Finite Group Databasebeta

← #15918290

Subgroups — #15918290

Order 3328.

1486 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Dic52C211yesmaximal, characteristic
2C2×#11718360C212yesmaximal, complemented
3C2×#11718777C211yesmaximal, characteristic
4C2×#11718400C212yesmaximal
5#13118480C212yesmaximal
6#13115938C214yesmaximal, complemented
7#13115697C212yesmaximal, complemented
8C8×C13×C22⋊C4C211yesmaximal, characteristic
9C23×Dic26C2212yes—
10C23×C8×C13C2211yescharacteristic
11C2×#88178C2212yes—
12C22×Dic52—24no—
13C22×Dic52—28no—
14C22×Dic52C2212yes—
15C22×Dic52C2211yescharacteristic
16C22×Dic52C2214yes—
17C22×Dic52C2212yes—
18C2×#88123C2214yes—
19C2×C13×C22⋊C8C2211yescharacteristic
20C2×C4×C8×C13C2212yes—
21C4×C13×C22⋊C4C2211yescharacteristic
22#11718777—24no—
23C2×#88058C2211yescharacteristic
24C2×#88056C2212yes—
25C2×#88056C2214yes—
26#11718611—24no—
27#11718590C2214yes—
28#11718400—28no—
29#11718360—28no—
30#11718317—24no—
31C13×#336C2212yes—
32C23×C4×C13C2311yescharacteristic
33C23×Dic13—22no—
34C22×Dic26—24no—
35C22×Dic26—28no—
36C22×Dic26—24no—
37C22×Dic26—22no—
38C22×Dic26—44no—
39C22×Dic26C2312yes—
40C22×Dic26C2314yes—
41C22×C8×C13—24no—
42C22×C8×C13D414yes—
43C22×C8×C13C2312yes—
44C22×C8×C13C2311yescharacteristic
45C22×C8×C13C2311yescharacteristic
46C2×C13×C22⋊C4D412yes—
47C2×C42×C13C2312yes—
48C2×C22⋊Dic13—22no—
49C2×C4⋊Dic13—28no—
50C2×C4⋊Dic13C2312yes—
51#88178—24no—
52#88178—24no—
53C2×#6101—22no—
54C2×Dic52—232no—
55C2×Dic52—28no—
56C2×Dic52—24no—
57C2×Dic52—24no—
58C2×Dic52—24no—
59C2×Dic52—28no—
60C2×Dic52—28no—
61C2×Dic52—22no—
62C2×Dic52—24no—
63C2×Dic52—44no—
64C2×Dic52D418yes—
65#88123—28no—
66#88123—44no—
67C13×C22⋊C8D414yes—
68C4×C8×C13—24no—
69C13×#95D412yes—
70#88071—28no—
71#88058—22no—
72#88056—28no—
73#88056—24no—
74#25780—131nomaximal, Sylow
75C24×C13C2×D411yescharacteristic
76C22×C4×C13—22no—
77C22×C4×C13—22no—
78C22×C4×C13C2×D411yescharacteristic
79C22×C4×C13C2×D412yes—
80C22×C4×C13C2×D412yes—
81C22×C4×C13C2×D414yes—
82C22×C4×C13C2411yescharacteristic
83C22×Dic13—24no—
84C22×Dic13—22no—
85C22×Dic13—22no—
86C22×Dic13—48no—
87C22×Dic13—44no—
88C2×Dic26—216no—
89C2×Dic26—24no—
90C2×Dic26—24no—
91C2×Dic26—24no—
92C2×Dic26—24no—
93C2×Dic26—24no—
94C2×Dic26—24no—
95C2×Dic26—22no—
96C2×Dic26—22no—
97C2×Dic26—24no—
98C2×Dic26—416no—
99C2×Dic26—48no—
100C2×Dic26—48no—
101C2×Dic26—44no—
102C2×Dic26C2×D418yes—
103C2×C8×C13—22no—
104C2×C8×C13—22no—
105C2×C8×C13—22no—
106C2×C8×C13—28no—
107C2×C8×C13C2×D414yes—
108C2×C8×C13C2×D412yes—
109C2×C8×C13C2×D412yes—
110C2×C8×C13C2×D414yes—
111C13×C22⋊C4Q1618yescomplemented
112C42×C13—22no—
113C42×C13—22no—
114C22⋊Dic13—44no—
115C4⋊Dic13—22no—
116C4⋊Dic13—22no—
117C4⋊Dic13—416no—
