Finite Group Databasebeta

← #16781027

Subgroups — #16781027

Order 3328.

1158 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13130726C212yesmaximal, complemented
2C2×#11718285C212yesmaximal, complemented
3C2×#11718305C212yesmaximal
4C22×#88180C211yesmaximal, characteristic
5C24×C4×C13C411yescharacteristic
6C22×C22⋊Dic13C411yescharacteristic
7C22×C22⋊Dic13C2211yescharacteristic
8C2×#88180—24no—
9C2×#88180—24no—
10C2×#88180—24no—
11C22×C4⋊Dic13C411yescharacteristic
12C22×C4⋊Dic13C2211yescharacteristic
13C22×C4×Dic13C411yescharacteristic
14C22×C4×Dic13C2211yescharacteristic
15C2×C22⋊#1057C2214yes—
16#11718305—24no—
17#11718290—24no—
18#11718285—24no—
19C25×C13C2×C411yescharacteristic
20C23×C4×C13—22no—
21C23×C4×C13—22no—
22C23×C4×C13—22no—
23C23×C4×C13—22no—
24C23×C4×C13—22no—
25C23×C4×C13—22no—
26C23×C4×C13C814yescomplemented
27C23×C4×C13C2×C411yescharacteristic
28C23×C4×C13C2×C411yescharacteristic
29C23×Dic13C2×C411yescharacteristic
30C23×Dic13C2×C411yescharacteristic
31C23×Dic13C2×C411yescharacteristic
32C23×Dic13C2311yescharacteristic
33C22×#1057—24no—
34C2×C22⋊Dic13—22no—
35C2×C22⋊Dic13—22no—
36C2×C22⋊Dic13—22no—
37C2×C22⋊Dic13—22no—
38C2×C22⋊Dic13—22no—
39C2×C22⋊Dic13D414yes—
40#88180—232no—
41C2×C4⋊Dic13—22no—
42C2×C4⋊Dic13—22no—
43C2×C4⋊Dic13—22no—
44C2×C4⋊Dic13—22no—
45C2×C4⋊Dic13—22no—
46C2×C4⋊Dic13D414yes—
47C2×C4×Dic13—22no—
48C2×C4×Dic13—22no—
49C2×C4×Dic13—22no—
50C2×C4×Dic13—22no—
51C2×C4×Dic13—22no—
52C2×C4×Dic13D414yes—
53C22⋊#1057—24no—
54C22⋊#1057—24no—
55#8300—131nomaximal, Sylow
56C24×C13—21no—
57C24×C13—21no—
58C24×C13—21no—
59C24×C13—21no—
60C24×C13—42no—
61C24×C13—42no—
62C24×C13C22⋊C412yes—
63C24×C13C2×C812yes—
64C24×C13M1612yes—
65C24×C13C22×C411yescharacteristic
66C22×C4×C13—28no—
67C22×C4×C13—28no—
68C22×C4×C13—28no—
69C22×C4×C13—28no—
70C22×C4×C13—28no—
71C22×C4×C13—21no—
72C22×C4×C13—21no—
73C22×C4×C13—21no—
74C22×C4×C13—21no—
75C22×C4×C13—21no—
76C22×C4×C13—21no—
77C22×C4×C13—21no—
78C22×C4×C13—21no—
79C22×C4×C13—28no—
80C22×C4×C13—28no—
81C22×C4×C13C22⋊C412yes—
82C22×C4×C13C22⋊C412yes—
83C22×C4×C13C2×C812yes—
84C22×C4×C13C2×C812yes—
85C22×C4×C13M1612yes—
86C22×C4×C13M1612yes—
87C22×Dic13—21no—
88C22×Dic13—21no—
89C22×Dic13—21no—
90C22×Dic13—21no—
91C22×Dic13—21no—
92C22×Dic13—21no—
93C22×Dic13—42no—
94C22×Dic13—42no—
95C22×Dic13—42no—
96C22×Dic13—42no—
97C22×Dic13C22⋊C412yes—
98C22×Dic13C22⋊C412yes—
99C22×Dic13C22⋊C412yes—
100C22×Dic13C2×D412yes—
101C22×Dic13C2×D412yes—
102C22×Dic13C2×D412yes—
103C2×#1057—48no—
104C2×#1057—44no—
105C22⋊Dic13—28no—
106C22⋊Dic13—28no—
107C4⋊Dic13—28no—
108C4⋊Dic13—28no—
109C4×Dic13—28no—
110C4×Dic13—28no—
111C22×#117—131no—
112#1817—132no—
113C2×#324—132no—
114C2×#323—132no—
115C23×C13—22no—
116C23×C13—22no—
117C23×C13—22no—
118C23×C13—22no—
119C23×C13—22no—
120C23×C13—22no—
121C23×C13—22no—
122C23×C13—22no—
123C23×C13—44no—
124C23×C13—44no—
