Finite Group Databasebeta

← #15848218

Subgroups — #15848218

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13124637C211yesmaximal, characteristic
2C2×#11718755C211yesmaximal, characteristic
3#13123916C211yesmaximal, characteristic
4#13114291C214yesmaximal, complemented
5#13114203C214yesmaximal, complemented
6#13113611C211yesmaximal, characteristic
7#13113521C211yesmaximal, characteristic
8#13113493C211yesmaximal, characteristic
9C13×#3209C211yesmaximal, characteristic
10C13×#533C2211yescharacteristic
11C2×C13×C4⋊Q8C2211yescharacteristic
12C2×C4×Dic26C2211yescharacteristic
13C13×#494C2212yes—
14C13×#487C2212yes—
15C13×#462—22no—
16C13×#451—22no—
17C13×#431C2211yescharacteristic
18#11718755—22no—
19#11718755C2212yes—
20#11718755C2214yes—
21#11718752C2212yes—
22#11718724—22no—
23#11718665—21no—
24#11718656—21no—
25C2×#88050—21no—
26C2×#88048—21no—
27#11718586C2211yescharacteristic
28#11718585C2211yescharacteristic
29C2×C4⋊#1055C2211yescharacteristic
30#11718491—22no—
31#11718379—22no—
32C8×Dic26C2214yes—
33#11718139C2212yes—
34#11718139C2214yes—
35#11718138C2212yes—
36#11718050C2214yes—
37C22×Q8×C13—21no—
38C22×Dic26D411yescharacteristic
39C2×C13×Q16—24no—
40C13×C4⋊Q8—21no—
41C13×C4⋊Q8—21no—
42C13×C4⋊Q8—21no—
43C13×C4⋊Q8C2311yescharacteristic
44C13×C4⋊Q8C2312yes—
45C13×C4⋊Q8C2311yescharacteristic
46C13×#125—21no—
47C13×C22⋊Q8—21no—
48C13×C22⋊Q8—21no—
49C4×Q8×C13—21no—
50C4×Q8×C13—21no—
51C13×#117—21no—
52C2×C13×C4⋊C4—21no—
53C2×C13×C4⋊C4—21no—
54C2×C42×C13C2311yescharacteristic
55C2×C4⋊Dic13D411yescharacteristic
56C2×#6101—21no—
57C2×C4×Dic13—21no—
58C2×#6099—21no—
59C22×#1055—21no—
60C4×Dic26—21no—
61C4×Dic26—21no—
62C4×Dic26D412yes—
63C4×Dic26C2312yes—
64C13×C4⋊C8D412yes—
65C13×#103—24no—
66C13×C22⋊C8D412yes—
67C4×C8×C13C2312yes—
68C22⋊#1055—21no—
69#88054—24no—
70C8×Dic13—24no—
71#88050—22no—
72#88050—22no—
73#88050—24no—
74#88048—22no—
75#88048—24no—
76#88047—22no—
77C4⋊#1055—21no—
78C4⋊#1055—21no—
79C4⋊#1055C2312yes—
80C4⋊#1055C2312yes—
81C4⋊#1055C2312yes—
82C4×#1055—21no—
83#19399—131nomaximal, Sylow
84C2×Q8×C13—22no—
85C2×Q8×C13—21no—
86C2×Q8×C13—21no—
87C2×Q8×C13—41no—
88C2×Q8×C13—82no—
89C22×C4×C13—41no—
90C22×C4×C13C2×D411yescharacteristic
91C22×C4×C13C2×D411yescharacteristic
92C22×C4×C13#4111yescharacteristic
93C22×Dic13—21no—
94C22×Dic13—21no—
95C2×Dic26—21no—
96C2×Dic26—22no—
97C2×Dic26Q1614yescomplemented
98C2×Dic26C2×D412yes—
99C13×Q16—84no—
100C2×C8×C13C2×D414yes—
101C13×C4⋊C4—21no—
102C13×C4⋊C4—22no—
103C13×C4⋊C4—21no—
104C13×C4⋊C4—21no—
105C13×C4⋊C4—22no—
106C13×C4⋊C4—21no—
107C13×C4⋊C4—41no—
108C13×C4⋊C4—41no—
109C13×C4⋊C4—41no—
110C13×C4⋊C4—41no—
111C13×C22⋊C4—41no—
112C42×C13—41no—
113C42×C13C2×D411yescharacteristic
114C42×C13C2×D411yescharacteristic
115C42×C13#4111yescharacteristic
116C42×C13C2411yescharacteristic
117C4⋊Dic13—21no—
118C4⋊Dic13C2×D412yes—
119#6101—22no—
120#6101—22no—
121C4×Dic13—22no—
122C4×Dic13—22no—
123#6099—42no—
124C2×#1055—22no—
125C2×#1055—22no—
126C2×#1055—22no—
127C2×#1055—42no—
128#3209D1311yescharacteristic
129#3078—131no—
130C2×#475—131no—
131#2982—131no—
132#1684—134no—
133#1670—134no—
134#1571—131no—
135#1549—131no—
136#1542—131no—
137C23×C13#12111yescharacteristic
138Q8×C13—42no—
139Q8×C13—44no—
140Q8×C13—82no—
141Q8×C13—81no—
142C2×C4×C13—22no—
143C2×C4×C13—41no—
144C2×C4×C13—41no—
145C2×C4×C13—41no—
146C2×C4×C13—42no—
147C2×C4×C13—82no—
148C2×C4×C13#12111yescharacteristic
149C2×C4×C13#12111yescharacteristic
150C2×C4×C13#12111yescharacteristic
151C2×C4×C13C2×Q1612yes—
152C2×C4×C13#13512yes—
153C2×C4×C13C22×D411yescharacteristic
154C2×C4×C13C22×D411yescharacteristic
155C2×C4×C13C2×#4111yescharacteristic
156C2×Dic13—22no—
157C2×Dic13—24no—
158C2×Dic13—22no—
159C2×Dic13—81no—
160Dic26—22no—
161Dic26—24no—
162Dic26C2×Q1614yes—
163C8×C13—24no—
164#1055—42no—
165#1055—42no—
166#1055—42no—
167#533C2×D1311yescharacteristic
168C2×C4⋊Q8C2×D1311yescharacteristic
169C2×C4×Q8—131no—