118#6101—44no—
119Dic52—216no—
120Dic52—216no—
121Dic52—416no—
122Dic52—44no—
123C23×Q16—131no—
124C2×#494—132no—
125C2×#467—131no—
126C2×#451—132no—
127#2434—132no—
128#2016—134no—
129#1969—132no—
130C8×C22⋊C4D1311yescharacteristic
131C23×C13—22no—
132C23×C13—22no—
133C23×C13#12712yes—
134C23×C13C2×Q1614yes—
135C23×C13C22×D411yescharacteristic
136C2×C4×C13—24no—
137C2×C4×C13—22no—
138C2×C4×C13—24no—
139C2×C4×C13—22no—
140C2×C4×C13—22no—
141C2×C4×C13—24no—
142C2×C4×C13#12712yes—
143C2×C4×C13C2×Q1618yescomplemented
144C2×C4×C13C22×D411yescharacteristic
145C2×C4×C13C22×D412yes—
146C2×C4×C13C22×D411yescharacteristic
147C2×C4×C13H(5,𝔽2)12yes—
148C2×Dic13—28no—
149C2×Dic13—416no—
150C2×Dic13—44no—
151C2×Dic13—44no—
152C2×Dic13—42no—
153C2×Dic13—42no—
154C2×Dic13—44no—
155C2×Dic13—44no—
156Dic26—28no—
157Dic26—28no—
158Dic26—416no—
159Dic26—44no—
160Dic26—44no—
161Dic26—42no—
162Dic26—42no—
163Dic26—44no—
164C8×C13—22no—
165C8×C13—24no—
166C8×C13#12014yes—
167C23×Q8—132no—
168C22×Q16—132no—
169C22×Q16—131no—
170C22×Q16—132no—
171C22×Q16—134no—
172C22×Q16—268no—
173C22×Q16—264no—
174C23×C8C2×D1311yescharacteristic
175C2×C4⋊Q8—134no—
176C2×C22⋊Q8—132no—
177#494—268no—
178#467—264no—
179#451—268no—
180C2×#107—131no—
181C2×#103—134no—
182C2×#103—132no—
183C2×C22⋊C8C2×D1311yescharacteristic
184C2×C4×C8C2×D1312yes—
185#395—264no—
186#391—134no—
187#385—264no—
188C4×C22⋊C4C2×D1311yescharacteristic
189#336C2×D1312yes—
190C22×C13—22no—
191C22×C13—22no—
192C22×C13—28no—
193C22×C13#49414yescomplemented
194C22×C13C2×#12711yescharacteristic
195C22×C13D4212yes—
196C22×C13C22×Q1612yes—
197C22×C13#57812yes—
198C4×C13—22no—
199C4×C13—28no—
200C4×C13C2×#12011yescharacteristic
201C4×C13#53511yescharacteristic
202C4×C13D4212yes—
203Dic13—48no—
204Dic13—82no—
205Dic13—82no—
206C22×Q8—134no—
207C22×Q8—132no—
208C22×Q8—268no—
209C22×Q8—262no—
210C22×Q8—264no—
211C22×Q8—264no—
212C22×Q8—524no—
213C23×C4—262no—
214C23×C4C22×D1311yescharacteristic
215C2×Q16—138no—
216C2×Q16—268no—
217C2×Q16—2632no—
218C2×Q16—268no—
219C2×Q16—264no—
220C2×Q16—264no—
221C2×Q16—268no—
222C2×Q16—262no—
223C2×Q16—264no—
224C2×Q16—264no—
225C2×Q16—524no—
226C22×C8—24no—
227C22×C8D5214yes—
228C22×C8C22×D1311yescharacteristic
229C22×C8C22×D1312yes—
230C22×C8C22×D1311yescharacteristic
231C4⋊Q8—268no—
232C4⋊Q8—524no—
233C22⋊Q8—264no—
234C22⋊Q8—264no—
235C2×C4⋊C4—132no—
236C2×C4⋊C4—262no—
237C2×C4⋊C4—268no—
238C2×C22⋊C4—262no—
239C2×C22⋊C4D5212yes—
240C2×C42C22×D1312yes—
241#107—262no—
242#103—268no—
243#103—264no—
244C22⋊C8D5214yes—
245C4×C8—24no—
246#95—268no—
247#95D5212yes—
248C2×C13—24no—
249C2×C13#243311yescharacteristic
250C2×C13C2×#49411yescharacteristic
251C2×C13#315611yescharacteristic