125C23×C13—42no—
126C23×C13—42no—
127C23×C13—42no—
128C23×C13—42no—
129C23×C13—44no—
130C23×C13—44no—
131C23×C13—42no—
132C23×C13—42no—
133C23×C13C22⋊C814yes—
134C23×C13C2×C22⋊C411yescharacteristic
135C23×C13C2×C22⋊C411yescharacteristic
136C23×C13C2×C22⋊C411yescharacteristic
137C23×C13C22×C811yescharacteristic
138C23×C13C2×M1611yescharacteristic
139C23×C13C2×M1611yescharacteristic
140C23×C13C2×M1611yescharacteristic
141C2×C4×C13—232no—
142C2×C4×C13—22no—
143C2×C4×C13—22no—
144C2×C4×C13—22no—
145C2×C4×C13—22no—
146C2×C4×C13—22no—
147C2×C4×C13—22no—
148C2×C4×C13—22no—
149C2×C4×C13—22no—
150C2×C4×C13—22no—
151C2×C4×C13—22no—
152C2×C4×C13—22no—
153C2×C4×C13—22no—
154C2×C4×C13C22⋊C814yes—
155C2×C4×C13C22⋊C814yes—
156C2×Dic13—24no—
157C2×Dic13—22no—
158C2×Dic13—22no—
159C2×Dic13—22no—
160C2×Dic13—24no—
161C2×Dic13—42no—
162C2×Dic13—42no—
163C2×Dic13—42no—
164C2×Dic13—42no—
165C2×Dic13—42no—
166C2×Dic13—42no—
167C2×Dic13—42no—
168C2×Dic13—42no—
169C2×Dic13—42no—
170C2×Dic13—42no—
171C2×Dic13#12014yes—
172#1057—84no—
173C24×C4#26111yescharacteristic
174C2×#117—264no—
175C2×#117—264no—
176C2×#117—264no—
177C22×C4⋊C4—131no—
178C22×C4⋊C4—131no—
179C22×C22⋊C4—131no—
180C22×C22⋊C4—131no—
181C22×C42—131no—
182C22×C42—131no—
183C2×C22⋊C8—134no—
184#336—264no—
185#324—264no—
186#323—264no—
187C22×C13—22no—
188C22×C13—22no—
189C22×C13—22no—
190C22×C13—22no—
191C22×C13—21no—
192C22×C13—21no—
193C22×C13—21no—
194C22×C13—21no—
195C22×C13—24no—
196C22×C13—44no—
197C22×C13—44no—
198C22×C13—44no—
199C22×C13—44no—
200C22×C13—44no—
201C22×C13—44no—
202C22×C13—42no—
203C22×C13—42no—
204C22×C13#37911yescharacteristic
205C22×C13C2×C22⋊C811yescharacteristic
206C22×C13C2×C22⋊C811yescharacteristic
207C22×C13C2×C22⋊C811yescharacteristic
208C22×C13#40711yescharacteristic
209C22×C13#40711yescharacteristic
210C22×C13#40711yescharacteristic
211C22×C13#41012yes—
212C22×C13#41312yes—
213C4×C13—24no—
214C4×C13—24no—
215Dic13—24no—
216Dic13—42no—
217Dic13—42no—
218Dic13—42no—
219C25C2×#26111yescharacteristic
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—131no—
227C23×C4—131no—
228C23×C4—131no—
229C23×C4—131no—
230C23×C4#105714yescomplemented
231C23×C4C2×#26111yescharacteristic
232C23×C4C2×#26111yescharacteristic
233C22×C8—264no—
234#117—2632no—
235C2×C4⋊C4—134no—
236C2×C4⋊C4—262no—
237C2×C4⋊C4—262no—
238C2×C4⋊C4—262no—
239C2×C4⋊C4—262no—
240C2×C4⋊C4—262no—
241C2×C22⋊C4—134no—
242C2×C22⋊C4—262no—
243C2×C22⋊C4—262no—
244C2×C22⋊C4—262no—
245C2×C22⋊C4—262no—
246C2×C22⋊C4—262no—
247C2×C42—134no—
248C2×C42—262no—
249C2×C42—262no—
250C2×C42—262no—
251C2×C42—262no—
252C2×C42—262no—
253C22⋊C8—264no—
254C22⋊C8—264no—
255C2×C13—21no—
256C2×C13—21no—
257C2×C13—21no—
258C2×C13—21no—
259C2×C13—44no—
260C2×C13#145812yes—
261C2×C13#146412yes—
262C2×C13#177911yescharacteristic
263C2×C13#178211yescharacteristic
264C2×C13#178511yescharacteristic
