Finite Group Databasebeta

← #16785086

Subgroups — #16785086

Order 3328.

664 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 122 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
1#13130690C211yesmaximal, characteristic
2#13130562C211yesmaximal, characteristic
3#13129674C212yesmaximal
4#13129674C212yesmaximal
5C2×#11718307C211yesmaximal, characteristic
6C23×Dic26C411yescharacteristic
7C22×#6101C411yescharacteristic
8C22×C4×Dic13C2211yescharacteristic
9C2×C22⋊#1057C2212yes—
10C2×C22⋊#1057C2212yes—
11C2×#88110—21no—
12C2×C4⋊#1057—21no—
13C2×C13×#95C411yescharacteristic
14C2×#88071C411yescharacteristic
15C2×#88071C2211yescharacteristic
16C2×#88071C2211yescharacteristic
17#11718307—21no—
18#11718307—21no—
19#11718307—21no—
20#11718307—21no—
21#11718290—21no—
22#11718290—21no—
23C23×C4×C13C2×C411yescharacteristic
24C23×C4×C13C2×C411yescharacteristic
25C23×C4×C13C2×C411yescharacteristic
26C23×Dic13C2×C411yescharacteristic
27C23×Dic13C2×C411yescharacteristic
28C23×Dic13C2×C411yescharacteristic
29C23×Dic13C2311yescharacteristic
30C22×Dic26—22no—
31C22×Dic26—21no—
32C22×Dic26—21no—
33C22×Dic26—22no—
34C22×Dic26—42no—
35C22×Dic26—42no—
36C22×#1057—22no—
37C22×#1057—22no—
38C2×#6101—21no—
39C2×#6101—21no—
40C2×#6101—42no—
41C2×C4×Dic13—42no—
42C2×C4×Dic13D411yescharacteristic
43C2×C4×Dic13D411yescharacteristic
44C2×C4×Dic13D411yescharacteristic
45C2×C4×Dic13D411yescharacteristic
46C22⋊#1057—22no—
47C22⋊#1057—24no—
48C22⋊#1057Q814yescomplemented
49#88110—42no—
50C4⋊#1057—42no—
51C13×#95—21no—
52C13×#95—21no—
53#88071—21no—
54#88071—21no—
55#88071—21no—
56#88071—21no—
57#88071—21no—
58#88071—21no—
59#9882—131nomaximal, Sylow
60C24×C13C22×C411yescharacteristic
61C22×C4×C13—21no—
62C22×C4×C13—21no—
63C22×C4×C13—22no—
64C22×C4×C13—22no—
65C22×C4×C13—21no—
66C22×C4×C13—22no—
67C22×C4×C13—41no—
68C22×C4×C13—41no—
69C22×C4×C13—41no—
70C22×C4×C13—41no—
71C22×C4×C13—41no—
72C22×C4×C13C22⋊C411yescharacteristic
73C22×C4×C13C22⋊C411yescharacteristic
74C22×C4×C13M1612yes—
75C22×Dic13—21no—
76C22×Dic13—21no—
77C22×Dic13—22no—
78C22×Dic13—22no—
79C22×Dic13—21no—
80C22×Dic13—21no—
81C22×Dic13—21no—
82C22×Dic13—22no—
83C22×Dic13—41no—
84C22×Dic13—41no—
85C22×Dic13—41no—
86C22×Dic13—41no—
87C22×Dic13—41no—
88C22×Dic13—41no—
89C22×Dic13C22⋊C411yescharacteristic
90C22×Dic13C22⋊C411yescharacteristic
91C22×Dic13M1612yes—
92C22×Dic13C2×D411yescharacteristic
93C22×Dic13C2×D411yescharacteristic
94C22×Dic13C2×Q812yes—
95C22×Dic13#4112yes—
96C2×Dic26—24no—
97C2×Dic26—24no—
98C2×Dic26—48no—
99C2×Dic26—48no—
100C2×Dic26—42no—
101C2×Dic26—42no—
102C2×Dic26—42no—
103C2×Dic26—42no—
104C2×#1057—24no—
105C2×#1057—42no—
106C2×#1057—42no—
107C2×#1057—42no—
108C2×#1057—42no—
109C2×#1057—42no—
110#6101—42no—
111#6101—42no—
112C4×Dic13—22no—
113C4×Dic13—22no—
114C4×Dic13—41no—
115C4×Dic13—41no—
116C2×#391—131no—
117#1797—131no—
118#1749—131no—
119#1280—132no—
