Finite Group Databasebeta

← #11922841

Subgroups — #11922841

Order 1152.

894 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 107 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
2C3×A4×C22⋊C4C211yesmaximal, characteristic
3#10593666C211yesmaximal, characteristic
4#10593645C212yesmaximal, complemented
5C4×C3⋊SL(2,ℤ/4)C212yesmaximal, complemented
6#85292—33nomaximal
7#82098—31nomaximal
8C23×C3×A4C2211yescharacteristic
9C2×C3×C4×A4C2212yes—
10C2×C3⋊SL(2,ℤ/4)C414yescomplemented
11C2×C3⋊SL(2,ℤ/4)C2212yes—
12C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
13C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
14#64086—41nomaximal
15C22×SL(2,ℤ/4)—33no—
16C22×C3×C22⋊C4D311yescharacteristic
17C22×#217—31no—
18C2×#877—31no—
19A4×C22⋊C4D313yes—
20#5432—33no—
21SL(2,ℤ/4)⋊C4—36no—
22C4×SL(2,ℤ/4)—36no—
23#5247—31no—
24#5210—32no—
25C4×#217—32no—
26#4945—32no—
27#4944—32no—
28#4943—31no—
29#4942—31no—
30Dic3×C22⋊C4—31no—
31C22×C3×A4—22no—
32C22×C3×A4C2×C412yes—
33C22×C3×A4C2311yescharacteristic
34C22×C3⋊Dic3—41no—
35C3×C4×A4—22no—
36C3⋊SL(2,ℤ/4)—22no—
37C3⋊SL(2,ℤ/4)C2×C412yes—
38C3⋊SL(2,ℤ/4)C2×C412yes—
39C32×C22⋊C4—41no—
40#3757—41no—
41#3750—42no—
42C4×C3⋊Dic3—42no—
43#2342—91noSylow
44C25×C3C2×D311yescharacteristic
45C23×A4C2×D313yes—
46C23×C3×C4C2×D312yes—
47C23×Dic3—31no—
48C23×Dic3—31no—
49C2×C4×A4C2×D316yes—
50C2×SL(2,ℤ/4)—312no—
51C2×SL(2,ℤ/4)—33no—
52C2×SL(2,ℤ/4)—33no—
53C2×SL(2,ℤ/4)—36no—
54C2×C3×C22⋊C4—31no—
55C2×C3×C22⋊C4—31no—
56C2×C3×C22⋊C4—32no—
57C2×#217—31no—
58C2×#217—31no—
59C2×#217—34no—
60C2×#217—34no—
61C2×#217—31no—
62C2×#217—31no—
63C2×#217—31no—
64C2×#217—31no—
65C2×#217—32no—
66C2×#217—32no—
67C2×Dic3⋊C4—32no—
68C2×Dic3⋊C4—32no—
69C2×C4×Dic3—32no—
70C2×C4×Dic3—32no—
71#877—61no—
72#877—62no—
73#877—61no—
74#877—123no—
75C3⋊#95—32no—
76C3⋊#95—32no—
77C3⋊#95—32no—
78C3⋊#95—32no—
79C23×C32—41no—
80C2×C3×A4—22no—
81C2×C3×A4C22×C411yescharacteristic
82C2×C3×A4#4112yes—
83C2×C32×C4—42no—
84C2×C3⋊Dic3—44no—
85C2×C3⋊Dic3—41no—
86C2×C3⋊Dic3—41no—
87C2×C3⋊Dic3—42no—
88C2×#117—91no—
89C22×C22⋊C4—91no—
90C22×C22⋊C4C3⋊D311yescharacteristic
91#388—92no—
92#388—92no—
93#385—92no—
94#381—91no—
95#380—91no—
96#379—91no—
97C4×C22⋊C4—91no—
98C4×C22⋊C4—92no—
99C24×C3—22no—
100C24×C3—31no—
101C24×C3—31no—
102C24×C3—32no—
103C24×C3—62no—
104C24×C3C4×D312yes—
105C24×C3C22×D311yescharacteristic
106C22×A4—26no—
107C22×A4C4×D316yes—
108C22×A4C22×D313yes—
109C22×C3×C4—22no—
110C22×C3×C4—32no—
111C22×C3×C4—32no—
112C22×C3×C4—62no—
113C22×Dic3—31no—
114C22×Dic3—31no—
115C22×Dic3—31no—
116C22×Dic3—31no—
117C22×Dic3—34no—
118C22×Dic3—34no—
119C22×Dic3—32no—
120C22×Dic3—32no—
121C22×Dic3—61no—
122C22×Dic3—62no—
123C22×Dic3—61no—
124C22×Dic3—123no—
125C22×Dic3—122no—
126C4×A4—26no—
127SL(2,ℤ/4)—36no—
