no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—81no—
208C4×D4—81no—
209C2×C42C2×D811yescharacteristic
210D16—22no—
211D16—22no—
212D16—22no—
213D16—22no—
214D16—81no—
215D16—81no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—22no—
219C2×C16—41no—
220C2×C16C4×D411yescharacteristic
221C2×C16C4×D411yescharacteristic
222C2×C16C23×C411yescharacteristic
223C2×C16C2×#4111yescharacteristic
224#107—22no—
225#107—22no—
226#107—41no—
227#107—41no—
228C4⋊C8—21no—
229C4⋊C8—41no—
230C4⋊C8—41no—
231C4⋊C8—41no—
232C4⋊C8—41no—
233C4⋊C8—81no—
234#102—42no—
235#102—42no—
236#102—42no—
237#102—42no—
238C22⋊C8—21no—
239C22⋊C8—82no—
240C22⋊C8—82no—
241C22⋊C8—82no—
242C22⋊C8—82no—
243C22⋊C8—81no—
244#97—21no—
245C4×C8—81no—
246C4×C8C4×D411yescharacteristic
247C4×C8C2×D812yes—
248C4×C8C2×D812yes—
249C4×C8C2×D811yescharacteristic
250C4×C8#13511yescharacteristic
251C4×C8C22×D411yescharacteristic
252C32—21no—
253C32—21no—
254C32—21no—
255C32—42no—
256C32#13112yescomplemented
257#41—162no—
258#41—162no—
259#41—321no—
260#41—321no—
261C2×Q8—81no—
262C2×Q8—81no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—81no—
268C2×D4—81no—
269C2×D4—81no—
270C2×D4—81no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4C4×D811yescharacteristic
274C22×C4#50911yescharacteristic
275C22×C4#50911yescharacteristic
276D8—42no—
277D8—42no—
278D8—42no—
279D8—42no—
280D8—161no—
281D8—161no—
282M16—42no—
283M16—164no—
284M16—322no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—41no—
290C2×C8—81no—
291C2×C8—81no—
292C2×C8—81no—
293C2×C8—81no—
294C2×C8—164no—
295C2×C8—162no—
296C2×C8—322no—
297C2×C8C4×D811yescharacteristic
298C2×C8C4×D811yescharacteristic
299C2×C8C2×C4×D411yescharacteristic
300C2×C8C2×#13511yescharacteristic
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C42C4×D811yescharacteristic
310C42#50911yescharacteristic
311C42#50911yescharacteristic
312C42C22×D811yescharacteristic
313C16—21no—
314C16—21no—
315C16—21no—
316C16—42no—
317C16C2×#13111yescharacteristic
318C16#56811yescharacteristic
319C23—82no—
320C23—82no—
321C23#217711yescharacteristic
322Q8—162no—
323Q8—321no—
324D4—82no—
325D4—82no—
326D4—82no—
327D4—82no—
328D4—164no—
329D4—162no—
330D4—162no—
331D4—321no—
332D4—321no—
333D4—322no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—162no—
341C2×C4—162no—
342C2×C4—162no—
343C2×C4—162no—
344C2×C4#217711yescharacteristic
345C2×C4#217711yescharacteristic
346C2×C4#217711yescharacteristic
347C2×C4C2×C4×D811yescharacteristic
348C2×C4C2×#50911yescharacteristic
349C2×C4C2×#50911yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—42no—
354C8—42no—
355C8—161no—
356C8—161no—
357C8—161no—
358C8—161no—
359C8#293511yescharacteristic
360C8#295711yescharacteristic
361C22—21no—
362C22—21no—
363C22—21no—
364C22—162no—
365C22—162no—
366C22—162no—
367C22—162no—
368C22—162no—
369C22—162no—
370C22C2×#217711yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—162no—
376C4—162no—
377C4#1365812yes—
378C4#1806611yescharacteristic
379C4#3346511yescharacteristic
380C2—41no—
381C2—322no—
382C2—322no—
383C2#48411811yescharacteristic
384C2C2×#1365811yescharacteristic
385C2#51278111yescharacteristic
subgroupG/H
center, socleC22C2×#2177
commutatorC2×C16C23×C4
Frattini, cosocleC4×C16C24
prime radical#513082C2
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ #11408 ◁ G
LECSG ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #484128C2semidirect
3. #484118C2semidirect
4. #151129C2semidirect
5. #151115C2semidirect
6. #151115C2semidirect
7. #151085C2semidirect
8. #151038C2semidirect
9. #150126C2semidirect
10. #150124C2semidirect
11. #149509C2semidirect
256. C32#131semidirect

Maximal subgroups

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

subgroupindex
#5130822
#4841282
#4841182
#1511292
#1511152
#1510852
#1510382
#1501262
#1501242
#1495092