Finite Group Databasebeta

← #11785909

Subgroups — #11785909

Order 1152.

697 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5798C211yesmaximal, characteristic
2C2×#63661C211yesmaximal, characteristic
3C2×C3×#834C211yesmaximal, characteristic
4#10589187C212yesmaximal, complemented
5#10589186C212yesmaximal, complemented
6#71653—31nomaximal
7#70020—31nomaximal
8C22×C32×D4C2211yescharacteristic
9C2×#3757C2212yes—
10C2×C3×#217C412yescomplemented
11C2×C3×#217C2212yes—
12C2×C3×#217C2211yescharacteristic
13C2×C3×#217C2211yescharacteristic
14C3×D4×Dic3—22no—
15C3×D4×Dic3—22no—
16C3×#926C412yescomplemented
17#63661—22no—
18C3×#834—22no—
19C3×#518—31no—
20#5798D311yescharacteristic
21C2×C3×#99—31no—
22C2×#834—31no—
23C2×#834—31no—
24C2×#834—61no—
25C2×#834D311yescharacteristic
26#4527—32no—
27#4527—32no—
28#4527—32no—
29#4526—32no—
30C24×C32C2×C411yescharacteristic
31C24×C32C2311yescharacteristic
32C2×C32×D4—22no—
33C2×C32×D4—22no—
34C2×C32×D4D412yes—
35C2×C32×D4D412yes—
36C22×C32×C4C2×C411yescharacteristic
37C22×C3⋊Dic3—22no—
38C22×C3×Dic3—22no—
39C22×C3×Dic3—21no—
40C22×C3×Dic3—21no—
41C22×C3×Dic3—22no—
42#3757—22no—
43#3757—22no—
44C3×#217—22no—
45C3×#217—21no—
46C3×#217—21no—
47C3×#217—22no—
48C3×#217—22no—
49C3×#217C2×C412yes—
50C3×#217C2×C412yes—
51C3×C4⋊Dic3—22no—
52C3×C4×Dic3—22no—
53#1828—91noSylow
54C22×C3×D4—21no—
55C22×C3×D4C2×D311yescharacteristic
56C22×C3×D4C2×D311yescharacteristic
57C3×C4×D4—62no—
58C3×C4×D4—62no—
59C3×#117—32no—
60C2×C3×C22⋊C4—32no—
61C2×C3×C22⋊C4—31no—
62C2×C3×C22⋊C4—31no—
63C2×C3×C22⋊C4—32no—
64C2×#217—32no—
65C2×#217—32no—
66C2×#217—62no—
67C2×#217Dic312yescomplemented
68C2×#217C2×D312yes—
69C2×#217C2×D311yescharacteristic
70C2×#217C2×D311yescharacteristic
71D4×Dic3—22no—
72D4×Dic3—22no—
73#926Dic312yescomplemented
74C3×#99—62no—
75#834—22no—
76#834—64no—
77#834—62no—
78#834—62no—
79C23×C32—21no—
80C23×C32—21no—
81C23×C32—81no—
82C23×C32—81no—
83C23×C32C22⋊C412yes—
84C23×C32C22×C411yescharacteristic
85C23×C32C2×D411yescharacteristic
86C23×C32C2×D411yescharacteristic
87C23×C32#4112yes—
88C23×C32#4112yes—
89C32×D4—24no—
90C32×D4—24no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4C22⋊C412yes—
94C2×C3⋊Dic3—42no—
95C2×C3⋊Dic3—82no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—81no—
100C2×C3×Dic3—81no—
101C2×C3×Dic3—82no—
102C2×C3×Dic3—82no—
103#518—31no—
104C2×#99—31no—
105C2×#99—91no—
106#342—92no—
107#342—92no—
108C24×C3—21no—
109C24×C3—21no—
110C24×C3C4×D311yescharacteristic
111C24×C3C2×Dic311yescharacteristic
112C24×C3C22×D311yescharacteristic
113C24×C3C22×D311yescharacteristic
114C2×C3×D4—22no—
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
118C2×C3×D4—22no—
119C2×C3×D4—22no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C2×C3×D4D1212yes—
123C2×C3×D4#6712yes—
124C2×C3×D4#6712yes—
125C2×C3×D4#6712yes—
126C22×C3×C4—21no—
127C22×C3×C4—62no—
128C22×C3×C4—61no—
129C22×C3×C4—61no—
