Finite Group Databasebeta

← #11921909

Subgroups — #11921909

Order 1152.

712 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 75 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×#64259C211yesmaximal, characteristic, complemented
2C2×C3×C4×S4C211yesmaximal, characteristic
3A4×C4⋊Dic3C211yesmaximal, characteristic, complemented
4(C2×S4)⋊Dic3C212yesmaximal
5Dic3⋊SL(2,ℤ/4)C212yesmaximal, complemented
6#85312—31nomaximal
7#82353—31nomaximal
8C22×C3×S4C2211yescharacteristic
9C2×C3×C4×A4C2211yescharacteristic
10C2×Dic3×A4C2212yes—
11C2×C3⋊SL(2,ℤ/4)C2212yes—
12#64259—22no—
13C2×C3×SL(2,ℤ/4)C2211yescharacteristic
14C3×C4×S4—22no—
15#63857—41nomaximal
16C2×C4×S4D311yescharacteristic
17C2×A4⋊Q8—31no—
18C2×A4⋊Q8—61no—
19C2×C3×C4×D4—31no—
20C22×C4⋊Dic3D311yescharacteristic, complemented
21C2×#924—31no—
22C2×#924—31no—
23A4×C4⋊C4—31no—
24(C2×S4)⋊C4—32no—
25C4⋊SL(2,ℤ/4)—32no—
26#5220—32no—
27#5212—31no—
28#4982—32no—
29#4948—32no—
30#4942—32no—
31#4929—31no—
32#4928—31no—
33C22×C3×A4C2311yescharacteristic
34C2×C3×S4—21no—
35C2×C3×S4—21no—
36C2×C3×S4Q812yescomplemented
37C2×#634—41no—
38C2×C3×C4×D3—41no—
39C3×C4×A4D412yes—
40Dic3×A4—22no—
41C3⋊SL(2,ℤ/4)—22no—
42C3×SL(2,ℤ/4)—21no—
43C3×C4⋊Dic3—41no—
44Dic3⋊Dic3—42no—
45(C2×D3)⋊Dic3—42no—
46#2417—91noSylow
47C22×S4C2×D311yescharacteristic
48C22×C3×D4—31no—
49C23×C3×C4—31no—
50C23×C3×C4C2×D311yescharacteristic
51C23×Dic3C2×D312yes—
52C22×Dic6—31no—
53C2×C4×A4—21no—
54C2×C4×A4—32no—
55C2×C4×A4C2×D311yescharacteristic
56C2×SL(2,ℤ/4)—32no—
57C2×SL(2,ℤ/4)—62no—
58C2×SL(2,ℤ/4)C2×D311yescharacteristic
59C4×S4—22no—
60A4⋊Q8—64no—
61A4⋊Q8—62no—
62C3×C4×D4—62no—
63C3×C4×D4—62no—
64C2×C3×C4⋊C4—31no—
65C2×C3×C22⋊C4—31no—
66C2×C3×C22⋊C4—31no—
67C2×C3×C42—31no—
68C2×#217—32no—
69C2×#217—32no—
70C2×#217—32no—
71C2×C4⋊Dic3—31no—
72C2×C4⋊Dic3—31no—
73C2×C4⋊Dic3—32no—
74C2×C4⋊Dic3—31no—
75#924—61no—
76#924—62no—
77#924—62no—
78#924—62no—
79#924—61no—
80C2×Dic3⋊C4—32no—
81C2×Dic3⋊C4—32no—
82C2×Dic3⋊C4—32no—
83#897—121no—
84C3⋊#95—32no—
85C3⋊#95—32no—
86C3⋊#95—32no—
87C3⋊#95—32no—
88C22×C3×D3—41no—
89C2×C3×A4C2×D411yescharacteristic
90C2×C3×A4C2×Q811yescharacteristic
91C2×C3×A4#4111yescharacteristic
92C3×S4—22no—
93C2×C32×C4—41no—
94C2×C3⋊Dic3—42no—
95#634—82no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—42no—
98C3×C4×D3—82no—
99C2×C22⋊Q8—91no—
100C2×C22⋊Q8—91no—
101C2×C4×D4—31no—
102C22×C4⋊C4—31no—
103#398—92no—
104#396—92no—
105#391—91no—
106#389—91no—
107#386—92no—
108#380—92no—
109#380—91no—
110C24×C3—31no—
111C24×C3C22×D311yescharacteristic
112C22×A4—21no—
113C22×A4C22×D311yescharacteristic
114C2×S4—21no—
115C2×S4—21no—
116C2×S4Dic612yescomplemented
117C2×C3×D4—32no—
118C2×C3×D4—32no—
119C2×C3×D4—61no—
120C2×C3×D4—61no—
121C2×C3×D4—61no—
122C2×C3×D4—61no—
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—32no—
126C22×C3×C4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—61no—
130C22×C3×C4—62no—
131C22×C3×C4—62no—
132C22×C3×C4—61no—
133C22×C3×C4D1212yes—
134C22×Dic3—22no—
135C22×Dic3—32no—
136C22×Dic3—32no—
137C22×Dic3—32no—
138C22×Dic3—32no—
139C22×Dic3—62no—
140C2×C4×D3—41no—
141C2×Dic6—61no—
142C2×Dic6—62no—
