Finite Group Databasebeta

← #15766549

Subgroups — #15766549

Order 3328.

408 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718668C211yesmaximal, characteristic
2#13114828C211yesmaximal, characteristic
3#13113611C211yesmaximal, characteristic, complemented
4#13113520C211yesmaximal, characteristic, complemented
5#13113516C211yesmaximal, characteristic, complemented
6#13112732C211yesmaximal, characteristic, complemented
7C13×#1478C211yesmaximal, characteristic
8C2×C4×D4×C13C2211yescharacteristic
9C2×#88123C2211yescharacteristic
10C13×#431C2211yescharacteristic
11C13×#431C2211yescharacteristic
12C2×#88073—21no—
13C2×#88072—21no—
14#11718668—22no—
15#11718668—22no—
16C2×#88047—21no—
17#11718585C2211yescharacteristic
18#11718585C2211yescharacteristic
19C2×C4⋊#1055C2211yescharacteristic
20C13×D4⋊C8—22no—
21#11718122—21no—
22#11718074—22no—
23#11718073—22no—
24#11718050—22no—
25C22×D4×C13D411yescharacteristic
26C23×C4×C13—21no—
27C22×Dic26—21no—
28C4×D4×C13—21no—
29C4×D4×C13—21no—
30C4×D4×C13—21no—
31C4×D4×C13—21no—
32C2×C13×C4⋊C4D411yescharacteristic
33C2×C13×C22⋊C4—21no—
34C2×C42×C13C2311yescharacteristic
35C2×C4⋊Dic13—21no—
36C2×C4⋊Dic13—21no—
37C2×#6099—21no—
38C2×#6099—21no—
39C22×#1055—21no—
40#88123—21no—
41#88123—21no—
42#88123—21no—
43#88123D412yes—
44C13×C4⋊C8—21no—
45C13×C4⋊C8D412yes—
46C13×C22⋊C8—21no—
47C13×C22⋊C8—21no—
48C4×C8×C13—21no—
49C4×C8×C13—21no—
50#88073—42no—
51#88072—24no—
52#88047—24no—
53C4⋊#1055—21no—
54C4⋊#1055—21no—
55C4⋊#1055—21no—
56C4⋊#1055—21no—
57C4⋊#1055—21no—
58C4⋊#1055D412yes—
59#10449—131nomaximal, Sylow
60C24×C13—21no—
61C2×D4×C13—21no—
62C2×D4×C13—21no—
63C2×D4×C13—21no—
64C2×D4×C13—21no—
65C2×D4×C13—21no—
66C2×D4×C13Q1612yes—
67C22×C4×C13—21no—
68C22×C4×C13—41no—
69C22×C4×C13—41no—
70C22×C4×C13—82no—
71C22×C4×C13C2×D411yescharacteristic
72C22×C4×C13#4111yescharacteristic
73C22×C4×C13#4111yescharacteristic
74C22×Dic13—41no—
75C2×Dic26—22no—
76C2×Dic26—41no—
77C2×Dic26—82no—
78C2×C8×C13—22no—
79C2×C8×C13—41no—
80C13×C4⋊C4—21no—
81C13×C4⋊C4—21no—
82C13×C22⋊C4—41no—
83C13×C22⋊C4—41no—
84C42×C13C2×D411yescharacteristic
85C42×C13C2×D411yescharacteristic
86C42×C13C2×D411yescharacteristic
87C42×C13#4111yescharacteristic
88C4⋊Dic13—22no—
89C4⋊Dic13—22no—
90C4⋊Dic13—41no—
91C4⋊Dic13—41no—
92#6099—41no—
93#6099—41no—
94#6099—41no—
95#6099—41no—
96C2×#1055—22no—
97C2×#1055—41no—
98C2×#1055—41no—
99C2×#1055—41no—
100C2×#1055—41no—
101C2×#474—131no—
102#1800—131no—
103#1571—131no—
104#1549—131no—
105#1548—131no—
106#1478D1311yescharacteristic
107#1301—131no—
108C23×C13—22no—
109C23×C13—41no—
110C23×C13—41no—
111C23×C13—81no—
112C23×C13—81no—
113C23×C13#12311yescharacteristic
114D4×C13—22no—
115D4×C13—22no—
116D4×C13—41no—
117D4×C13—41no—
118C2×C4×C13—22no—
119C2×C4×C13—41no—
120C2×C4×C13—41no—
121C2×C4×C13—82no—
122C2×C4×C13—82no—
123C2×C4×C13—81no—
124C2×C4×C13#12011yescharacteristic
125C2×C4×C13#12111yescharacteristic
126C2×C4×C13#12111yescharacteristic
127C2×C4×C13#12111yescharacteristic
128C2×C4×C13#12311yescharacteristic
129C2×C4×C13#12411yescharacteristic
130C2×C4×C13C2×Q1611yescharacteristic
131C2×C4×C13#13511yescharacteristic
132C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C13#13711yescharacteristic
134C2×Dic13—42no—
135C2×Dic13—82no—
136Dic26—82no—
137Dic26—82no—
138C8×C13—42no—
139C8×C13—81no—
140#1055—42no—
141#1055—81no—
142#1055—81no—
143C2×C4⋊Q8—131no—
144C2×C4×D4C2×D1311yescharacteristic
145#474—262no—
146#474—262no—
147#431C2×D1311yescharacteristic
148#431C2×D1311yescharacteristic
149C2×#107—261no—
150#423—131no—
151#423—131no—
152C2×C4⋊C8—131no—
153C2×#102—261no—
154C2×#100—261no—
155#343—261no—
156#335—262no—
157#333—262no—
158#332—262no—
159D4⋊C8—22no—
160C22×C13—21no—
161C22×C13—21no—
162C22×C13—42no—
