Finite Group Databasebeta

← #28316411

Subgroups — #28316411

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291608C211yesmaximal, characteristic, complemented
2#20291562C211yesmaximal, characteristic
3SD16⋊#88324C214yesmaximal, complemented
4SD16×#88324C214yesmaximal, complemented
5#20290262C211yesmaximal, characteristic, complemented
6#20290017C211yesmaximal, characteristic, complemented
7#20290014C211yesmaximal, characteristic
8#20289918C211yesmaximal, characteristic, complemented
9C2×SD16×Dic53C211yesmaximal, characteristic
10Q8×#88324—22no—
11Q8×#88324C2214yescomplemented
12D4×#88324—22no—
13D4×#88324C2214yes—
14#16811639—22no—
15#16811639C2211yescharacteristic
16#16811639C2211yescharacteristic
17#16811639C2211yescharacteristic
18#16811639C2211yescharacteristic
19C2×C22⋊#88324—22no—
20#16811486C2212yescomplemented
21#16811484C2212yescomplemented
22Q8⋊#88324C2214yescomplemented
23D4⋊#88324C2214yes—
24#16811464C2212yescomplemented
25C8×#88324C2212yescomplemented
26C2×Q8×Dic53C2211yescharacteristic
27C2×D4×Dic53C2211yescharacteristic
28SD16×Dic53C412yes—
29SD16×Dic53C412yes—
30SD16×Dic53C2212yes—
31SD16×Dic53C2212yes—
32C2×#13133149C411yescharacteristic
33C2×#13133141C411yescharacteristic
34C2×#13133138C411yescharacteristic
35C2×C8×Dic53C2211yescharacteristic
36C22×SD16×C53C411yescharacteristic
37C22×#88324—24no—
38C4⋊#88324—21no—
39C4⋊#88324—24no—
40C4⋊#88324D412yes—
41C4⋊#88324C2312yes—
42C4⋊#88324C2312yes—
43C22⋊#88324—21no—
44C22⋊#88324—21no—
45C22⋊#88324—21no—
46C22⋊#88324—24no—
47C22⋊#88324D412yes—
48C4×#88324—21no—
49C4×#88324—21no—
50C4×#88324—24no—
51C4×#88324C2312yes—
52C4×#88324C2312yes—
53C23×Dic53—21no—
54C22×#88322C2×C411yescharacteristic
55Q8×Dic53—21no—
56Q8×Dic53—21no—
57Q8×Dic53C2×C412yes—
58Q8×Dic53C2312yes—
59D4×Dic53—21no—
60D4×Dic53—21no—
61D4×Dic53C2×C412yes—
62D4×Dic53C2312yes—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53C2×C411yescharacteristic
65C2×C4⋊Dic53C2×C411yescharacteristic
66C2×C22⋊Dic53—21no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53C2311yescharacteristic
69#13133149C2×C412yes—
70#13133149C2×C412yes—
71#13133141C2×C412yes—
72#13133141C2×C412yes—
73#13133138C2×C412yes—
74#13133138C2×C412yes—
75C8×Dic53C2×C411yescharacteristic
76C8×Dic53C2×C411yescharacteristic
77C8×Dic53C2311yescharacteristic
78C8×Dic53C2311yescharacteristic
79C22×Q8×C53C2×C411yescharacteristic
80C22×D4×C53C2×C411yescharacteristic
81C2×SD16×C53C818yescomplemented
82C2×SD16×C53C2×C412yes—
83C2×SD16×C53C2×C412yes—
84C22×C8×C53C2×C411yescharacteristic
85C24×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—22no—
88C2×Q8×C53C2×C814yescomplemented
89C2×Q8×C53C22×C412yes—
90C2×D4×C53—21no—
91C2×D4×C53—22no—
92C2×D4×C53C2×C814yes—
93C2×D4×C53C22×C412yes—
94C22×C4×C53—21no—
95C22×C4×C53C22×C411yescharacteristic
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—22no—
99C22×Dic53—22no—
100C22×Dic53—82no—
101C22×Dic53C2×D411yescharacteristic
102C22×Dic53#4111yescharacteristic
103C2×#88324—22no—
104C2×#88324—22no—
105C2×#88324—22no—
106C2×#88324—84no—
107C2×#88324C2×D414yes—
108SD16×C53C2×C818yescomplemented
109SD16×C53C2×C818yescomplemented
110C2×C8×C53C2×C814yescomplemented
111C2×C8×C53C22×C411yescharacteristic
112C2×C8×C53C22×C411yescharacteristic
113C22⋊Dic53—22no—