128SL(2,ℤ/4)—36no—
129SL(2,ℤ/4)—66no—
130C3×C22⋊C4—31no—
131C3×C22⋊C4—32no—
132C3×C22⋊C4—43no—
133C3×C22⋊C4—61no—
134C3×C22⋊C4S411yescharacteristic
135#217—32no—
136#217—32no—
137#217—34no—
138#217—34no—
139#217—32no—
140#217—32no—
141#217—61no—
142#217—62no—
143#217—62no—
144#217—62no—
145#217—61no—
146#217—123no—
147#217—122no—
148Dic3⋊C4—62no—
149Dic3⋊C4—62no—
150Dic3⋊C4—126no—
151Dic3⋊C4—122no—
152C4×Dic3—62no—
153C4×Dic3—62no—
154C4×Dic3—126no—
155C4×Dic3—122no—
156C22×C32—42no—
157C22×C32—41no—
158C22×C32—82no—
159C3×A4#11711yescharacteristic, complemented
160C32×C4—82no—
161C3⋊Dic3—42no—
162C3⋊Dic3—42no—
163C3⋊Dic3—82no—
164C25C2×C3⋊D311yescharacteristic
165C23×C4—91no—
166C23×C4—91no—
167C23×C4C2×C3⋊D312yes—
168#117—181no—
169#117—182no—
170#117—181no—
171C2×C4⋊C4—92no—
172C2×C4⋊C4—92no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—31no—
175C2×C22⋊C4—32no—
176C2×C22⋊C4—91no—
177C2×C22⋊C4—91no—
178C2×C22⋊C4—94no—
179C2×C22⋊C4—94no—
180C2×C22⋊C4—91no—
181C2×C22⋊C4—91no—
182C2×C22⋊C4—91no—
183C2×C22⋊C4—91no—
184C2×C22⋊C4—92no—
185C2×C22⋊C4—92no—
186C2×C42—92no—
187C2×C42—92no—
188#95—92no—
189#95—92no—
190#95—92no—
191#95—92no—
192C23×C3—22no—
193C23×C3—31no—
194C23×C3—31no—
195C23×C3—32no—
196C23×C3—32no—
197C23×C3—32no—
198C23×C3—32no—
199C23×C3—31no—
200C23×C3—43no—
201C23×C3—62no—
202C23×C3—62no—
203C23×C3—61no—
204C23×C3—62no—
205C23×C3—62no—
206C23×C3—64no—
207C23×C3—62no—
208C23×C3—122no—
209C23×C3C2×C4×D311yescharacteristic
210C23×C3#23712yes—
211C23×C3C2×S411yescharacteristic
212C2×A4—26no—
213C2×A4C2×C4×D313yes—
214C2×A4#23716yes—
215C2×C3×C4—32no—
216C2×C3×C4—46no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—62no—
221C2×C3×C4C2×S412yes—
222C2×Dic3—32no—
223C2×Dic3—32no—
224C2×Dic3—32no—
225C2×Dic3—32no—
226C2×Dic3—62no—
227C2×Dic3—62no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—64no—
231C2×Dic3—64no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—123no—
241C2×Dic3—123no—
242C2×Dic3—1212no—
243C2×Dic3—126no—
244C2×Dic3—124no—
245C2×C32—41no—
246C2×C32—42no—
247C2×C32—82no—
248C24—22no—
249C24—31no—
250C24—31no—
251C24—32no—
252C24—62no—
253C24C4×C3⋊D312yes—
254C24C22×C3⋊D311yescharacteristic
255C22×C4—22no—
256C22×C4—32no—
257C22×C4—32no—
258C22×C4—62no—
259C22×C4—91no—
260C22×C4—91no—
261C22×C4—91no—
262C22×C4—91no—
263C22×C4—94no—
264C22×C4—94no—
265C22×C4—92no—
266C22×C4—92no—
267C22×C4—181no—
268C22×C4—182no—
269C22×C4—181no—
270C22×C4—362no—
271C4⋊C4—182no—
272C4⋊C4—182no—
273C4⋊C4—362no—
274C22⋊C4—31no—
275C22⋊C4—32no—
276C22⋊C4—61no—
277C22⋊C4—92no—
278C22⋊C4—92no—
279C22⋊C4—94no—
280C22⋊C4—94no—
281C22⋊C4—92no—
282C22⋊C4—92no—
283C22⋊C4—181no—
284C22⋊C4—182no—
285C22⋊C4—182no—
286C22⋊C4—181no—
287C22⋊C4—182no—
288C22⋊C4—362no—