252C2×C13D4×Q1614yes—
253C24C2×D5211yescharacteristic
254C2×Q8—138no—
255C2×Q8—2616no—
256C2×Q8—264no—
257C2×Q8—264no—
258C2×Q8—264no—
259C2×Q8—262no—
260C2×Q8—262no—
261C2×Q8—264no—
262C2×Q8—264no—
263C2×Q8—264no—
264C2×Q8—264no—
265C2×Q8—528no—
266C2×Q8—5216no—
267C2×Q8—528no—
268C2×Q8—524no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—264no—
272C22×C4—262no—
273C22×C4—262no—
274C22×C4—528no—
275C22×C4—524no—
276C22×C4C2×D5211yescharacteristic
277C22×C4C2×D5212yes—
278C22×C4C2×D5212yes—
279C22×C4C2×D5214yes—
280C22×C4C23×D1311yescharacteristic
281Q16—2616no—
282Q16—2616no—
283Q16—5216no—
284Q16—524no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—28no—
289C2×C8C2×D5214yes—
290C2×C8C2×D5212yes—
291C2×C8C2×D5212yes—
292C2×C8D4×D1314yes—
293C4⋊C4—262no—
294C4⋊C4—262no—
295C4⋊C4—5216no—
296C4⋊C4—524no—
297C22⋊C4—524no—
298C22⋊C4Dic5218yescomplemented
299C42—22no—
300C42—22no—
301C13#2578011yescharacteristic, complemented, Sylow
302C23—22no—
303C23—22no—
304C23#8812812yes—
305C23C2×Dic5214yes—
306C23C22×D5211yescharacteristic
307Q8—268no—
308Q8—268no—
309Q8—5216no—
310Q8—524no—
311Q8—524no—
312Q8—522no—
313Q8—522no—
314Q8—524no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—268no—
322C2×C4—5216no—
323C2×C4—524no—
324C2×C4—524no—
325C2×C4—522no—
326C2×C4—522no—
327C2×C4—524no—
328C2×C4—524no—
329C2×C4#8812812yes—
330C2×C4C2×Dic5218yescomplemented
331C2×C4C22×D5211yescharacteristic
332C2×C4C22×D5211yescharacteristic
333C2×C4C2×D4×D1312yes—
334C2×C4#8825612yes—
335C8—22no—
336C8—24no—
337C8#8813614yes—
338C22—22no—
339C22—28no—
340C22—22no—
341C22#1171836014yescomplemented
342C22C2×#8812811yescharacteristic
343C22D4×D5212yes—
344C22C22×Dic5212yes—
345C22#1171945812yes—
346C4—22no—
347C4—28no—
348C4—528no—
349C4—1042no—
350C4—1042no—
351C4C2×#8813611yescharacteristic
352C4#1171923011yescharacteristic
353C4D4×D5212yes—
354C2—24no—
355C2#1311847311yescharacteristic
356C2C2×#1171836011yescharacteristic
357C2#1312533011yescharacteristic
358C2D4×Dic5214yes—
subgroupG/H
centerC23C22×D52
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC8×C13×C22⋊C4C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenterC8×C22⋊C4D13
hypocenterC13#25780
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#11718360C2semidirect
6. #13115938C2semidirect
7. #13115697C2semidirect
111. C13×C22⋊C4Q16semidirect
143. C2×C4×C13C2×Q16semidirect
193. C22×C13#494semidirect
298. C22⋊C4Dic52semidirect
301. C13#25780semidirect
330. C2×C4C2×Dic52semidirect
341. C22#11718360semidirect

Maximal subgroups

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

subgroupindex
C23×Dic522
C2×#117183602
C2×#117187772
C2×#117184002
#131184802
#131159382
#131156972
C8×C13×C22⋊C42
#2578013