120#1280—132no—
121C23×C13—21no—
122C23×C13—21no—
123C23×C13—21no—
124C23×C13—21no—
125C23×C13C2×C22⋊C411yescharacteristic
126C23×C13C4×Q812yes—
127C23×C13C2×M1611yescharacteristic
128C23×C13C2×M1611yescharacteristic
129C23×C13#13112yes—
130C2×C4×C13—21no—
131C2×C4×C13—21no—
132C2×C4×C13—22no—
133C2×C4×C13—21no—
134C2×C4×C13—21no—
135C2×C4×C13—21no—
136C2×C4×C13—44no—
137C2×C4×C13—42no—
138C2×C4×C13—41no—
139C2×C4×C13—41no—
140C2×C4×C13—42no—
141C2×C4×C13—41no—
142C2×C4×C13—41no—
143C2×C4×C13—41no—
144C2×C4×C13—82no—
145C2×C4×C13#10112yes—
146C2×Dic13—24no—
147C2×Dic13—24no—
148C2×Dic13—21no—
149C2×Dic13—21no—
150C2×Dic13—21no—
151C2×Dic13—21no—
152C2×Dic13—21no—
153C2×Dic13—21no—
154C2×Dic13—21no—
155C2×Dic13—21no—
156C2×Dic13—22no—
157C2×Dic13—42no—
158C2×Dic13—42no—
159C2×Dic13—42no—
160C2×Dic13—41no—
161C2×Dic13—41no—
162C2×Dic13—42no—
163C2×Dic13—41no—
164C2×Dic13—41no—
165C2×Dic13—41no—
166C2×Dic13—82no—
167C2×Dic13#9912yes—
168C2×Dic13#10012yes—
169C2×Dic13C22⋊Q812yes—
170C2×Dic13#12411yescharacteristic
171C2×Dic13C4⋊Q811yescharacteristic
172Dic26—48no—
173Dic26—48no—
174#1057—44no—
175#1057—82no—
176C23×Q8—131no—
177C22×C4⋊C4—131no—
178C22×C42—131no—
179C2×C4⋊C8—261no—
180C2×C22⋊C8—132no—
181C2×C22⋊C8—132no—
182C2×#97—261no—
183#391—261no—
184#391—261no—
185#391—261no—
186#391—261no—
187C2×#95—131no—
188C2×#95—131no—
189C2×#95—131no—
190C2×#95#26111yescharacteristic
191#336—261no—
192#336—261no—
193C22×C13—21no—
194C22×C13—21no—
195C22×C13—21no—
196C22×C13—21no—
197C22×C13—22no—
198C22×C13—22no—
199C22×C13—22no—
200C22×C13—22no—
201C22×C13#39111yescharacteristic
202C22×C13#40711yescharacteristic
203C22×C13#40811yescharacteristic
204C22×C13C2×#9911yescharacteristic
205C22×C13#41011yescharacteristic
206C22×C13C2×#10011yescharacteristic
207C22×C13C2×#10111yescharacteristic
208C22×C13#44612yes—
209C22×C13#44612yes—
210C4×C13—22no—
211C4×C13—42no—
212C4×C13—41no—
213C4×C13—81no—
214Dic13—22no—
215Dic13—22no—
216Dic13—22no—
217Dic13—21no—
218Dic13—21no—
219Dic13—42no—
220Dic13—81no—
221C22×Q8—261no—
222C22×Q8—261no—
223C22×Q8—262no—
224C22×Q8—262no—
225C22×Q8—522no—
226C22×Q8—522no—
227C23×C4—131no—
228C23×C4—131no—
229C23×C4—131no—
230C23×C4—131no—
231C23×C4C2×#26111yescharacteristic
232C23×C4C2×#26111yescharacteristic
233C23×C4C2×#26111yescharacteristic
234C22×C8—262no—
235C22×C8—262no—
236C2×C4⋊C4—261no—
237C2×C4⋊C4—261no—
238C2×C4⋊C4—522no—
239C2×C42—131no—
240C2×C42—131no—
241C2×C42—131no—
242C2×C42—131no—
243C2×C42—522no—
244C4⋊C8—522no—
245C22⋊C8—134no—
246C22⋊C8—262no—
247C22⋊C8—264no—
248#97—522no—
249#95—21no—
250#95—21no—
251#95—261no—
252#95—261no—
253#95—261no—
254#95—261no—
255#95—261no—
256#95—261no—
257C2×C13—22no—
258C2×C13—22no—
259C2×C13#146211yescharacteristic
260C2×C13#146711yescharacteristic
261C2×C13#146911yescharacteristic
262C2×C13#147211yescharacteristic
263C2×C13#198511yescharacteristic
264C2×C13#199611yescharacteristic
