Finite Group Databasebeta

← #11921100

Subgroups — #11921100

Order 1152.

630 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 91 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×D4×A4C211yesmaximal, characteristic
2C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
3C2×C3⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4#73144—33nomaximal
5#72128—31nomaximal
6C23×C3×A4—21no—
7C3×D4×A4—21no—
8C3×D4×A4C412yescomplemented
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3⋊SL(2,ℤ/4)—21no—
11C3⋊(A4⋊C8)—21no—
12#63659—41nomaximal
13C23×C3×D4D311yescharacteristic
14C2×D4×A4D313yes—
15C4⋊SL(2,ℤ/4)—33no—
16C2×A4⋊C8—33no—
17C2×#832—31no—
18C2×#832—31no—
19C2×#832—31no—
20C2×#832—31no—
21#5212—31no—
22C2×#830—31no—
23C22×C3×A4—22no—
24C22×C3×A4—41no—
25C22×C3×A4—41no—
26C22×C3×A4D411yescharacteristic
27C2×C32×D4—41no—
28C3×C4×A4C2×C411yescharacteristic
29C3×C4×A4D411yescharacteristic
30C3⋊SL(2,ℤ/4)—41no—
31C4⋊(C3⋊Dic3)—41no—
32C2×C3⋊(C3⋊C8)—41no—
33#1786—91noSylow
34C25×C3—21no—
35C23×A4—23no—
36C22×C3×D4—21no—
37C22×C3×D4—31no—
38C22×C3×D4—31no—
39C22×C3×D4—32no—
40C22×C3×D4—61no—
41C22×C3×D4—61no—
42C22×C3×D4Dic312yescomplemented
43C23×C3×C4C2×D311yescharacteristic
44D4×A4—23no—
45D4×A4Dic316yescomplemented
46C2×C4×A4C2×D313yes—
47C2×SL(2,ℤ/4)—63no—
48C2×#217—61no—
49C2×C4⋊Dic3—31no—
50C2×C4⋊Dic3—31no—
51C22×C3⋊C8—31no—
52C22×C3⋊C8—31no—
53A4⋊C8—63no—
54#832—62no—
55#832—62no—
56#832—62no—
57#832—61no—
58#832—61no—
59#832—123no—
60C3⋊#95—61no—
61#830—61no—
62#830—61no—
63C23×C32—81no—
64C2×C3×A4—42no—
65C2×C3×A4C22⋊C411yescharacteristic
66C2×C3×A4D811yescharacteristic
67C2×C3×A4SD1611yescharacteristic
68C32×D4—42no—
69C32×D4—81no—
70C2×C32×C4—41no—
71C2×C3⋊Dic3—81no—
72C3⋊(C3⋊C8)—81no—
73C23×D4C3⋊D311yescharacteristic
74C2×#102—91no—
75C2×#102—91no—
76C2×#102—91no—
77C2×#102—91no—
78C2×C22⋊C8—91no—
79#380—91no—
80C24×C3—22no—
81C24×C3—41no—
82C24×C3—41no—
83C24×C3—61no—
84C24×C3—61no—
85C24×C3—62no—
86C24×C3—121no—
87C24×C3—121no—
88C24×C3#6711yescharacteristic
89C22×A4—26no—
90C22×A4—43no—
91C22×A4—43no—
92C22×A4#6713yes—
93C2×C3×D4—32no—
94C2×C3×D4—32no—
95C2×C3×D4—32no—
96C2×C3×D4—32no—
97C2×C3×D4—32no—
98C2×C3×D4—31no—
99C2×C3×D4—43no—
100C2×C3×D4—61no—
101C2×C3×D4—61no—
102C2×C3×D4—61no—
103C2×C3×D4—62no—
104C2×C3×D4—61no—
105C2×C3×D4—61no—
106C2×C3×D4—62no—
107C2×C3×D4—64no—
108C2×C3×D4S411yescharacteristic
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4—31no—
113C22×C3×C4C2×Dic311yescharacteristic
114C22×C3×C4#6711yescharacteristic
115C22×Dic3—61no—
116C22×Dic3—61no—
117C4×A4C2×Dic313yes—
118C4×A4#6713yes—
119SL(2,ℤ/4)—123no—
120#217—121no—
121#217—121no—
122C4⋊Dic3—61no—
123C4⋊Dic3—61no—
124C4⋊Dic3—61no—
125C4⋊Dic3—61no—
126C4⋊Dic3—123no—
127C2×C3⋊C8—61no—
128C2×C3⋊C8—61no—
129C2×C3⋊C8—61no—
130C2×C3⋊C8—61no—
131C2×C3⋊C8—61no—
132C2×C3⋊C8—61no—
133C2×C3⋊C8—123no—
