Finite Group Databasebeta

← #11922501

Subgroups — #11922501

Order 1152.

776 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C22×C3×SL(2,ℤ/4)C211yesmaximal, characteristic
2A4×#217C211yesmaximal, characteristic
3#10593666C211yesmaximal, characteristic
4SL(2,ℤ/4)⋊Dic3C212yesmaximal, complemented
5Dic3×SL(2,ℤ/4)C212yesmaximal, complemented
6#85292—31nomaximal
7#82097—31nomaximal
8C23×C3×A4C2211yescharacteristic
9C2×Dic3×A4C2212yes—
10C2×C3⋊SL(2,ℤ/4)C2212yes—
11C2×C3×SL(2,ℤ/4)C414yescomplemented
12C2×C3×SL(2,ℤ/4)C2211yescharacteristic
13C2×C3×SL(2,ℤ/4)C2211yescharacteristic
14#63955—41nomaximal
15C22×SL(2,ℤ/4)D311yescharacteristic
16C22×C3×C22⋊C4—31no—
17C22×#217D311yescharacteristic
18C2×#926—31no—
19A4×C22⋊C4—31no—
20#5432—31no—
21#5432—61no—
22SL(2,ℤ/4)⋊C4—32no—
23C4×SL(2,ℤ/4)—32no—
24#5247—31no—
25#5212—31no—
26#5211—31no—
27C4×#217—31no—
28#4951—32no—
29#4951—32no—
30#4950—32no—
31Dic3×C22⋊C4—32no—
32C22×C3×A4—22no—
33C22×C3×A4C2×C412yes—
34C22×C3×A4C2311yescharacteristic
35C22×C3×Dic3—41no—
36Dic3×A4—22no—
37C3⋊SL(2,ℤ/4)—22no—
38C3×SL(2,ℤ/4)C2×C412yes—
39C3×SL(2,ℤ/4)C2×C412yes—
40#3757—41no—
41C3×#217—41no—
42Dic3⋊Dic3—42no—
43Dic32—42no—
44#2342—91noSylow
45C25×C3C2×D311yescharacteristic
46C23×A4—21no—
47C23×A4C2×D311yescharacteristic
48C23×C3×C4—31no—
49C23×C3×C4—31no—
50C23×Dic3C2×D312yes—
51C2×C4×A4—32no—
52C2×SL(2,ℤ/4)—32no—
53C2×SL(2,ℤ/4)—62no—
54C2×SL(2,ℤ/4)Dic314yescomplemented
55C2×SL(2,ℤ/4)C2×D311yescharacteristic
56C2×SL(2,ℤ/4)C2×D311yescharacteristic
57C2×C3×C22⋊C4—31no—
58C2×C3×C22⋊C4—31no—
59C2×C3×C22⋊C4—34no—
60C2×C3×C22⋊C4—34no—
61C2×C3×C22⋊C4—31no—
62C2×C3×C22⋊C4—31no—
63C2×#217—31no—
64C2×#217—32no—
65C2×#217—31no—
66C2×#217—31no—
67C2×#217—31no—
68C2×#217—32no—
69C2×#217—32no—
70#926—61no—
71#926—62no—
72#926—61no—
73C2×C4⋊Dic3—32no—
74C2×C4⋊Dic3—32no—
75C2×C4×Dic3—32no—
76C2×C4×Dic3—32no—
77#877—121no—
78C3⋊#95—32no—
79C3⋊#95—32no—
80C3⋊#95—32no—
81C3⋊#95—32no—
82C23×C32—41no—
83C2×C3×A4—22no—
84C2×C3×A4C22×C411yescharacteristic
85C2×C3×A4#4112yes—
86C2×C3⋊Dic3—42no—
87C2×C3×Dic3—44no—
88C2×C3×Dic3—41no—
89C2×C3×Dic3—41no—
90C2×C3×Dic3—42no—
91C2×#117—91no—
92C22×C22⋊C4—31no—
93C22×C22⋊C4—31no—
94#388—92no—
95#388—92no—
96#385—92no—
97#381—91no—
98#380—91no—
99#379—91no—
100C4×C22⋊C4—92no—
101C4×C22⋊C4—91no—
102C24×C3—22no—
103C24×C3—31no—
104C24×C3—31no—
105C24×C3—32no—
106C24×C3—62no—
107C24×C3C4×D312yes—
108C24×C3C22×D311yescharacteristic
109C22×A4—22no—
110C22×A4—22no—
111C22×A4—21no—
112C22×A4—42no—
113C22×A4C2×Dic312yes—
114C22×A4C22×D311yescharacteristic
115C22×C3×C4—31no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C22×C3×C4—34no—
120C22×C3×C4—34no—
121C22×C3×C4—61no—
122C22×C3×C4—62no—
123C22×C3×C4—61no—
124C22×C3×C4—122no—
125C22×Dic3—22no—
126C22×Dic3—32no—
127C22×Dic3—32no—
128C22×Dic3—32no—
129C22×Dic3—32no—
130C22×Dic3—41no—
131C22×Dic3—62no—
132C4×A4—62no—
133SL(2,ℤ/4)—62no—
134SL(2,ℤ/4)—122no—
135SL(2,ℤ/4)C2×Dic312yes—