163C22×C13—82no—
164C22×C13—82no—
165C22×C13—81no—
166C22×C13—81no—
167C22×C13#39411yescharacteristic
168C22×C13#48411yescharacteristic
169C22×C13#48511yescharacteristic
170C4×C13—21no—
171C4×C13—21no—
172C4×C13—21no—
173C4×C13—81no—
174C4×C13#44911yescharacteristic
175C4×C13#45111yescharacteristic
176C4×C13#46711yescharacteristic
177C4×C13#46811yescharacteristic
178C4×C13#47212yes—
179Dic13—82no—
180C22×Q8—261no—
181C22×D4D5211yescharacteristic
182C23×C4—21no—
183C2×M16—261no—
184C2×M16—261no—
185C22×C8—261no—
186C4⋊Q8—132no—
187C4⋊Q8—261no—
188C4⋊Q8—261no—
189C4⋊Q8—261no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C2×C4⋊C4—261no—
195C2×C4⋊C4—261no—
196C2×C4⋊C4D5211yescharacteristic
197C2×C22⋊C4—21no—
198C2×C42C22×D1311yescharacteristic
199#107—264no—
200C4⋊C8—21no—
201C4⋊C8—132no—
202C4⋊C8—261no—
203C4⋊C8—261no—
204C4⋊C8—261no—
205C4⋊C8—261no—
206C4⋊C8—261no—
207C4⋊C8C22⋊D1312yes—
208#102—264no—
209#100—522no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C2×C13—21no—
215C2×C13—21no—
216C2×C13—82no—
217C2×C13#203611yescharacteristic
218C2×C13#204511yescharacteristic
219C2×C13#204911yescharacteristic
220C24—21no—
221C2×Q8—262no—
222C2×Q8—521no—
223C2×Q8—1042no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4Dic5212yes—
230C22×C4—21no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—82no—
234C22×C4—521no—
235C22×C4C2×D5211yescharacteristic
236C22×C4#612911yescharacteristic
237C22×C4#612911yescharacteristic
238M16—521no—
239M16—521no—
240M16—521no—
241M16—521no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—262no—
245C2×C8—521no—
246C2×C8—521no—
247C2×C8—521no—
248C2×C8—521no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—262no—
252C4⋊C4—262no—
253C4⋊C4—521no—
254C4⋊C4—521no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C42#612711yescharacteristic
258C42D4×D1311yescharacteristic
259C42D4×D1311yescharacteristic
260C42C2×C22⋊D1311yescharacteristic
261C13#1044911yescharacteristic, complemented, Sylow
262C23—22no—
263C23—41no—
264C23—41no—
265C23—81no—
266C23—81no—
267C23#8814011yescharacteristic
268Q8—1042no—
269Q8—1042no—
270D4—22no—
271D4—22no—
272D4—41no—
273D4—41no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—522no—
281C2×C4—1042no—
282C2×C4#8813611yescharacteristic
283C2×C4#8813711yescharacteristic
284C2×C4#8813911yescharacteristic
285C2×C4#8815811yescharacteristic
286C2×C4C2×Dic5211yescharacteristic
287C2×C4#8816211yescharacteristic
288C2×C4#8816311yescharacteristic
289C2×C4#8818411yescharacteristic
290C2×C4#8819211yescharacteristic
291C2×C4#8819311yescharacteristic
292C8—42no—
293C8—81no—
294C8—522no—
295C8—1041no—
296C8—1041no—
297C22—21no—
298C22—21no—
299C22—42no—
300C22—82no—
301C22—82no—
302C22—81no—
303C22—81no—
304C22#1171838311yescharacteristic
305C22#1171838411yescharacteristic
306C22#1171862211yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—81no—
311C4—1042no—
312C4#1171839511yescharacteristic
313C4#1171840011yescharacteristic
314C4#1171877711yescharacteristic
315C4#1171879511yescharacteristic
316C4#1171886012yes—
317C2—21no—
318C2—21no—
319C2—82no—
320C2#1311604011yescharacteristic
321C2#1311609111yescharacteristic
322C2#1311611711yescharacteristic
subgroupG/H
centerC22#11718622
commutator, cosocleC2×C42×C13C23
FrattiniC2×C42C22×D13
FittingC13×#1478C2
socleC22×C13#394
prime radicalC22×D4×C13D4
hypercenter#1478D13
hypocenterC13#10449
LCSG ▷ C2×C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
3. #13113611C2semidirect
4. #13113520C2semidirect
5. #13113516C2semidirect
6. #13112732C2semidirect
261. C13#10449semidirect

Maximal subgroups

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

subgroupindex
C2×#117186682
#131148282
#131136112
#131135202
#131135162
#131127322
C13×#14782
#1044913