Finite Group Databasebeta

← #72260471

Subgroups — #72260471

Order 1024.

616 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#2263C211yesmaximal, characteristic
2#483828C211yesmaximal, characteristic, complemented
3#483508C211yesmaximal, characteristic, complemented
4#151117C214yesmaximal, complemented
5#151115C214yesmaximal, complemented
6#151038C211yesmaximal, characteristic, complemented
7#150062C211yesmaximal, characteristic, complemented
8C2×#7385C211yesmaximal, characteristic, complemented
9#149501C211yesmaximal, characteristic, complemented
10C2×#2263C411yescharacteristic
11C4×#508C2211yescharacteristic
12C2×C4×D16C2211yescharacteristic
13#17895—21no—
14#17857—21no—
15#13654C2212yes—
16#13652C2212yes—
17C4×SD64C412yes—
18C4×D32C412yes—
19#13634C411yescharacteristic
20#13617C411yescharacteristic
21#13616C411yescharacteristic
22C4×M64C2211yescharacteristic
23C2×#1397C2211yescharacteristic
24#12086C2214yes—
25#12081—22no—
26#12045C2212yescomplemented
27#12044C2212yescomplemented
28C8×SD32—22no—
29#12008C2211yescharacteristic
30C2×#1372C2211yescharacteristic
31#11593—21no—
32#11592—21no—
33#11399C2211yescharacteristic
34#7463C2214yescomplemented
35#7386C2212yescomplemented
36#7385C2214yes—
37#7385C2212yescomplemented
38#7332C2214yescomplemented
39C2×#508C2×C411yescharacteristic
40C22×D16C2×C411yescharacteristic
41#2988—41no—
42#2975—41no—
43C4×#135—21no—
44C2×C4×D8—21no—
45#2263—24no—
46C2×SD64C2×C412yes—
47C2×D32C2×C412yes—
48C2×M64C2×C411yescharacteristic
49C2×M64C2×C411yescharacteristic
50C22×C32—21no—
51C4×Q32C2×C411yescharacteristic
52C4×Q32C2311yescharacteristic
53C4×SD32—21no—
54C4×SD32—21no—
55C4×D16—21no—
56C4×D16—21no—
57C4×D16C2×C411yescharacteristic
58C4×D16C2×C412yes—
59C4×D16C2311yescharacteristic
60C4×D16C2312yes—
61#2158C2×C411yescharacteristic
62C2×#366C2×C411yescharacteristic
63#2138—21no—
64C2×#357—21no—
65C4×M32D411yescharacteristic
66C2×C4×C16C2311yescharacteristic
67#1594—42no—
68#1589—24no—
69SD16⋊C8—42no—
70C8×Q16—22no—
71C8×D8—22no—
72#1564—21no—
73C2×#335—21no—
74#1478—41no—
75#1477—41no—
76#1450D411yescharacteristic
77#1424C2×C412yes—
78#1423C2×C412yes—
79#1416C2×C412yes—
80#1415C2×C412yes—
81#1415C2×C412yes—
82#1415C2×C412yes—
83#1397—21no—
84#1397C2312yes—
85#1397C2312yes—
86C4×C32C2×C412yes—
87#1372—21no—
88#1372C2312yes—
89#1372C2312yes—
90#1371—21no—
91#1332—22no—
92#1331—22no—
93#1331—24no—
94#1311C2312yes—
95C8×C16D412yes—
96C2×#135—21no—
97C22×D8—21no—
98C4×#41—41no—
99C2×C4×D4—41no—
100#508—22no—
101#508M1614yescomplemented
102C2×Q32C22×C411yescharacteristic
103C2×SD32—21no—
104C2×D16—21no—
105C2×D16—22no—
106C2×D16—22no—
107C2×D16C22×C412yes—
108C2×D16C22×C411yescharacteristic
109C2×M32—21no—
110C22×C16C22×C411yescharacteristic
111C22×C16#4111yescharacteristic
112C8×Q8—42no—
113C4×Q16—21no—
114C4×Q16—21no—
115C4×SD16—41no—
116C4×SD16—41no—