134C22×C32—41no—
135C22×C32—82no—
136C22×C32—161no—
137C22×C32—161no—
138C3×A4#10211yescharacteristic, complemented
139C32×C4—41no—
140C32×C4—41no—
141C3⋊Dic3—161no—
142C25—21no—
143C22×D4—21no—
144C22×D4—31no—
145C22×D4—31no—
146C22×D4—32no—
147C22×D4—61no—
148C22×D4—61no—
149C22×D4C3⋊Dic312yescomplemented
150C23×C4C2×C3⋊D311yescharacteristic
151C22×C8—91no—
152C22×C8—91no—
153C2×C4⋊C4—91no—
154C2×C4⋊C4—91no—
155C2×C22⋊C4—181no—
156#102—182no—
157#102—182no—
158#102—182no—
159#102—181no—
160#102—181no—
161C22⋊C8—181no—
162C22⋊C8—181no—
163#95—181no—
164C23×C3—21no—
165C23×C3—31no—
166C23×C3—31no—
167C23×C3—31no—
168C23×C3—31no—
169C23×C3—42no—
170C23×C3—62no—
171C23×C3—62no—
172C23×C3—62no—
173C23×C3—62no—
174C23×C3—61no—
175C23×C3—83no—
176C23×C3—121no—
177C23×C3—121no—
178C23×C3—121no—
179C23×C3—121no—
180C23×C3—122no—
181C23×C3—122no—
182C23×C3—122no—
183C23×C3—121no—
184C23×C3—121no—
185C23×C3—121no—
186C23×C3—121no—
187C23×C3—124no—
188C23×C3#21311yescharacteristic
189C23×C3#21411yescharacteristic
190C23×C3#21711yescharacteristic
191C2×A4—46no—
192C2×A4#21313yes—
193C2×A4#21413yes—
194C2×A4#21713yes—
195C3×D4—21no—
196C3×D4—34no—
197C3×D4—32no—
198C3×D4—46no—
199C3×D4—62no—
200C3×D4—62no—
201C3×D4—62no—
202C3×D4—61no—
203C3×D4—83no—
204C3×D4SL(2,ℤ/4)12yescomplemented
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—43no—
213C2×C3×C4—61no—
214C2×C3×C4C2×S411yescharacteristic
215C2×Dic3—121no—
216C2×Dic3—121no—
217C2×Dic3—121no—
218C2×Dic3—121no—
219C2×Dic3—121no—
220C2×Dic3—121no—
221C2×Dic3—243no—
222C3⋊C8—121no—
223C3⋊C8—121no—
224C3⋊C8—243no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—162no—
229C24—22no—
230C24—41no—
231C24—41no—
232C24—61no—
233C24—61no—
234C24—62no—
235C24—121no—
236C24—121no—
237C24#63811yescharacteristic
238C2×D4—32no—
239C2×D4—32no—
240C2×D4—32no—
241C2×D4—32no—
242C2×D4—32no—
243C2×D4—31no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—62no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—62no—
251C2×D4—64no—
252C2×D4C3⋊S411yescharacteristic
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—31no—
257C22×C4—181no—
258C22×C4—181no—
259C22×C4C2×C3⋊Dic311yescharacteristic
260C22×C4#63811yescharacteristic
261C2×C8—181no—
262C2×C8—181no—
263C2×C8—181no—
264C2×C8—181no—
265C2×C8—181no—
266C2×C8—181no—
267C4⋊C4—181no—
268C4⋊C4—181no—
269C4⋊C4—181no—
270C4⋊C4—181no—
271C22⋊C4—361no—
272C22⋊C4—361no—
273C22×C3—22no—
274C22×C3—31no—
275C22×C3—31no—
276C22×C3—31no—
277C22×C3—31no—
278C22×C3—31no—
279C22×C3—31no—
280C22×C3—31no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—43no—
286C22×C3—61no—
287C22×C3—62no—
288C22×C3—86no—
289C22×C3—122no—
290C22×C3—122no—
291C22×C3—122no—
292C22×C3—122no—
293C22×C3—122no—
294C22×C3—122no—
295C22×C3—122no—
296C22×C3—122no—
297C22×C3—121no—
298C22×C3—121no—
299C22×C3—163no—
300C22×C3—163no—