130C22×C3×C4—62no—
131C22×C3×C4C4×D311yescharacteristic
132C22×C3×C4C2×Dic311yescharacteristic
133C22×Dic3—22no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—22no—
137C22×Dic3—62no—
138C22×Dic3—62no—
139C22×Dic3—122no—
140C3×C4⋊C4—62no—
141C3×C22⋊C4—32no—
142C3×C22⋊C4—32no—
143C3×C22⋊C4—62no—
144C3×C22⋊C4—61no—
145C3×C22⋊C4—61no—
146C3×C22⋊C4—62no—
147C3×C22⋊C4—62no—
148C3×C42—62no—
149#217—22no—
150#217—21no—
151#217—21no—
152#217—22no—
153#217—22no—
154#217—64no—
155#217—62no—
156#217—62no—
157#217—62no—
158#217—62no—
159#217—122no—
160#217C2×Dic312yes—
161#217C2×Dic312yes—
162C4⋊Dic3—22no—
163C4×Dic3—22no—
164C22×C32—22no—
165C22×C32—42no—
166C22×C32—41no—
167C22×C32—41no—
168C22×C32—41no—
169C22×C32—42no—
170C22×C32—42no—
171C22×C32—81no—
172C22×C32—81no—
173C22×C32C2×C22⋊C411yescharacteristic
174C22×C32#11711yescharacteristic
175C22×C32#11711yescharacteristic
176C22×C32#12312yes—
177C22×C32#12312yes—
178C32×C4—22no—
179C3⋊Dic3—82no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C3×Dic3—82no—
183C22×D4D3211yescharacteristic
184C4×D4—62no—
185C4×D4—62no—
186#117—32no—
187C2×C22⋊C4—32no—
188C2×C22⋊C4—31no—
189C2×C22⋊C4—31no—
190C2×C22⋊C4—32no—
191C2×C22⋊C4—92no—
192#99—62no—
193#99—182no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—22no—
199C23×C3—21no—
200C23×C3—22no—
201C23×C3—22no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—81no—
207C23×C3—81no—
208C23×C3—81no—
209C23×C3—81no—
210C23×C3—82no—
211C23×C3—82no—
212C23×C3(C2×D3)⋊C412yes—
213C23×C3#21712yes—
214C23×C3C2×C4×D311yescharacteristic
215C23×C3C2×D1211yescharacteristic
216C23×C3#23512yes—
217C23×C3#23512yes—
218C23×C3#23712yes—
219C23×C3#23712yes—
220C23×C3C22×Dic311yescharacteristic
221C23×C3C2×#6711yescharacteristic
222C23×C3C2×#6711yescharacteristic
223C23×C3C2×#6711yescharacteristic
224C3×D4—24no—
225C3×D4—24no—
226C3×D4—24no—
227C3×D4—24no—
228C3×D4—44no—
229C3×D4—44no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—122no—
240C2×C3×C4—241no—
241C2×C3×C4—241no—
242C2×C3×C4—242no—
243C2×C3×C4—242no—
244C2×C3×C4(C2×D3)⋊C412yes—
245C2×C3×C4#21712yes—
246C2×Dic3—22no—
247C2×Dic3—22no—
248C2×Dic3—42no—
249C2×Dic3—81no—
250C2×Dic3—81no—
251C2×Dic3—82no—
252C2×Dic3—82no—
253C2×Dic3—124no—
254C2×Dic3—122no—
255C2×Dic3—122no—
256C2×Dic3—244no—
257C2×Dic3—242no—
258C2×Dic3—242no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—22no—
262C2×C32—22no—
263C2×C32—21no—
264C2×C32—42no—
265C2×C32#38111yescharacteristic
266C24D3×Dic311yescharacteristic
267C24C2×D3211yescharacteristic
268C2×D4—22no—
269C2×D4—22no—
270C2×D4#62512yes—
271C2×D4#62612yes—
272C22×C4—62no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4—182no—
277C22×C4D3×Dic311yescharacteristic
278C4⋊C4—62no—
279C22⋊C4—32no—
280C22⋊C4—32no—
281C22⋊C4—62no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—62no—
285C22⋊C4—62no—
286C22⋊C4—182no—
287C22⋊C4—182no—
