Finite Group Databasebeta

← #11921895

Subgroups — #11921895

Order 1152.

668 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 89 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×C3×A4⋊Q8C211yesmaximal, characteristic
2A4×C4⋊Dic3C211yesmaximal, characteristic
3C4×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic
4SL(2,ℤ/4)⋊Dic3C212yesmaximal
5Dic3×SL(2,ℤ/4)C212yesmaximal
6#85302—31nomaximal
7#82747—31nomaximal
8C2×C3×C4×A4C2211yescharacteristic
9C2×Dic3×A4C2212yes—
10C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
11C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
12C2×C3×SL(2,ℤ/4)C2212yes—
13C3×A4⋊Q8C414yescomplemented
14#63843—41nomaximal
15C2×A4⋊Q8D311yescharacteristic
16C2×C3×C22⋊Q8—31no—
17C2×Q8×Dic3—31no—
18C22×C4⋊Dic3D311yescharacteristic
19A4×C4⋊C4—31no—
20SL(2,ℤ/4)⋊C4—32no—
21C4×SL(2,ℤ/4)—31no—
22C4×SL(2,ℤ/4)—32no—
23C4×SL(2,ℤ/4)—61no—
24#5229—31no—
25C4×#217—31no—
26#4984—32no—
27#4984—31no—
28#4980—32no—
29Dic3×C4⋊C4—31no—
30#4950—32no—
31Dic3×C22⋊C4—32no—
32C22×C3×A4C2311yescharacteristic
33C2×C3×Dic6—41no—
34C3×C4×A4C2×C412yes—
35Dic3×A4—22no—
36C3⋊SL(2,ℤ/4)—21no—
37C3⋊SL(2,ℤ/4)Q812yescomplemented
38C3×SL(2,ℤ/4)C2×C414yes—
39C4×C3⋊Dic3—41no—
40C3×C4⋊Dic3—41no—
41Dic3⋊Dic3—42no—
42Dic32—42no—
43#2360—91noSylow
44C22×C3×Q8—31no—
45C23×C3×C4C2×D311yescharacteristic
46C23×Dic3C2×D312yes—
47C2×C4×A4—21no—
48C2×C4×A4—32no—
49C2×C4×A4C2×D311yescharacteristic
50C2×SL(2,ℤ/4)—31no—
51C2×SL(2,ℤ/4)—31no—
52C2×SL(2,ℤ/4)—61no—
53C2×SL(2,ℤ/4)—61no—
54C2×SL(2,ℤ/4)C2×D312yes—
55A4⋊Q8Dic314yescomplemented
56C3×C22⋊Q8—34no—
57C3×C22⋊Q8—34no—
58C2×C3×C4⋊C4—31no—
59C2×C3×C4⋊C4—32no—
60C2×C3×C22⋊C4—32no—
61C2×#217—31no—
62C2×#217—31no—
63Q8×Dic3—61no—
64Q8×Dic3—62no—
65Q8×Dic3—61no—
66C2×C4⋊Dic3—31no—
67C2×C4⋊Dic3—31no—
68C2×C4⋊Dic3—32no—
69C2×C4⋊Dic3—32no—
70C2×C4⋊Dic3—32no—
71C2×C4×Dic3—32no—
72C2×C4×Dic3—31no—
73C2×C4×Dic3—31no—
74C2×C4×Dic3—32no—
75#887—121no—
76C3⋊#95—32no—
77C3⋊#95—32no—
78C3⋊#95—32no—
79C3⋊#95—32no—
80C3⋊#95—31no—
81C3⋊#95—31no—
82C2×C3×A4C22×C411yescharacteristic
83C2×C3×A4C2×Q811yescharacteristic
84C2×C3×A4#4111yescharacteristic
85C2×C32×C4—41no—
86C2×C3⋊Dic3—41no—
87C2×C3⋊Dic3—41no—
88C2×C3×Dic3—42no—
89C2×C3×Dic3—42no—
90C3×Dic6—44no—
91C2×C22⋊Q8—31no—
92C2×C4×Q8—91no—
93C22×C4⋊C4—31no—
94#391—91no—
95#389—91no—
96#389—92no—
97#387—92no—
98#385—92no—
99C4×C4⋊C4—91no—
100C4×C22⋊C4—92no—
101C4×C22⋊C4—91no—
102C24×C3C22×D311yescharacteristic
103C22×A4—21no—
104C22×A4C22×D311yescharacteristic
105C2×C3×Q8—34no—
106C2×C3×Q8—61no—
107C2×C3×Q8—62no—
108C2×C3×Q8—61no—
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—32no—
112C22×C3×C4—32no—
113C22×C3×C4—32no—
114C22×C3×C4C4×D312yes—
115C22×Dic3—22no—
116C22×Dic3—31no—
