Finite Group Databasebeta

← #11785728

Subgroups — #11785728

Order 1152.

681 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×C4⋊C4C211yesmaximal, characteristic
2#10591069C212yesmaximal
3#10589396C211yesmaximal, characteristic, complemented
4C3×#4530C211yesmaximal, characteristic, complemented
5#10589167C212yesmaximal, complemented
6#71562—31nomaximal
7#68990—31nomaximal
8C22×C3×C4×D3C412yes—
9C22×C3×C4×D3C2211yescharacteristic
10C2×C32×C4⋊C4C2211yescharacteristic
11C2×#3748C2212yes—
12C2×C3×C4⋊Dic3C2211yescharacteristic
13C2×C3×Dic3⋊C4C412yes—
14C2×C3×#208C2212yes—
15C3×D3×C4⋊C4—22no—
16C3×D3×C4⋊C4—22no—
17(C2×D3)⋊(C3⋊C8)—22no—
18(C2×D3)⋊(C3⋊C8)—22no—
19C22×C3×C4⋊C4—31no—
20C2×D3×C4⋊C4D311yescharacteristic
21#5207—32no—
22#5126—32no—
23C3×#343—31no—
24#4554—32no—
25#4530—31no—
26#4530—31no—
27#4530—32no—
28#4530—61no—
29#4530D311yescharacteristic, complemented
30C23×C3×D3C2×C411yescharacteristic
31C22×C32×C4C2×C412yes—
32C22×C32×C4C2311yescharacteristic
33C22×C3×Dic3C2×C411yescharacteristic
34C22×C3×Dic3C2×C412yes—
35C2×C3×C4×D3—22no—
36C2×C3×C4×D3—21no—
37C2×C3×C4×D3—21no—
38C2×C3×C4×D3—22no—
39C2×C3×C4×D3—22no—
40C2×C3×C4×D3—44no—
41C2×C3×C4×D3D412yes—
42C2×C3×C4×D3D412yes—
43C32×C4⋊C4—22no—
44#3748—22no—
45#3748—22no—
46C2×C3⋊(C3⋊C8)—22no—
47C3×C4⋊Dic3—22no—
48C3×Dic3⋊C4—24no—
49C3×#208—22no—
50C3×#208—22no—
51C2×C3×C3⋊C8—22no—
52#1821—91noSylow
53C23×C3×C4—31no—
54C23×C3×C4—32no—
55C22×C4×D3Dic312yes—
56C22×C4×D3C2×D311yescharacteristic
57C2×C3×M16—32no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—32no—
60C2×C3×C4⋊C4—31no—
61C2×C3×C4⋊C4—62no—
62C2×C3×C4⋊C4—62no—
63C2×C3×C4⋊C4C2×D311yescharacteristic
64C2×C3×C4⋊C4C2×D311yescharacteristic
65C2×C4⋊Dic3C2×D311yescharacteristic
66C2×Dic3⋊C4Dic312yes—
67C2×#208—32no—
68C2×#208—32no—
69C2×#208—62no—
70C2×#208C2×D312yes—
71D3×C4⋊C4—22no—
72D3×C4⋊C4—22no—
73#830—62no—
74#830—62no—
75(C2×D3)⋊C8—62no—
76(C2×D3)⋊C8—62no—
77C23×C32C22×C411yescharacteristic
78C22×C3×D3—21no—
79C22×C3×D3—21no—
80C22×C3×D3—22no—
81C22×C3×D3—22no—
82C22×C3×D3C22⋊C411yescharacteristic
83C22×C3×D3C22⋊C411yescharacteristic
84C2×C32×C4—22no—
85C2×C32×C4—42no—
86C2×C32×C4C2×D412yes—
87C2×C32×C4#4112yes—
88C2×C32×C4#4111yescharacteristic
89C2×C32×C4#4111yescharacteristic
90C2×C3×Dic3—21no—
91C2×C3×Dic3—21no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—42no—
94C2×C3×Dic3C22⋊C412yes—
95C3×C4×D3—22no—
96C3×C4×D3—22no—
97C3×C4×D3—22no—
98C3×C4×D3—22no—
99C3×C4×D3—44no—
100C3×C4×D3—44no—
101C3⋊(C3⋊C8)—42no—
102C3×C3⋊C8—42no—
103C22×C4⋊C4—31no—
104#407—92no—
105#343—31no—
106#343—92no—
107#343—91no—
108C24×C3—31no—
109C23×D3C2×Dic311yescharacteristic
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—31no—
113C22×C3×C4—32no—
114C22×C3×C4—32no—
115C22×C3×C4—32no—
116C22×C3×C4—62no—
117C22×C3×C4—62no—
118C22×C3×C4—62no—
119C22×C3×C4—61no—