288C42—62no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—42no—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3—41no—
303C22×C3—42no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—42no—
308C22×C3—42no—
309C22×C3—44no—
310C22×C3—42no—
311C22×C3—82no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3—82no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—82no—
320C22×C3—82no—
321C22×C3#87711yescharacteristic
322C22×C3#87711yescharacteristic
323C22×C3#88612yes—
324C22×C3#92611yescharacteristic
325C22×C3#92611yescharacteristic
326C22×C3C2×(C2×D3)⋊C411yescharacteristic
327C22×C3#92912yes—
328C22×C3#92912yes—
329C22×C3#93512yes—
330C22×C3C2×#21711yescharacteristic
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—42no—
334C3×C4—122no—
335C3×C4—122no—
336C3×C4—242no—
337Dic3—42no—
338Dic3—42no—
339Dic3—82no—
340Dic3—242no—
341Dic3—242no—
342Dic3—244no—
343C32#182811yescharacteristic, complemented, Sylow
344C23—21no—
345C23—21no—
346C23—81no—
347C23—81no—
348C23(C2×D3)⋊Dic312yes—
349C23C2×D3×Dic311yescharacteristic
350C23#380412yes—
351C23#380412yes—
352C23C2×#62511yescharacteristic
353C23C2×#62611yescharacteristic
354D4—24no—
355D4—24no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—62no—
359C2×C4—62no—
360C2×C4—122no—
361C2×C4—241no—
362C2×C4—241no—
363C2×C4—242no—
364C2×C4—242no—
365C2×C4—362no—
366C2×C4—722no—
367C2×C4(C2×D3)⋊Dic312yes—
368C2×C3—22no—
369C2×C3—22no—
370C2×C3—22no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—22no—
376C2×C3—22no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—42no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—42no—
384C2×C3—42no—
385C2×C3—42no—
386C2×C3—82no—
387C2×C3#494311yescharacteristic
388C2×C3#521111yescharacteristic
389C22—22no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—42no—
394C22—41no—
395C22—42no—
396C22—81no—
397C22—81no—
398C22#6395511yescharacteristic
399C22#6395511yescharacteristic
400C22C2×(C2×D3)⋊Dic311yescharacteristic
401C22#6396112yes—
402C22#6396612yes—
403C4—22no—
404C4—122no—
405C4—122no—
406C4—242no—
407C4—722no—
408C3—21no—
409C3#7002011yescharacteristic, complemented
410C3#7165311yescharacteristic, complemented
411C2—21no—
412C2—22no—
413C2—22no—
414C2—21no—
415C2—21no—
416C2—42no—
417C2#1059169611yescharacteristic
subgroupG/H
centerC2#10591696
commutatorC23×C32C22×C4
FrattiniC24C2×D32
Fitting, prime radicalC22×C32×D4C22
socleC2×C32#381
cosocleC24×C32C23
hypercenterC22×D4D32
hypocenterC32#1828
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
4. #10589187C2semidirect
5. #10589186C2semidirect
10. C2×C3×#217C4semidirect
16. C3×#926C4semidirect
67. C2×#217Dic3semidirect
73. #926Dic3semidirect
343. C32#1828semidirect
409. C3#70020semidirect
410. C3#71653semidirect

Maximal subgroups

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

subgroupindex
C3×#57982
C2×#636612
C2×C3×#8342
#105891872
#105891862
#716533
#700203