117C4×D8—22no—
118C4×D8—21no—
119C4×D8—21no—
120C4×D8—22no—
121C4×D8—41no—
122C4×D8—41no—
123#436—44no—
124#436—82no—
125C8×D4—42no—
126C8×D4—82no—
127#431—41no—
128#431—81no—
129#431D811yescharacteristic
130#431D811yescharacteristic
131#427—21no—
132C2×#107—21no—
133C2×C4⋊C8—41no—
134C2×C4⋊C8—81no—
135#416—41no—
136C2×#102—41no—
137#407—82no—
138C2×C4×C8C2×D411yescharacteristic
139SD64—24no—
140D32—24no—
141M64—22no—
142M64—22no—
143C2×C32—22no—
144C2×C32—42no—
145C2×C32C22×C412yes—
146C2×C32C22×C412yes—
147#367—21no—
148#366—21no—
149#366C22×C412yes—
150#366C22×C411yescharacteristic
151#366C22×C411yescharacteristic
152#358—21no—
153#358—21no—
154#357—22no—
155#357—21no—
156#357—22no—
157#357—21no—
158#346—21no—
159C4×C16C22×C411yescharacteristic
160C4×C16C2×D412yes—
161C4×C16C2×D411yescharacteristic
162C4×C16#4111yescharacteristic
163C4×C16C2411yescharacteristic
164#335—22no—
165#335—22no—
166#335—41no—
167#334—41no—
168Q8⋊C8—42no—
169D4⋊C8—42no—
170D4⋊C8—44no—
171#322D812yes—
172C82C2×D412yes—
173C2×#41—41no—
174C22×D4—41no—
175C23×C4—81no—
176#135—24no—
177#135—42no—
178C2×Q16—21no—
179C2×SD16—41no—
180C2×D8—21no—
181C2×D8—22no—
182C2×D8—42no—
183C2×D8—41no—
184C2×D8—42no—
185C2×M16—84no—
186C22×C8—82no—
187C22×C8—81no—
188C22×C8C4×D411yescharacteristic
189C22×C8#13511yescharacteristic
190C4×Q8—41no—
191C4×Q8—41no—
192C4×D4—42no—
193C4×D4—41no—
194C4×D4—42no—
195C4×D4—41no—
196C4×D4—81no—
197C4×D4—81no—
198C4×D4—81no—
199C4×D4—81no—
200#117—41no—
201#117—81no—
202C2×C4⋊C4—41no—
203C2×C22⋊C4—81no—
204C2×C42—81no—
205C2×C42C2×D811yescharacteristic
206Q32—21no—
207Q32C2×M1612yes—
208SD32—24no—
209D16—22no—
210D16—21no—
211D16—22no—
212D16—42no—
213D16C2×M1612yes—
214M32—24no—
215C2×C16—22no—
216C2×C16—22no—
217C2×C16C4×D411yescharacteristic
218C2×C16C4×D411yescharacteristic
219C2×C16C2×M1612yes—
220C2×C16#13112yes—
221C2×C16C23×C411yescharacteristic
222C2×C16C2×#4111yescharacteristic
223#107—22no—
224#107—21no—
225#107—21no—
226#107—41no—
227#106—41no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—82no—
233C4⋊C8—81no—
234#103—41no—
235#103—41no—
236#102—42no—
237#102—41no—
238#102—42no—
239#102—41no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—84no—
243C22⋊C8—82no—
244C22⋊C8—81no—
245C4×C8—82no—
246C4×C8—81no—
247C4×C8C4×D411yescharacteristic
248C4×C8C2×D812yes—
249C4×C8C2×D812yes—
250C4×C8C2×D811yescharacteristic
251C4×C8#13511yescharacteristic
252C4×C8C22×D411yescharacteristic
253C32—22no—
254C32—22no—
255C32—42no—
256C24—81no—
257#41—44no—
258#41—82no—
259C2×Q8—41no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—81no—
263C2×D4—82no—
264C2×D4—81no—
265C2×D4—82no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—82no—
270C22×C4—81no—
271C22×C4—82no—
272C22×C4—322no—
273C22×C4C4×D811yescharacteristic
274C22×C4#50911yescharacteristic
275C22×C4#50911yescharacteristic