117C22×Dic3—31no—
118C22×Dic3—32no—
119C22×Dic3—32no—
120C22×Dic3—31no—
121C22×Dic3—31no—
122C22×Dic3—62no—
123C2×Dic6—41no—
124C4×A4—22no—
125C4×A4—62no—
126C4×A4C2×Dic312yes—
127SL(2,ℤ/4)—32no—
128SL(2,ℤ/4)—62no—
129SL(2,ℤ/4)—62no—
130SL(2,ℤ/4)—61no—
131SL(2,ℤ/4)C2×Dic314yes—
132C3×C4⋊C4—34no—
133C3×C4⋊C4—38no—
134C3×C4⋊C4—121no—
135C3×C22⋊C4—34no—
136C3×C22⋊C4—34no—
137#217—32no—
138#217—32no—
139#217—61no—
140#217—61no—
141C4⋊Dic3—31no—
142C4⋊Dic3—32no—
143C4⋊Dic3—61no—
144C4⋊Dic3—62no—
145C4⋊Dic3—62no—
146C4⋊Dic3—122no—
147C4⋊Dic3S411yescharacteristic
148Dic3⋊C4—122no—
149C4×Dic3—62no—
150C4×Dic3—62no—
151C4×Dic3—61no—
152C4×Dic3—61no—
153C4×Dic3—62no—
154C4×Dic3—122no—
155C4×Dic3—121no—
156C4×Dic3—122no—
157C4×Dic3—241no—
158C22×C32—41no—
159C3×A4C4×Q811yescharacteristic, complemented
160C32×C4—42no—
161C3⋊Dic3—42no—
162C3⋊Dic3—81no—
163C3×Dic3—44no—
164C3×Dic3—82no—
165C22×Q8—31no—
166C23×C4—32no—
167C23×C4D3211yescharacteristic
168C22⋊Q8—34no—
169C22⋊Q8—34no—
170C4×Q8—181no—
171C4×Q8—182no—
172C4×Q8—181no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—91no—
176C2×C4⋊C4—92no—
177C2×C4⋊C4—91no—
178C2×C4⋊C4—92no—
179C2×C4⋊C4—92no—
180C2×C22⋊C4—32no—
181C2×C22⋊C4—91no—
182C2×C22⋊C4—91no—
183C2×C42—92no—
184C2×C42—91no—
185C2×C42—91no—
186C2×C42—92no—
187#95—91no—
188#95—92no—
189#95—91no—
190#95—92no—
191#95—92no—
192#95—92no—
193C23×C3—31no—
194C23×C3—31no—
195C23×C3—31no—
196C23×C3—31no—
197C23×C3C2×C4×D311yescharacteristic
198C23×C3#23711yescharacteristic
199C23×C3D3×Q811yescharacteristic
200C2×A4—21no—
201C2×A4—21no—
202C2×A4—21no—
203C2×A4D3×Q811yescharacteristic
204C2×A4#23911yescharacteristic
205C2×A4C22×Dic311yescharacteristic
206C3×Q8—64no—
207C3×Q8—64no—
208C2×C3×C4—31no—
209C2×C3×C4—32no—
210C2×C3×C4—32no—
211C2×C3×C4—32no—
212C2×C3×C4—34no—
213C2×C3×C4—34no—
214C2×C3×C4—41no—
215C2×C3×C4—61no—
216C2×C3×C4—62no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—81no—
221C2×C3×C4—122no—
222C2×C3×C4C2×S411yescharacteristic
223C2×Dic3—32no—
224C2×Dic3—32no—
225C2×Dic3—32no—
226C2×Dic3—42no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—121no—
242C2×Dic3—121no—
243C2×Dic3—241no—
244C2×Dic3—241no—
245C2×Dic3C2×S412yes—
246Dic6—44no—
247C2×C32—41no—
248C2×C32—41no—
249C2×C32—41no—
250C24C2×D3211yescharacteristic
251C2×Q8—34no—
252C2×Q8—61no—
253C2×Q8—62no—
254C2×Q8—61no—
255C22×C4—31no—
256C22×C4—31no—
257C22×C4—32no—
258C22×C4—32no—
259C22×C4—32no—
260C22×C4—62no—
261C22×C4—92no—
262C22×C4—91no—
263C22×C4—91no—
264C22×C4—92no—
265C22×C4—91no—
266C22×C4—91no—
267C22×C4—182no—
268C22×C4D3×Dic312yes—
269C4⋊C4—34no—
270C4⋊C4—38no—
271C4⋊C4—31no—
272C4⋊C4—92no—
273C4⋊C4—91no—
274C4⋊C4—182no—
275C4⋊C4—181no—
276C4⋊C4—182no—
277C4⋊C4—362no—
278C22⋊C4—34no—
