Finite Group Databasebeta

← #72322310

Subgroups — #72322310

Order 1024.

645 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
1C4×#2263C211yesmaximal, characteristic
2#483823C211yesmaximal, characteristic, complemented
3C2×#12045C211yesmaximal, characteristic, complemented
4#151116C214yesmaximal, complemented
5#151115C214yesmaximal, complemented
6#151038C211yesmaximal, characteristic, complemented
7#150064C211yesmaximal, characteristic, complemented
8#150063C211yesmaximal, characteristic, complemented
9#149502C211yesmaximal, characteristic, complemented
10C2×#2263C411yescharacteristic
11C4×#508C2211yescharacteristic
12C2×C4×D16C2211yescharacteristic
13#17886—21no—
14C2×C8×D8—21no—
15#13654C2212yes—
16#13652C2212yes—
17C4×SD64C412yes—
18C4×D32C412yes—
19#13634C411yescharacteristic
20#13617C411yescharacteristic
21#13616C411yescharacteristic
22C4×M64C2211yescharacteristic
23C4×M64C2211yescharacteristic
24#12086C2212yescomplemented
25#12085C2212yescomplemented
26#12084—22no—
27#12045—22no—
28#12045C2214yes—
29#12008C2211yescharacteristic
30C2×#1372C2211yescharacteristic
31#11591—21no—
32C2×#1331—21no—
33C2×#1311C2211yescharacteristic
34#7463C2214yescomplemented
35#7386C2212yescomplemented
36#7385C2214yes—
37#7385C2212yescomplemented
38#7332C2214yescomplemented
39C2×#508C2×C411yescharacteristic
40C22×D16C2×C411yescharacteristic
41#2965—41no—
42C4×#135—21no—
43C2×C4×D8—21no—
44C2×C8×D4—41no—
45#2263—24no—
46C2×SD64C2×C412yes—
47C2×D32C2×C412yes—
48C2×M64—21no—
49C2×M64C2×C411yescharacteristic
50C2×M64C2×C411yescharacteristic
51C4×Q32C2×C411yescharacteristic
52C4×Q32C2311yescharacteristic
53C4×SD32—21no—
54C4×SD32—21no—
55C4×D16—21no—
56C4×D16—21no—
57C4×D16C2×C411yescharacteristic
58C4×D16C2×C412yes—
59C4×D16C2312yes—
60C4×D16C2311yescharacteristic
61#2158C2×C411yescharacteristic
62C2×#366C2×C411yescharacteristic
63#2138—21no—
64C2×#357—21no—
65C2×C4×C16D411yescharacteristic
66C2×C4×C16C2311yescharacteristic
67#1591—22no—
68#1590—42no—
69#1589—22no—
70C8×D8—24no—
71C8×D8—42no—
72#1564—21no—
73C2×#335—21no—
74#1476—41no—
75C2×D4⋊C8—41no—
76C2×C82D411yescharacteristic
77#1424C2×C412yes—
78#1423C2×C412yes—
79#1416C2×C412yes—
80#1415C2×C412yes—
81#1415C2×C412yes—
82#1415C2×C412yes—
83#1397C2312yes—
84#1397C2312yes—
85C4×C32—21no—
86C4×C32C2×C412yes—
87#1372—21no—
88#1372C2312yes—
89#1372C2312yes—
90#1371—21no—
91#1332—22no—
92#1331—24no—
93#1331—22no—
94#1311D412yes—
95#1311C2312yes—
96C2×#135—21no—
97C22×D8—21no—
98C23×C8—81no—
99C4×#41—41no—
100C2×C4×D4—41no—
101#508—22no—
102#508—22no—
103C2×Q32C22×C411yescharacteristic
104C2×SD32—21no—
105C2×D16—21no—
106C2×D16—22no—
107C2×D16M1614yescomplemented
108C2×D16C22×C411yescharacteristic
109C2×D16C22×C412yes—
110C22×C16—21no—
111C22×C16C22×C411yescharacteristic
112C22×C16#4111yescharacteristic
113#446—42no—
114C4×Q16—21no—
115C4×Q16—21no—
116C4×SD16—41no—
117C4×SD16—41no—
118C4×D8—22no—