170#494C2×D1312yes—
171#487C2×D1312yes—
172#479—262no—
173#477—132no—
174#475—134no—
175#475—132no—
176#475—262no—
177#462—22no—
178#451—22no—
179C8×Q8—134no—
180C8×Q8—262no—
181#431—131no—
182#431—262no—
183#431C2×D1311yescharacteristic
184#428—261no—
185C2×#106—261no—
186#423—131no—
187C2×C4⋊C8—131no—
188#419—261no—
189C2×#103—261no—
190#335—134no—
191#333—132no—
192#332—132no—
193#332—134no—
194C22×C13—21no—
195C22×C13—21no—
196C22×C13#46712yes—
197C22×C13C2×#12111yescharacteristic
198C4×C13—21no—
199C4×C13—21no—
200C4×C13—21no—
201C4×C13—82no—
202C4×C13—81no—
203C4×C13—81no—
204C4×C13#46712yes—
205C4×C13C22×Q1611yescharacteristic
206C4×C13C2×#13511yescharacteristic
207C4×C13#57712yes—
208Dic13—24no—
209Dic13—42no—
210C22×Q8—21no—
211C22×Q8—131no—
212C2×Q16—24no—
213C2×M16—261no—
214C22×C8—261no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8C22×D1312yes—
219C4⋊Q8C22×D1311yescharacteristic
220C4⋊Q8C22×D1311yescharacteristic
221#125—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—132no—
227C4×Q8—132no—
228C4×Q8—261no—
229C4×Q8—261no—
230#117—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—131no—
234C2×C4⋊C4—261no—
235C2×C42—261no—
236C2×C42C22×D1311yescharacteristic
237#107—262no—
238#106—264no—
239#106—262no—
240C4⋊C8—132no—
241C4⋊C8—132no—
242C4⋊C8—132no—
243C4⋊C8—264no—
244C4⋊C8—261no—
245C4⋊C8—261no—
246C4⋊C8C22⋊D1312yes—
247#103—24no—
248#103—264no—
249#103—262no—
250#103—262no—
251C22⋊C8—261no—
252C22⋊C8C22⋊D1312yes—
253C4×C8—264no—
254C4×C8—261no—
255C4×C8C22×D1312yes—
256C2×C13—22no—
257C2×C13#304311yescharacteristic
258C2×C13C2×#46711yescharacteristic
259C2×C13#306911yescharacteristic
260C2×Q8—21no—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—41no—
264C2×Q8—82no—
265C2×Q8—134no—
266C2×Q8—132no—
267C2×Q8—262no—
268C2×Q8—261no—
269C22×C4—41no—
270C22×C4—261no—
271C22×C4—261no—
272C22×C4D4×D1311yescharacteristic
273C22×C4#612911yescharacteristic
274C22×C4C2×C22⋊D1311yescharacteristic
275Q16—84no—
276M16—522no—
277C2×C8—262no—
278C2×C8—262no—
279C2×C8—262no—
280C2×C8—522no—
281C2×C8C2×C22⋊D1314yes—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—132no—
293C4⋊C4—261no—
294C4⋊C4—262no—
295C4⋊C4—262no—
296C22⋊C4—41no—
297C42—41no—
298C42—262no—
299C42—262no—
300C42D4×D1311yescharacteristic
301C42#612911yescharacteristic
302C42C2×C22⋊D1311yescharacteristic
303C42C23×D1311yescharacteristic
304C13#1939911yescharacteristic, complemented, Sylow
305C23#8819211yescharacteristic
306Q8—42no—
307Q8—44no—
308Q8—81no—
309Q8—82no—
310Q8—134no—
311Q8—262no—
312Q8—264no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—82no—
319C2×C4—262no—
320C2×C4—264no—
321C2×C4—262no—
322C2×C4—1041no—
323C2×C4Q16×D1312yes—
324C2×C4#8817312yes—
325C2×C4#8819211yescharacteristic
326C2×C4#8819211yescharacteristic
327C2×C4#8819211yescharacteristic
328C2×C4C2×D4×D1311yescharacteristic
329C2×C4C2×#612911yescharacteristic
330C2×C4C22×C22⋊D1311yescharacteristic
331C8—24no—
332C8—522no—
333C8—522no—
334C8—522no—
335C22—21no—
336C22—21no—
337C22#1171884912yes—
338C22C2×#8819211yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—82no—
343C4—81no—
344C4—81no—
345C4—264no—
346C4—522no—
347C4#1171884912yes—
348C4C2×Q16×D1311yescharacteristic
349C4C2×#8817311yescharacteristic
350C4#1171954112yes—
351C2—22no—
352C2#1312435311yescharacteristic
353C2C2×#1171884911yescharacteristic
354C2#1312456911yescharacteristic
subgroupG/H
centerC22C2×#88192
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3209C2
socleC22×C13C2×#121
prime radicalC23×C13#121
hypercenter#3209D13
hypocenterC13#19399
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
4. #13114291C2semidirect
5. #13114203C2semidirect
97. C2×Dic26Q16semidirect
304. C13#19399semidirect

Maximal subgroups

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

subgroupindex
#131246372
C2×#117187552
#131239162
#131142912
#131142032
#131136112
#131135212
#131134932
C13×#32092
#1939913