301C22×C3#83211yescharacteristic, complemented
302C22×C3GL(2,ℤ/4)11yescharacteristic
303A4#83213yescomplemented
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—43no—
307C3×C4—43no—
308C3×C4C2×SL(2,ℤ/4)11yescharacteristic
309C3×C4GL(2,ℤ/4)11yescharacteristic
310Dic3—241no—
311Dic3—241no—
312Dic3—483no—
313C32—41noSylow
314C23—21no—
315C23—31no—
316C23—31no—
317C23—31no—
318C23—31no—
319C23—42no—
320C23—62no—
321C23—62no—
322C23—62no—
323C23—62no—
324C23—61no—
325C23—121no—
326C23—121no—
327C23—121no—
328C23—121no—
329C23—122no—
330C23—122no—
331C23—122no—
332C23—121no—
333C23—121no—
334C23—121no—
335C23—121no—
336C23—124no—
337C23#375311yescharacteristic
338C23#375411yescharacteristic
339C23#375711yescharacteristic
340D4—21no—
341D4—34no—
342D4—32no—
343D4—62no—
344D4—62no—
345D4—62no—
346D4—61no—
347D4C3⋊SL(2,ℤ/4)12yescomplemented
348C2×C4—31no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—31no—
352C2×C4—31no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—61no—
356C2×C4—361no—
357C2×C4—361no—
358C2×C4—361no—
359C2×C4—361no—
360C2×C4—361no—
361C2×C4—361no—
362C2×C4C2×C3⋊S411yescharacteristic
363C8—361no—
364C8—361no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—42no—
370C2×C3—43no—
371C2×C3—43no—
372C2×C3—43no—
373C2×C3—122no—
374C2×C3—166no—
375C2×C3#541411yescharacteristic
376C2×C3#541511yescharacteristic
377C2×C3#543211yescharacteristic
378C22—22no—
379C22—31no—
380C22—31no—
381C22—31no—
382C22—31no—
383C22—31no—
384C22—31no—
385C22—31no—
386C22—31no—
387C22—31no—
388C22—41no—
389C22—41no—
390C22—61no—
391C22—62no—
392C22—122no—
393C22—122no—
394C22—122no—
395C22—122no—
396C22—122no—
397C22—122no—
398C22—122no—
399C22—122no—
400C22—121no—
401C22—121no—
402C22#6365911yescharacteristic, complemented
403C22#6426911yescharacteristic
404C4—31no—
405C4—31no—
406C4—721no—
407C4—721no—
408C4C2×C3⋊SL(2,ℤ/4)11yescharacteristic
409C4#6426911yescharacteristic
410C3—43no—
411C3#7314411yescharacteristic, complemented
412C2—31no—
413C2—31no—
414C2—31no—
415C2—31no—
416C2—42no—
417C2—122no—
418C2#1059364811yescharacteristic
419C2#1059364911yescharacteristic
420C2#1059366611yescharacteristic
subgroupG/H
centerC22#64269
commutatorC3×C4×A4C2×C4
FrattiniC2×C4C2×C3⋊S4
FittingC23×C3×D4D3
socleC24×C3#67
prime radicalC2×C3×D4×A4C2
cosocleC2×C3×C4×A4C22
hypercenterC2×D4C3⋊S4
hypocenterC3×A4#102
LCSG ▷ C3×C4×A4 ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C4⋊(C3⋊SL(2,ℤ/4))C2semidirect
3. C2×C3⋊(A4⋊C8)C2semidirect
8. C3×D4×A4C4semidirect
42. C22×C3×D4Dic3semidirect
45. D4×A4Dic3semidirect
138. C3×A4#102semidirect
149. C22×D4C3⋊Dic3semidirect
204. C3×D4SL(2,ℤ/4)semidirect
301. C22×C3#832semidirect
303. A4#832semidirect
347. D4C3⋊SL(2,ℤ/4)semidirect
402. C22#63659semidirect
411. C3#73144semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×A42
C4⋊(C3⋊SL(2,ℤ/4))2
C2×C3⋊(A4⋊C8)2
#731443
#721283
#636594