120C22×C3×C4—61no—
121C22×C3×C4—124no—
122C22×C3×C4C4×D312yes—
123C22×C3×C4C2×Dic312yes—
124C22×C3×C4C22×D311yescharacteristic
125C22×C3×C4C22×D311yescharacteristic
126C22×Dic3C2×Dic311yescharacteristic
127C22×Dic3C2×Dic312yes—
128C2×C4×D3—21no—
129C2×C4×D3—22no—
130C2×C4×D3—21no—
131C2×C4×D3—22no—
132C2×C4×D3—22no—
133C2×C4×D3—44no—
134C2×C4×D3#6712yes—
135C2×C4×D3#6712yes—
136C3×M16—62no—
137C3×M16—62no—
138C2×C3×C8—62no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—42no—
142C3×C4⋊C4—64no—
143C3×C4⋊C4—62no—
144C4⋊Dic3—22no—
145Dic3⋊C4—24no—
146#208—22no—
147#208—22no—
148#208—64no—
149#208—62no—
150#208—62no—
151#208—62no—
152#208—62no—
153#208—122no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—62no—
156C2×C3⋊C8—62no—
157C2×C3⋊C8—122no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32C2×C22⋊C411yescharacteristic
161C22×C32#11711yescharacteristic
162C22×C32#11711yescharacteristic
163C2×C3×D3—24no—
164C2×C3×D3—24no—
165C2×C3×D3—21no—
166C2×C3×D3—21no—
167C2×C3×D3—21no—
168C2×C3×D3—21no—
169C2×C3×D3#10012yes—
170C2×C3×D3#10112yes—
171C32×C4—42no—
172C32×C4#12312yes—
173C32×C4#12312yes—
174C3×Dic3—22no—
175C3×Dic3—42no—
176C23×C4—31no—
177C23×C4—32no—
178C2×M16—32no—
179C2×M16—92no—
180C2×C4⋊C4—32no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4D3211yescharacteristic
185C22⋊C8—182no—
186C22⋊C8—182no—
187C23×C3—21no—
188C23×C3—31no—
189C23×C3—31no—
190C23×C3—62no—
191C23×C3—62no—
192C23×C3—61no—
193C23×C3—61no—
194C23×C3C2×C4×D311yescharacteristic
195C23×C3C22×Dic311yescharacteristic
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—22no—
199C22×D3—22no—
200C22×D3#21711yescharacteristic
201C22×D3#21711yescharacteristic
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—32no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—61no—
213C2×C3×C4—61no—
214C2×C3×C4—62no—
215C2×C3×C4—62no—
216C2×C3×C4—62no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—82no—
220C2×C3×C4—124no—
221C2×C3×C4—124no—
222C2×C3×C4—122no—
223C2×C3×C4#23511yescharacteristic
224C2×C3×C4#23512yes—
225C2×C3×C4#23512yes—
226C2×C3×C4#23511yescharacteristic
227C2×C3×C4D3×D412yes—
228C2×C3×C4#23711yescharacteristic
229C2×C3×C4#23911yescharacteristic
230C2×C3×C4C2×#6712yes—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—22no—
234C2×Dic3—42no—
235C2×Dic3#21712yes—
236C4×D3—22no—
237C4×D3—22no—
238C4×D3—22no—
239C4×D3—22no—
240C4×D3—44no—
241C4×D3—44no—
242C3×C8—122no—
243C3⋊C8—42no—
244C3⋊C8—124no—
245C3⋊C8—122no—
246C3⋊C8—122no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32#38111yescharacteristic
250C2×C32C2×#10011yescharacteristic
251C2×C32C2×#10111yescharacteristic
252C3×D3—22no—
253C3×D3—22no—
254C24—31no—
255C22×C4—31no—
256C22×C4—32no—
257C22×C4—32no—
258C22×C4—32no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4—61no—
262C22×C4—62no—
263C22×C4—61no—
264C22×C4—124no—