119C4×D8—21no—
120C4×D8—22no—
121C4×D8—21no—
122C4×D8—41no—
123C4×D8—41no—
124#436—42no—
125#435—82no—
126C8×D4—44no—
127C8×D4—82no—
128#431—41no—
129#431—81no—
130#427—21no—
131C2×#107—21no—
132C2×C4⋊C8—41no—
133#416—41no—
134C2×#102—41no—
135C2×C22⋊C8—81no—
136C4×M16—81no—
137C2×C4×C8D811yescharacteristic
138C2×C4×C8D811yescharacteristic
139C2×C4×C8C2×D411yescharacteristic
140SD64—24no—
141D32—24no—
142M64—22no—
143M64—22no—
144M64—42no—
145C2×C32—22no—
146C2×C32C22×C412yes—
147C2×C32C22×C412yes—
148#367—21no—
149#366—21no—
150#366C22×C412yes—
151#366C22×C411yescharacteristic
152#366C22×C411yescharacteristic
153#358—21no—
154#358—21no—
155#357—22no—
156#357—21no—
157#357—22no—
158#357—21no—
159C4×C16—21no—
160C4×C16C22×C411yescharacteristic
161C4×C16C2×D412yes—
162C4×C16C2×D411yescharacteristic
163C4×C16#4111yescharacteristic
164C4×C16C2411yescharacteristic
165#335—22no—
166#335—22no—
167#335—41no—
168#334—41no—
169Q8⋊C8—42no—
170D4⋊C8—44no—
171D4⋊C8—42no—
172C82D812yes—
173C82C2×D412yes—
174C2×#41—41no—
175C22×D4—41no—
176C23×C4—81no—
177#135—42no—
178#135—42no—
179C2×Q16—21no—
180C2×SD16—41no—
181C2×D8—22no—
182C2×D8—21no—
183C2×D8—24no—
184C2×D8—41no—
185C2×D8—42no—
186C2×M16—82no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—84no—
190C22×C8—81no—
191C22×C8—164no—
192C22×C8C4×D411yescharacteristic
193C22×C8#13511yescharacteristic
194C4×Q8—41no—
195C4×Q8—41no—
196C4×D4—42no—
197C4×D4—41no—
198C4×D4—42no—
199C4×D4—41no—
200C4×D4—81no—
201C4×D4—81no—
202C4×D4—81no—
203C4×D4—81no—
204#117—41no—
205#117—81no—
206C2×C4⋊C4—41no—
207C2×C22⋊C4—81no—
208C2×C42—81no—
209C2×C42C2×D811yescharacteristic
210Q32—21no—
211Q32—21no—
212SD32—42no—
213D16—21no—
214D16—22no—
215D16—21no—
216D16—24no—
217D16C2×M1614yes—
218C2×C16—24no—
219C2×C16—22no—
220C2×C16—22no—
221C2×C16C4×D411yescharacteristic
222C2×C16C4×D411yescharacteristic
223C2×C16C2×M1612yes—
224C2×C16#13112yes—
225C2×C16C23×C411yescharacteristic
226C2×C16C2×#4111yescharacteristic
227#107—22no—
228#107—21no—
229#107—21no—
230#107—41no—
231#106—41no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—82no—
235C4⋊C8—81no—
236#103—41no—
237#103—41no—
238#102—42no—
239#102—41no—
240#102—42no—
241#102—41no—
242C22⋊C8—84no—
243C22⋊C8—82no—
244C22⋊C8—81no—
245#97—82no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—81no—
249C4×C8C4×D411yescharacteristic
250C4×C8C2×D812yes—
251C4×C8C2×D812yes—
252C4×C8C2×D811yescharacteristic
253C4×C8#13511yescharacteristic
254C4×C8C22×D411yescharacteristic
255C32—22no—
256C32—22no—
257C32—42no—
258C24—81no—
259#41—82no—
260#41—82no—
261C2×Q8—41no—
262C2×D4—44no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—81no—
266C2×D4—81no—
267C2×D4—82no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—82no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—82no—
274C22×C4—162no—
275C22×C4—162no—
276C22×C4C4×D811yescharacteristic
277C22×C4#50911yescharacteristic