136SL(2,ℤ/4)C2×Dic312yes—
137C3×C22⋊C4—32no—
138C3×C22⋊C4—32no—
139C3×C22⋊C4—34no—
140C3×C22⋊C4—34no—
141C3×C22⋊C4—32no—
142C3×C22⋊C4—32no—
143C3×C22⋊C4—121no—
144#217—31no—
145#217—32no—
146#217—61no—
147#217—61no—
148#217—62no—
149#217—62no—
150#217—62no—
151#217—61no—
152#217—121no—
153#217—122no—
154#217—241no—
155#217S411yescharacteristic
156C4⋊Dic3—62no—
157C4⋊Dic3—62no—
158C4⋊Dic3—122no—
159Dic3⋊C4—122no—
160C4×Dic3—62no—
161C4×Dic3—62no—
162C4×Dic3—122no—
163C4×Dic3—122no—
164C22×C32—42no—
165C22×C32—41no—
166C22×C32—82no—
167C3×A4#11711yescharacteristic, complemented
168C3⋊Dic3—82no—
169C3×Dic3—42no—
170C3×Dic3—42no—
171C3×Dic3—82no—
172C25D3211yescharacteristic
173C23×C4—31no—
174C23×C4—31no—
175C23×C4—32no—
176#117—181no—
177#117—182no—
178#117—181no—
179C2×C4⋊C4—92no—
180C2×C4⋊C4—92no—
181C2×C22⋊C4—31no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—34no—
184C2×C22⋊C4—34no—
185C2×C22⋊C4—31no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—91no—
188C2×C22⋊C4—91no—
189C2×C22⋊C4—92no—
190C2×C22⋊C4—91no—
191C2×C22⋊C4—91no—
192C2×C22⋊C4—92no—
193C2×C22⋊C4—92no—
194C2×C42—92no—
195C2×C42—92no—
196#95—92no—
197#95—92no—
198#95—92no—
199#95—92no—
200C23×C3—22no—
201C23×C3—31no—
202C23×C3—32no—
203C23×C3—31no—
204C23×C3—32no—
205C23×C3—32no—
206C23×C3—32no—
207C23×C3—31no—
208C23×C3—41no—
209C23×C3—62no—
210C23×C3—62no—
211C23×C3—61no—
212C23×C3—62no—
213C23×C3—62no—
214C23×C3—64no—
215C23×C3—62no—
216C23×C3—81no—
217C23×C3—122no—
218C23×C3C2×C4×D311yescharacteristic
219C23×C3#23712yes—
220C23×C3C2×S411yescharacteristic
221C2×A4—22no—
222C2×A4—21no—
223C2×A4—22no—
224C2×A4—42no—
225C2×A4#23512yes—
226C2×A4C22×Dic311yescharacteristic
227C2×C3×C4—32no—
228C2×C3×C4—32no—
229C2×C3×C4—32no—
230C2×C3×C4—32no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—64no—
236C2×C3×C4—64no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—122no—
240C2×C3×C4—124no—
241C2×Dic3—32no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—44no—
245C2×Dic3—62no—
246C2×Dic3—62no—
247C2×Dic3—62no—
248C2×Dic3—62no—
249C2×Dic3—62no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—122no—
256C2×Dic3—242no—
257C2×Dic3C2×S412yes—
258C2×C32—41no—
259C2×C32—42no—
260C2×C32—82no—
261C24—22no—
262C24—31no—
263C24—31no—
264C24—32no—
265C24—62no—
266C24D3×Dic312yes—
267C24C2×D3211yescharacteristic
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—34no—
273C22×C4—34no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4—61no—
277C22×C4—62no—
278C22×C4—92no—
279C22×C4—92no—
280C22×C4—92no—
281C22×C4—92no—
282C22×C4—122no—
283C22×C4—182no—
284C4⋊C4—182no—
285C4⋊C4—182no—
286C4⋊C4—362no—
287C22⋊C4—32no—
288C22⋊C4—32no—
289C22⋊C4—34no—
290C22⋊C4—34no—
291C22⋊C4—31no—
292C22⋊C4—32no—
293C22⋊C4—32no—
294C22⋊C4—92no—
295C22⋊C4—91no—
296C22⋊C4—181no—
297C22⋊C4—181no—
298C22⋊C4—182no—
299C22⋊C4—182no—
300C22⋊C4—181no—
301C22⋊C4—182no—
302C22⋊C4—362no—
303C42—182no—
304C42—182no—