276Q16—22no—
277Q16—41no—
278SD16—44no—
279D8—22no—
280D8—42no—
281D8—41no—
282D8—42no—
283D8—82no—
284M16—324no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—82no—
289C2×C8—82no—
290C2×C8—322no—
291C2×C8—321no—
292C2×C8#43512yes—
293C2×C8C4×D811yescharacteristic
294C2×C8C4×D811yescharacteristic
295C2×C8#44412yes—
296C2×C8C2×C4×D411yescharacteristic
297C2×C8C2×#13511yescharacteristic
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C22⋊C4—81no—
305C22⋊C4—82no—
306C22⋊C4—81no—
307C22⋊C4—82no—
308C22⋊C4—81no—
309C42—81no—
310C42—81no—
311C42—81no—
312C42C4×D811yescharacteristic
313C42#50911yescharacteristic
314C42#50911yescharacteristic
315C42C22×D811yescharacteristic
316C16—24no—
317C16—21no—
318C16—21no—
319C16#43512yes—
320C16C22×M1611yescharacteristic
321C16C2×#13111yescharacteristic
322C23—81no—
323C23—82no—
324C23—162no—
325C23—321no—
326C23—321no—
327C23#217711yescharacteristic
328Q8—42no—
329Q8—81no—
330D4—42no—
331D4—84no—
332D4—81no—
333D4—82no—
334D4—82no—
335D4—162no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—81no—
339C2×C4—84no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—161no—
344C2×C4—164no—
345C2×C4—161no—
346C2×C4—321no—
347C2×C4—322no—
348C2×C4—321no—
349C2×C4#158112yes—
350C2×C4#217711yescharacteristic
351C2×C4#217711yescharacteristic
352C2×C4#217711yescharacteristic
353C2×C4#217812yes—
354C2×C4C2×C4×D811yescharacteristic
355C2×C4C2×#50911yescharacteristic
356C2×C4C2×#50911yescharacteristic
357C8—21no—
358C8—21no—
359C8—42no—
360C8—42no—
361C8—162no—
362C8—162no—
363C8#158112yes—
364C8C2×#43511yescharacteristic
365C8C2×#44411yescharacteristic
366C22—21no—
367C22—21no—
368C22—82no—
369C22—161no—
370C22—161no—
371C22—162no—
372C22—162no—
373C22—162no—
374C22—324no—
375C22#1207312yes—
376C22C2×#217711yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—161no—
383C4—162no—
384C4—161no—
385C4#1207312yes—
386C4C2×#158111yescharacteristic
387C4C2×#217811yescharacteristic
388C2—22no—
389C2—162no—
390C2—322no—
391C2C2×#1207311yescharacteristic
392C2#51277911yescharacteristic
393C2#51278111yescharacteristic
subgroupG/H
center, socleC22C2×#2177
commutatorC2×C16C23×C4
Frattini, cosocleC4×C16C24
prime radicalC2×#2263C4
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ #11399 ◁ G
LECSG ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #483828C2semidirect
3. #483508C2semidirect
4. #151117C2semidirect
5. #151115C2semidirect
6. #151038C2semidirect
7. #150062C2semidirect
8. C2×#7385C2semidirect
9. #149501C2semidirect
26. #12045C22semidirect
27. #12044C22semidirect
34. #7463C22semidirect
35. #7386C22semidirect
37. #7385C22semidirect
38. #7332C22semidirect
101. #508M16semidirect

Maximal subgroups

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

subgroupindex
C4×#22632
#4838282
#4835082
#1511172
#1511152
#1510382
#1500622
C2×#73852
#1495012