265C2×C13#199711yescharacteristic
266C24C22×#26111yescharacteristic
267C2×Q8—264no—
268C2×Q8—264no—
269C2×Q8—522no—
270C2×Q8—522no—
271C2×Q8—528no—
272C2×Q8—522no—
273C2×Q8—522no—
274C2×Q8—528no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—131no—
287C22×C4—131no—
288C22×C4—131no—
289C22×C4—131no—
290C22×C4—132no—
291C22×C4—132no—
292C22×C4—132no—
293C22×C4—262no—
294C22×C4—262no—
295C22×C4—261no—
296C22×C4—261no—
297C22×C4—261no—
298C22×C4—261no—
299C22×C4—261no—
300C22×C4—262no—
301C22×C4—521no—
302C22×C4—521no—
303C22×C4—521no—
304C22×C4—521no—
305C22×C4—521no—
306C22×C4—521no—
307C22×C4#611812yes—
308C22×C4C22⋊#26111yescharacteristic
309C22×C4C22⋊#26111yescharacteristic
310C2×C8—264no—
311C2×C8—522no—
312C2×C8—522no—
313C2×C8—522no—
314C2×C8—522no—
315C2×C8—522no—
316C4⋊C4—522no—
317C4⋊C4—522no—
318C42—262no—
319C42—262no—
320C42—521no—
321C42—521no—
322C13#988211yescharacteristic, complemented, Sylow
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23C2×#611811yescharacteristic
328C23D13⋊M1611yescharacteristic
329C23#8823812yes—
330C23Q8×#26112yes—
331C23C2×C22⋊#26111yescharacteristic
332Q8—528no—
333Q8—528no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—132no—
350C2×C4—132no—
351C2×C4—132no—
352C2×C4—131no—
353C2×C4—131no—
354C2×C4—264no—
355C2×C4—264no—
356C2×C4—261no—
357C2×C4—261no—
358C2×C4—261no—
359C2×C4—261no—
360C2×C4—261no—
361C2×C4—261no—
362C2×C4—261no—
363C2×C4—261no—
364C2×C4—262no—
365C2×C4—522no—
366C2×C4—522no—
367C2×C4—522no—
368C2×C4—521no—
369C2×C4—521no—
370C2×C4—522no—
371C2×C4—521no—
372C2×C4—521no—
373C2×C4—521no—
374C2×C4—1042no—
375C2×C4#8811312yes—
376C8—524no—
377C8—1042no—
378C22—22no—
379C22—22no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22#1171907812yes—
387C22#1171908012yes—
388C22#1171911611yescharacteristic
389C22#1171912711yescharacteristic
390C22C2×#8811311yescharacteristic
391C22D13⋊#9911yescharacteristic
392C22#1171914611yescharacteristic
393C22D13⋊#10011yescharacteristic
394C22(C2×#261)⋊Q811yescharacteristic
395C4—22no—
396C4—42no—
397C4—41no—
398C4—81no—
399C4—262no—
400C4—261no—
401C4—261no—
402C4—262no—
403C4—262no—
404C4—522no—
405C4—1041no—
406C2—22no—
407C2—22no—
408C2#1313008211yescharacteristic
409C2#1313009811yescharacteristic
410C2#1313010511yescharacteristic
411C2#1313011511yescharacteristic
412C2#1313100911yescharacteristic
413C2#1313102711yescharacteristic
414C2#1313102911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutator, prime radicalC24×C13C22×C4
FrattiniC24C22×#261
FittingC2×C13×#95C4
socleC23×C13C2×C22⋊C4
cosocleC23×Dic13C23
hypercenterC2×#95#261
hypocenterC13#9882
LCSG ▷ C24×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
48. C22⋊#1057Q8semidirect
322. C13#9882semidirect

Maximal subgroups

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

subgroupindex
#131306902
#131305622
#131296742
C2×#117183072
#988213