114C22⋊Dic53—22no—
115C4⋊Dic53—21no—
116C4⋊Dic53—21no—
117C4⋊Dic53—22no—
118C4⋊Dic53—22no—
119C4⋊Dic53C22×C412yes—
120C4⋊Dic53C22×C412yes—
121C4×Dic53—22no—
122C4×Dic53—22no—
123C4×Dic53C22×C411yescharacteristic
124C4×Dic53C2×D411yescharacteristic
125C4×Dic53#4111yescharacteristic
126C4×Dic53C2411yescharacteristic
127C2×#88322—22no—
128C2×#88322C22×C411yescharacteristic
129C2×#88322C22×C411yescharacteristic
130C23×C53—22no—
131C23×C53—24no—
132C23×C53—81no—
133C23×C53—81no—
134C23×C53C4×D411yescharacteristic
135Q8×C53—22no—
136Q8×C53—24no—
137Q8×C53C22×C814yes—
138D4×C53—22no—
139D4×C53—24no—
140D4×C53C22×C814yes—
141C2×C4×C53—22no—
142C2×C4×C53—24no—
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53C22×C812yes—
145C2×C4×C53#13112yes—
146C2×C4×C53C23×C411yescharacteristic
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—22no—
150C2×Dic53—41no—
151C2×Dic53—44no—
152C2×Dic53—82no—
153C2×Dic53—81no—
154C2×Dic53—81no—
155C2×Dic53C4×D411yescharacteristic
156C2×Dic53C4×D411yescharacteristic
157C2×Dic53C22×D411yescharacteristic
158C2×Dic53C2×#4111yescharacteristic
159#88324—24no—
160#88324—42no—
161#88324—42no—
162#88324—42no—
163C8×C53C22×C812yes—
164C8×C53C22×C812yes—
165#88322—21no—
166#88322—21no—
167#17855—531nomaximal, Sylow
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—24no—
171C22×C53—42no—
172C22×C53—42no—
173C22×C53—84no—
174C22×C53C8×D412yes—
175C22×C53C2×C4×D411yescharacteristic
176C4×C53—24no—
177C4×C53C8×D412yes—
178C4×C53C23×C811yescharacteristic
179C4×C53C2×#13111yescharacteristic
180Dic53—21no—
181Dic53—21no—
182Dic53—21no—
183Dic53—42no—
184Dic53—42no—
185Dic53#57712yes—
186#2982—531no—
187#2972—531no—
188C2×C4×SD16—531no—
189SD16⋊C8—534no—
190C8×SD16—534no—
191#1564—531no—
192#1479—531no—
193#1478—531no—
194#1450—531no—
195C2×C53—22no—
196C2×C53—44no—
197C2×C53C2×C8×D411yescharacteristic
198C2×C53#294911yescharacteristic
199C2×C53#295711yescharacteristic
200C22×SD16#615711yescharacteristic
201C2×C4×Q8—531no—
202C2×C4×D4—531no—
203C8×Q8—534no—
204C8×Q8—1062no—
205C4×SD16—532no—
206C4×SD16—532no—
207C4×SD16—532no—
208C4×SD16—532no—
209C8×D4—534no—
210C8×D4—1062no—
211#431—531no—
212#431—531no—
213#431—531no—
214#431—531no—
215#431—1062no—
216C2×#106—531no—
217C2×#103—531no—
218C2×#102—531no—
219C2×C22⋊C8—1062no—
220C2×C4×C8—531no—
221#335—532no—
222#334—532no—
223Q8⋊C8—534no—
224D4⋊C8—534no—
225#322—532no—
226C82—532no—
227C53#1785511yescharacteristic, complemented, Sylow
228C22×Q8C2×#615711yescharacteristic
229C22×D4C2×#615711yescharacteristic
230C23×C4—1061no—
231C2×SD16#8832418yescomplemented
232C2×SD16C2×#615712yes—
233C2×SD16C2×#615712yes—
234C22×C8—531no—
235C22×C8—1064no—
236C22×C8C2×#615711yescharacteristic
237C4×Q8—532no—
238C4×Q8—532no—
239C4×Q8—1061no—
240C4×Q8—1061no—
241C4×D4—532no—
242C4×D4—532no—
243C4×D4—1061no—
244C4×D4—1061no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—1061no—
248C2×C22⋊C4—1061no—
249C2×C42—531no—
250C2×C42—1061no—
251#106—532no—
252#106—532no—
253C4⋊C8—532no—
254C4⋊C8—532no—
255C4⋊C8—532no—
256C4⋊C8—1061no—
257C4⋊C8—1064no—
258#103—532no—
259#103—532no—
260#102—532no—
261#102—532no—
262C22⋊C8—532no—
263C22⋊C8—1064no—
264C22⋊C8—1061no—
265C22⋊C8—1061no—
266C22⋊C8—1061no—
267C4×C8—532no—
268C4×C8—531no—
269C4×C8—531no—