279C22⋊C4—34no—
280C22⋊C4—92no—
281C22⋊C4—92no—
282C22⋊C4—181no—
283C22⋊C4—181no—
284C42—182no—
285C42—181no—
286C42—182no—
287C42—182no—
288C42—181no—
289C42—362no—
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—41no—
300C22×C3—61no—
301C22×C3—81no—
302C22×C3#88711yescharacteristic, complemented
303C22×C3C22×S411yescharacteristic
304A4—21no—
305A4Q8×Dic311yescharacteristic, complemented
306C3×C4—32no—
307C3×C4—42no—
308C3×C4—64no—
309C3×C4—64no—
310C3×C4—82no—
311C3×C4—242no—
312C3×C4C4×S412yes—
313Dic3—22no—
314Dic3—44no—
315Dic3—62no—
316Dic3—62no—
317Dic3—62no—
318Dic3—122no—
319Dic3—121no—
320Dic3—121no—
321Dic3—242no—
322Dic3—241no—
323Dic3—242no—
324C32—41noSylow
325C23—31no—
326C23—31no—
327C23—31no—
328C23—31no—
329C23#379611yescharacteristic
330C23#379711yescharacteristic
331C23C2×D3×Dic311yescharacteristic
332Q8—64no—
333Q8—64no—
334C2×C4—31no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—34no—
339C2×C4—34no—
340C2×C4—32no—
341C2×C4—61no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—92no—
347C2×C4—92no—
348C2×C4—92no—
349C2×C4—182no—
350C2×C4—182no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4—181no—
355C2×C4—182no—
356C2×C4—182no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—181no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4D3×S411yescharacteristic
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—81no—
372C2×C3—81no—
373C2×C3—81no—
374C2×C3C2×C4×S411yescharacteristic
375C2×C3#591411yescharacteristic
376C2×C3Q8×S411yescharacteristic
377C22—31no—
378C22—31no—
379C22—31no—
380C22—31no—
381C22—31no—
382C22—31no—
383C22—31no—
384C22—31no—
385C22—31no—
386C22—61no—
387C22#6384311yescharacteristic, complemented
388C22C2×D3×S411yescharacteristic
389C4—32no—
390C4—64no—
391C4—62no—
392C4—64no—
393C4—182no—
394C4—182no—
395C4—182no—
396C4—361no—
397C4—361no—
398C4Dic3×S412yes—
399C3—41no—
400C3—81no—
401C3#8530211yescharacteristic, complemented
402C2—31no—
403C2—31no—
404C2—31no—
405C2—31no—
406C2(C3⋊S4)⋊Q811yescharacteristic
407C2#1059642511yescharacteristic
408C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutatorC2×C3×A4C22×C4
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radical, cosocleC22×C3×A4C23
hypercenterC2×C4D3×S4
hypocenterC3×A4C4×Q8
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
13. C3×A4⋊Q8C4semidirect
37. C3⋊SL(2,ℤ/4)Q8semidirect
55. A4⋊Q8Dic3semidirect
159. C3×A4C4×Q8semidirect
302. C22×C3#887semidirect
305. A4Q8×Dic3semidirect
387. C22#63843semidirect
401. C3#85302semidirect

Maximal subgroups

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

subgroupindex
C2×C3×A4⋊Q82
A4×C4⋊Dic32
C4×C3⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊Dic32
Dic3×SL(2,ℤ/4)2
#853023
#827473
#638434