305C42—362no—
306C22×C3—22no—
307C22×C3—31no—
308C22×C3—32no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—32no—
312C22×C3—32no—
313C22×C3—32no—
314C22×C3—41no—
315C22×C3—42no—
316C22×C3—62no—
317C22×C3—62no—
318C22×C3—64no—
319C22×C3—61no—
320C22×C3—62no—
321C22×C3—62no—
322C22×C3—62no—
323C22×C3—82no—
324C22×C3—81no—
325C22×C3—82no—
326C22×C3—122no—
327C22×C3—162no—
328C22×C3#87711yescharacteristic, complemented
329C22×C3C4×S412yes—
330C22×C3C22×S411yescharacteristic
331A4—21no—
332A4#92611yescharacteristic, complemented
333C3×C4—62no—
334C3×C4—62no—
335C3×C4—62no—
336C3×C4—62no—
337C3×C4—242no—
338Dic3—22no—
339Dic3—42no—
340Dic3—42no—
341Dic3—62no—
342Dic3—122no—
343Dic3—122no—
344Dic3—242no—
345Dic3—482no—
346C32—41noSylow
347C23—22no—
348C23—31no—
349C23—31no—
350C23—32no—
351C23—32no—
352C23—31no—
353C23—32no—
354C23—32no—
355C23—62no—
356C23—62no—
357C23—61no—
358C23—62no—
359C23—62no—
360C23—64no—
361C23—62no—
362C23—122no—
363C23C2×D3×Dic311yescharacteristic
364C23#380412yes—
365C23D3×S411yescharacteristic
366C2×C4—32no—
367C2×C4—32no—
368C2×C4—32no—
369C2×C4—32no—
370C2×C4—32no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—64no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—64no—
379C2×C4—92no—
380C2×C4—124no—
381C2×C4—182no—
382C2×C4—182no—
383C2×C4—182no—
384C2×C4—182no—
385C2×C4—182no—
386C2×C4—182no—
387C2×C4—182no—
388C2×C4—182no—
389C2×C4—182no—
390C2×C4—182no—
391C2×C3—22no—
392C2×C3—32no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—41no—
396C2×C3—42no—
397C2×C3—62no—
398C2×C3—82no—
399C2×C3—81no—
400C2×C3—82no—
401C2×C3—162no—
402C2×C3C2×C4×S411yescharacteristic
403C2×C3#591412yes—
404C22—22no—
405C22—31no—
406C22—32no—
407C22—31no—
408C22—31no—
409C22—32no—
410C22—32no—
411C22—32no—
412C22—62no—
413C22—62no—
414C22—64no—
415C22—61no—
416C22—62no—
417C22—62no—
418C22—62no—
419C22—122no—
420C22#6395511yescharacteristic, complemented
421C22Dic3×S412yes—
422C22C2×D3×S411yescharacteristic
423C4—62no—
424C4—62no—
425C4—62no—
426C4—62no—
427C4—62no—
428C4—182no—
429C4—362no—
430C4—362no—
431C3—41no—
432C3—81no—
433C3#8529211yescharacteristic, complemented
434C2—22no—
435C2—31no—
436C2—32no—
437C2—31no—
438C2—62no—
439C2C2×Dic3×S411yescharacteristic
440C2#1059646712yes—
subgroupG/H
center, FrattiniC22C2×D3×S4
commutatorC2×C3×A4C22×C4
FittingC25×C3C2×D3
socleC24×C3C22×D3
prime radicalC23×C3×A4C22
cosocleC22×C3×A4C23
hypercenterC23D3×S4
hypocenterC3×A4#117
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. SL(2,ℤ/4)⋊Dic3C2semidirect
5. Dic3×SL(2,ℤ/4)C2semidirect
11. C2×C3×SL(2,ℤ/4)C4semidirect
54. C2×SL(2,ℤ/4)Dic3semidirect
167. C3×A4#117semidirect
328. C22×C3#877semidirect
332. A4#926semidirect
420. C22#63955semidirect
433. C3#85292semidirect

Maximal subgroups

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

subgroupindex
C22×C3×SL(2,ℤ/4)2
A4×#2172
#105936662
SL(2,ℤ/4)⋊Dic32
Dic3×SL(2,ℤ/4)2
#852923
#820973
#639554