270C4×C8—531no—
271C4×C8—531no—
272C4×C8—532no—
273C4×C8—1061no—
274C4×C8—1061no—
275C4×C8—1064no—
276C24—21no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8C2×#8832414yescomplemented
280C2×Q8C22×#615712yes—
281C2×D4—21no—
282C2×D4—22no—
283C2×D4C2×#8832414yes—
284C2×D4C22×#615712yes—
285C22×C4—21no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—1062no—
289C22×C4—1062no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—4242no—
293C22×C4C22×#615711yescharacteristic
294SD16C2×#8832418yescomplemented
295SD16C2×#8832418yescomplemented
296C2×C8—534no—
297C2×C8—531no—
298C2×C8—531no—
299C2×C8—1062no—
300C2×C8—1062no—
301C2×C8—1062no—
302C2×C8—1062no—
303C2×C8—4244no—
304C2×C8C2×#8832414yescomplemented
305C2×C8C22×#615711yescharacteristic
306C2×C8C22×#615711yescharacteristic
307C4⋊C4—532no—
308C4⋊C4—532no—
309C4⋊C4—1062no—
310C4⋊C4—1062no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C22⋊C4—1062no—
314C22⋊C4—1062no—
315C42—531no—
316C42—531no—
317C42—531no—
318C42—531no—
319C42—1062no—
320C42—1062no—
321C23—22no—
322C23—24no—
323C23—81no—
324C23—81no—
325C23D4×#615711yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8C22×#8832414yes—
329D4—24no—
330D4—22no—
331D4C22×#8832414yes—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—531no—
335C2×C4—531no—
336C2×C4—531no—
337C2×C4—531no—
338C2×C4—1062no—
339C2×C4—1062no—
340C2×C4—1062no—
341C2×C4—4242no—
342C2×C4—4241no—
343C2×C4—4241no—
344C2×C4—2124no—
345C2×C4—2121no—
346C2×C4D4×#615711yescharacteristic
347C2×C4C22×#8832412yes—
348C2×C4#1313330412yes—
349C2×C4C23×#615711yescharacteristic
350C8—1064no—
351C8—1061no—
352C8—1061no—
353C8—2122no—
354C8—2122no—
355C8—2122no—
356C8C22×#8832412yes—
357C8C22×#8832412yes—
358C22—21no—
359C22—21no—
360C22—24no—
361C22—42no—
362C22—42no—
363C22—84no—
364C22D4×#8832412yes—
365C22C2×D4×#615711yescharacteristic
366C4—24no—
367C4—532no—
368C4—1061no—
369C4—1061no—
370C4—1061no—
371C4—2122no—
372C4—2122no—
373C4D4×#8832412yes—
374C4C23×#8832411yescharacteristic
375C4C2×#1313330411yescharacteristic
376C2—22no—
377C2—44no—
378C2C2×D4×#8832411yescharacteristic
379C2#2029147511yescharacteristic
380C2#2029149311yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC4×C53C23×C8
FrattiniC2×C4C23×#6157
FittingC22×SD16×C53C4
socleC22×C53C2×C4×D4
prime radicalC22×D4×C53C2×C4
cosocleC4×Dic53C24
hypercenterC22×SD16#6157
hypocenterC53#17855
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. #20291608C2semidirect
3. SD16⋊#88324C2semidirect
4. SD16×#88324C2semidirect
5. #20290262C2semidirect
6. #20290017C2semidirect
8. #20289918C2semidirect
11. Q8×#88324C22semidirect
20. #16811486C22semidirect
21. #16811484C22semidirect
22. Q8⋊#88324C22semidirect
24. #16811464C22semidirect
25. C8×#88324C22semidirect
81. C2×SD16×C53C8semidirect
88. C2×Q8×C53C2×C8semidirect
108. SD16×C53C2×C8semidirect
109. SD16×C53C2×C8semidirect
110. C2×C8×C53C2×C8semidirect
227. C53#17855semidirect
231. C2×SD16#88324semidirect
279. C2×Q8C2×#88324semidirect
294. SD16C2×#88324semidirect
295. SD16C2×#88324semidirect
304. C2×C8C2×#88324semidirect

Maximal subgroups

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

subgroupindex
#202916082
#202915622
SD16⋊#883242
SD16×#883242
#202902622
#202900172
#202900142
#202899182
C2×SD16×Dic532
#1785553