265C22×C4D3×Dic312yes—
266C22×C4C2×D3211yescharacteristic
267M16—62no—
268M16—62no—
269M16—182no—
270M16—182no—
271C2×C8—62no—
272C2×C8—182no—
273C4⋊C4—22no—
274C4⋊C4—64no—
275C4⋊C4—62no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—32no—
284C22×C3—32no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—64no—
288C22×C3—64no—
289C22×C3—61no—
290C22×C3—61no—
291C22×C3—61no—
292C22×C3—61no—
293C22×C3#87211yescharacteristic
294C22×C3D3×C22⋊C411yescharacteristic
295C22×C3#89211yescharacteristic
296C22×C3#92611yescharacteristic
297C22×C3#92611yescharacteristic
298C22×C3C2×#21711yescharacteristic
299C2×D3—24no—
300C2×D3—24no—
301C2×D3—21no—
302C2×D3—21no—
303C2×D3—21no—
304C2×D3—21no—
305C2×D3#83312yes—
306C2×D3#83612yes—
307C3×C4—22no—
308C3×C4—22no—
309C3×C4—42no—
310C3×C4—42no—
311C3×C4—62no—
312C3×C4—82no—
313C3×C4—122no—
314C3×C4#88312yes—
315C3×C4#89412yes—
316C3×C4#92912yes—
317C3×C4#92912yes—
318Dic3—22no—
319Dic3—42no—
320C32#182111yescharacteristic, complemented, Sylow
321C23—31no—
322C23—31no—
323C23—62no—
324C23—62no—
325C23—61no—
326C23—61no—
327C23C2×D3×Dic311yescharacteristic
328C2×C4—22no—
329C2×C4—32no—
330C2×C4—42no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—62no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—124no—
339C2×C4—124no—
340C2×C4—122no—
341C2×C4#379511yescharacteristic
342C2×C4#379812yes—
343C2×C4#379911yescharacteristic
344C2×C4D3×#6712yes—
345C8—122no—
346C8—362no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—62no—
357C2×C3—62no—
358C2×C3#466411yescharacteristic
359C2×C3D3×#10011yescharacteristic
360C2×C3D3×#10111yescharacteristic
361C2×C3#521111yescharacteristic
362C2×C3C2×#83311yescharacteristic
363C2×C3C2×#83611yescharacteristic
364D3—22no—
365D3—22no—
366C22—21no—
367C22—21no—
368C22—32no—
369C22—32no—
370C22—64no—
371C22—64no—
372C22—61no—
373C22—61no—
374C22—61no—
375C22—61no—
376C22#6384111yescharacteristic
377C22#6385511yescharacteristic
378C22D3×#21711yescharacteristic
379C4—42no—
380C4—62no—
381C4—122no—
382C4#6385012yes—
383C4#6390512yes—
384C3—21no—
385C3#6899011yescharacteristic, complemented
386C3#7156211yescharacteristic, complemented
387C2—21no—
388C2—21no—
389C2—62no—
390C2—62no—
391C2#1059129411yescharacteristic
392C2D3×#83311yescharacteristic
393C2D3×#83611yescharacteristic
subgroupG/H
centerC22D3×#217
commutatorC23×C32C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC22×C32C2×C22⋊C4
prime radicalC23×C3×D3C2×C4
cosocleC22×C32×C4C23
hypercenterC2×C4⋊C4D32
hypocenterC32#1821
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #10589396C2semidirect
4. C3×#4530C2semidirect
5. #10589167C2semidirect
29. #4530D3semidirect
320. C32#1821semidirect
385. C3#68990semidirect
386. C3#71562semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×C4⋊C42
#105910692
#105893962
C3×#45302
#105891672
#715623
#689903