289C22⋊C4C3⋊S411yescharacteristic
290C42—182no—
291C42—182no—
292C42—362no—
293C22×C3—22no—
294C22×C3—31no—
295C22×C3—32no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—32no—
299C22×C3—32no—
300C22×C3—32no—
301C22×C3—46no—
302C22×C3—43no—
303C22×C3—62no—
304C22×C3—62no—
305C22×C3—64no—
306C22×C3—61no—
307C22×C3—62no—
308C22×C3—62no—
309C22×C3—62no—
310C22×C3—86no—
311C22×C3—122no—
312C22×C3#87711yescharacteristic, complemented
313C22×C3C4×S412yes—
314C22×C3C22×S411yescharacteristic
315A4#87713yescomplemented
316C3×C4—22no—
317C3×C4—62no—
318C3×C4—86no—
319Dic3—62no—
320Dic3—62no—
321Dic3—62no—
322Dic3—62no—
323Dic3—126no—
324Dic3—126no—
325Dic3—122no—
326Dic3—122no—
327Dic3—246no—
328C32—41noSylow
329C23—22no—
330C23—31no—
331C23—31no—
332C23—32no—
333C23—32no—
334C23—32no—
335C23—32no—
336C23—31no—
337C23—62no—
338C23—62no—
339C23—61no—
340C23—62no—
341C23—62no—
342C23—64no—
343C23—62no—
344C23—122no—
345C23C2×C4×C3⋊D311yescharacteristic
346C23#383012yes—
347C23C2×C3⋊S411yescharacteristic
348C2×C4—32no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—62no—
352C2×C4—62no—
353C2×C4—92no—
354C2×C4—92no—
355C2×C4—92no—
356C2×C4—92no—
357C2×C4—182no—
358C2×C4—182no—
359C2×C4—181no—
360C2×C4—181no—
361C2×C4—184no—
362C2×C4—184no—
363C2×C4—181no—
364C2×C4—181no—
365C2×C4—182no—
366C2×C4—182no—
367C2×C4—182no—
368C2×C4—182no—
369C2×C4—182no—
370C2×C4—182no—
371C2×C4—364no—
372C2×C4C2×C3⋊S412yes—
373C2×C3—22no—
374C2×C3—31no—
375C2×C3—32no—
376C2×C3—31no—
377C2×C3—43no—
378C2×C3—46no—
379C2×C3—62no—
380C2×C3—86no—
381C2×C3C2×C4×S411yescharacteristic
382C2×C3#591412yes—
383C22—22no—
384C22—31no—
385C22—32no—
386C22—31no—
387C22—31no—
388C22—32no—
389C22—32no—
390C22—32no—
391C22—62no—
392C22—62no—
393C22—64no—
394C22—61no—
395C22—62no—
396C22—62no—
397C22—62no—
398C22—122no—
399C22#6408611yescharacteristic, complemented
400C22C4×C3⋊S412yes—
401C22C22×C3⋊S411yescharacteristic
402C4—22no—
403C4—62no—
404C4—182no—
405C4—182no—
406C4—182no—
407C4—182no—
408C4—362no—
409C4—362no—
410C3—43no—
411C3#8529211yescharacteristic, complemented
412C2—22no—
413C2—31no—
414C2—32no—
415C2—31no—
416C2—62no—
417C2C2×C4×C3⋊S411yescharacteristic
418C2#1059657712yes—
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutatorC2×C3×A4C22×C4
FittingC22×C3×C22⋊C4D3
socleC24×C3C22×D3
prime radicalC23×C3×A4C22
cosocleC22×C3×A4C23
hypercenterC22⋊C4C3⋊S4
hypocenterC3×A4#117
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #10593645C2semidirect
5. C4×C3⋊SL(2,ℤ/4)C2semidirect
10. C2×C3⋊SL(2,ℤ/4)C4semidirect
159. C3×A4#117semidirect
312. C22×C3#877semidirect
315. A4#877semidirect
399. C22#64086semidirect
411. 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
C3×A4×C22⋊C42
#105936662
#105936452
C4×C3⋊SL(2,ℤ/4)2
#852923
#820983
#640864