265C24—21no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—42no—
270C24—42no—
271C24C2×#105712yes—
272C24#612212yes—
273C24C22⋊#26112yes—
274C24C22×#26111yescharacteristic
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—28no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—132no—
291C22×C4—132no—
292C22×C4—132no—
293C22×C4—132no—
294C22×C4—132no—
295C22×C4—132no—
296C22×C4—261no—
297C22×C4—261no—
298C22×C4—261no—
299C22×C4—261no—
300C22×C4—261no—
301C22×C4—261no—
302C22×C4—522no—
303C22×C4—522no—
304C22×C4—522no—
305C22×C4—522no—
306C22×C4C2×#105712yes—
307C22×C4C2×#105712yes—
308C22×C4#612212yes—
309C22×C4#612212yes—
310C22×C4C22⋊#26112yes—
311C22×C4C22⋊#26112yes—
312C2×C8—524no—
313C2×C8—528no—
314C4⋊C4—268no—
315C4⋊C4—268no—
316C22⋊C4—268no—
317C22⋊C4—268no—
318C42—268no—
319C42—268no—
320C13#830011yescharacteristic, complemented, Sylow
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—42no—
333C23—44no—
334C23—44no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23C22⋊#105714yes—
340C23C22×#105711yescharacteristic
341C23C2×#612211yescharacteristic
342C23C2×#612211yescharacteristic
343C23C2×#612211yescharacteristic
344C23C2×C22⋊#26111yescharacteristic
345C23C2×C22⋊#26111yescharacteristic
346C23C2×C22⋊#26111yescharacteristic
347C2×C4—232no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—134no—
361C2×C4—264no—
362C2×C4—262no—
363C2×C4—262no—
364C2×C4—262no—
365C2×C4—264no—
366C2×C4—522no—
367C2×C4—522no—
368C2×C4—522no—
369C2×C4—522no—
370C2×C4—522no—
371C2×C4—522no—
372C2×C4—522no—
373C2×C4—522no—
374C2×C4—522no—
375C2×C4—522no—
376C2×C4C22⋊#105714yes—
377C2×C4C22⋊#105714yes—
378C8—1044no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—24no—
388C22—44no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—42no—
395C22—42no—
396C22#1171911612yes—
397C22#1171913212yes—
398C22C2×C22⋊#105711yescharacteristic
399C22C2×C22⋊#105711yescharacteristic
400C22C2×C22⋊#105711yescharacteristic
401C22#1171916911yescharacteristic
402C22#1171916911yescharacteristic
403C22#1171916911yescharacteristic
404C22#1171917111yescharacteristic
405C4—24no—
406C4—24no—
407C4—264no—
408C4—522no—
409C4—522no—
410C4—522no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—44no—
416C2#1313006712yes—
417C2#1313008712yes—
418C2#1313065811yescharacteristic
419C2#1313066511yescharacteristic
420C2#1313067211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC23×C13C22×C8
FrattiniC24C22×#261
FittingC24×C4×C13C4
socleC23×C13C2×C22⋊C4
prime radicalC25×C13C2×C4
cosocleC23×Dic13C23
hypercenterC24×C4#261
hypocenterC13#8300
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #13130726C2semidirect
2. C2×#11718285C2semidirect
26. C23×C4×C13C8semidirect
230. C23×C4#1057semidirect
320. C13#8300semidirect

Maximal subgroups

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

subgroupindex
#131307262
C2×#117182852
C2×#117183052
C22×#881802
#830013