278C22×C4#50911yescharacteristic
279Q16—41no—
280Q16—41no—
281SD16—82no—
282D8—24no—
283D8—44no—
284D8—41no—
285D8—42no—
286D8—41no—
287M16—164no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—42no—
292C2×C8—81no—
293C2×C8—82no—
294C2×C8—82no—
295C2×C8—168no—
296C2×C8—162no—
297C2×C8#43512yes—
298C2×C8C4×D811yescharacteristic
299C2×C8C4×D811yescharacteristic
300C2×C8#44412yes—
301C2×C8C2×C4×D411yescharacteristic
302C2×C8C2×#13511yescharacteristic
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C22⋊C4—81no—
310C22⋊C4—82no—
311C22⋊C4—81no—
312C22⋊C4—82no—
313C22⋊C4—81no—
314C42—81no—
315C42—81no—
316C42—81no—
317C42C4×D811yescharacteristic
318C42#50911yescharacteristic
319C42#50911yescharacteristic
320C42C22×D811yescharacteristic
321C16—24no—
322C16—21no—
323C16—21no—
324C16#43512yes—
325C16C22×M1611yescharacteristic
326C16C2×#13111yescharacteristic
327C23—84no—
328C23—82no—
329C23—81no—
330C23—162no—
331C23—162no—
332C23#217711yescharacteristic
333Q8—81no—
334Q8—81no—
335D4—44no—
336D4—84no—
337D4—82no—
338D4—81no—
339D4—81no—
340D4—162no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—162no—
348C2×C4—162no—
349C2×C4—162no—
350C2×C4—162no—
351C2×C4—162no—
352C2×C4—162no—
353C2×C4—164no—
354C2×C4#158112yes—
355C2×C4#217711yescharacteristic
356C2×C4#217711yescharacteristic
357C2×C4#217711yescharacteristic
358C2×C4#217812yes—
359C2×C4C2×C4×D811yescharacteristic
360C2×C4C2×#50911yescharacteristic
361C2×C4C2×#50911yescharacteristic
362C8—21no—
363C8—21no—
364C8—42no—
365C8—42no—
366C8—162no—
367C8—162no—
368C8#158112yes—
369C8C2×#43511yescharacteristic
370C8C2×#44411yescharacteristic
371C22—21no—
372C22—21no—
373C22—84no—
374C22—168no—
375C22—162no—
376C22—162no—
377C22—161no—
378C22—161no—
379C22—161no—
380C22#1207312yes—
381C22C2×#217711yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—161no—
386C4—162no—
387C4—161no—
388C4—161no—
389C4#1207312yes—
390C4#1365812yes—
391C4C2×#158111yescharacteristic
392C4C2×#217811yescharacteristic
393C2—22no—
394C2—164no—
395C2—321no—
396C2C2×#1207311yescharacteristic
397C2C2×#1365811yescharacteristic
398C2#51277811yescharacteristic
subgroupG/H
centerC2×C4#2177
commutatorC2×C16C23×C4
Frattini, cosocleC4×C16C24
socleC22C2×#2177
prime radicalC2×#2263C4
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ C2×#1311 ◁ G
LECSG ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #483823C2semidirect
3. C2×#12045C2semidirect
4. #151116C2semidirect
5. #151115C2semidirect
6. #151038C2semidirect
7. #150064C2semidirect
8. #150063C2semidirect
9. #149502C2semidirect
24. #12086C22semidirect
25. #12085C22semidirect
34. #7463C22semidirect
35. #7386C22semidirect
37. #7385C22semidirect
38. #7332C22semidirect
107. C2×D16M16semidirect

Maximal subgroups

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

subgroupindex
C4×#22632
#4838232
C2×#120452
#1511162
#1511152
#1510382
#1500642
#1500632
#1495022