143C2×Dic6—62no—
144C2×Dic6—61no—
145C2×Dic6—121no—
146C2×Dic6—241no—
147C4×A4—22no—
148C4×A4—62no—
149C4×A4#6712yes—
150SL(2,ℤ/4)—21no—
151SL(2,ℤ/4)—64no—
152SL(2,ℤ/4)—62no—
153C3×C4⋊C4—62no—
154C3×C4⋊C4—121no—
155C3×C22⋊C4—62no—
156C3×C22⋊C4—61no—
157C3×C22⋊C4—61no—
158C3×C42—62no—
159#217—62no—
160#217—62no—
161#217—62no—
162#217—62no—
163#217—122no—
164(C2×D3)⋊C4—122no—
165C4⋊Dic3—31no—
166C4⋊Dic3—32no—
167C4⋊Dic3—61no—
168C4⋊Dic3—62no—
169C4⋊Dic3—122no—
170C4⋊Dic3S411yescharacteristic, complemented
171Dic3⋊C4—62no—
172Dic3⋊C4—64no—
173Dic3⋊C4—62no—
174Dic3⋊C4—122no—
175C22×C32—41no—
176C2×C3×D3—42no—
177C2×C3×D3—81no—
178C2×C3×D3—81no—
179C3×A4C22⋊Q811yescharacteristic, complemented
180C32×C4—42no—
181C3⋊Dic3—82no—
182C3×Dic3—82no—
183C3×Dic3—81no—
184C22×Q8—91no—
185C22×D4—31no—
186C23×C4—31no—
187C23×C4—32no—
188C23×C4D3211yescharacteristic
189C22⋊Q8—181no—
190C22⋊Q8—182no—
191C22⋊Q8—181no—
192C22⋊Q8—182no—
193C22⋊Q8—182no—
194C4×D4—62no—
195C4×D4—62no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—91no—
198C2×C4⋊C4—91no—
199C2×C4⋊C4—92no—
200C2×C4⋊C4—91no—
201C2×C4⋊C4—92no—
202C2×C4⋊C4—92no—
203C2×C4⋊C4—92no—
204C2×C22⋊C4—31no—
205C2×C22⋊C4—31no—
206C2×C22⋊C4—92no—
207C2×C22⋊C4—92no—
208C2×C22⋊C4—92no—
209C2×C42—31no—
210#95—92no—
211#95—92no—
212#95—92no—
213#95—92no—
214C23×C3—31no—
215C23×C3—31no—
216C23×C3—31no—
217C23×C3—31no—
218C23×C3—32no—
219C23×C3—61no—
220C23×C3—61no—
221C23×C3—61no—
222C23×C3—61no—
223C23×C3—61no—
224C23×C3—61no—
225C23×C3C2×D1211yescharacteristic
226C23×C3#23711yescharacteristic
227C23×C3D3×Q811yescharacteristic
228C22×D3—41no—
229C2×A4—21no—
230C2×A4—21no—
231C2×A4—21no—
232C2×A4C2×Dic611yescharacteristic
233C2×A4#23511yescharacteristic
234C2×A4C2×#6711yescharacteristic
235S4—22no—
236C3×D4—62no—
237C3×D4—62no—
238C3×D4—62no—
239C3×D4—62no—
240C2×C3×C4—31no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—32no—
244C2×C3×C4—32no—
245C2×C3×C4—41no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—81no—
256C2×C3×C4—122no—
257C2×C3×C4—122no—
258C2×C3×C4—122no—
259C2×C3×C4C2×S411yescharacteristic
260C2×Dic3—32no—
261C2×Dic3—41no—
262C2×Dic3—62no—
263C2×Dic3—62no—
264C2×Dic3—62no—
265C2×Dic3—62no—
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—62no—
269C2×Dic3—62no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—122no—
273C2×Dic3—242no—
274C2×Dic3C2×S412yes—
275C4×D3—82no—
276Dic6—122no—
277Dic6—122no—
278Dic6—242no—
279Dic6—244no—
280C2×C32—41no—
281C2×C32—41no—
282C2×C32—41no—
283C3×D3—82no—
284C24—31no—
285C24C2×D3211yescharacteristic
286C2×Q8—181no—
287C2×Q8—182no—
288C2×Q8—182no—
289C2×Q8—181no—
290C2×D4—32no—
291C2×D4—32no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—61no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—32no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—62no—
303C22×C4—61no—
304C22×C4—62no—
305C22×C4—61no—
306C22×C4—62no—
307C22×C4—92no—
308C22×C4—92no—
309C22×C4—92no—
310C22×C4—92no—
311C22×C4—182no—
312C22×C4#62612yes—
313C4⋊C4—31no—
314C4⋊C4—62no—
315C4⋊C4—92no—
316C4⋊C4—91no—
317C4⋊C4—182no—
318C4⋊C4—181no—
319C4⋊C4—184no—
320C4⋊C4—182no—
321C4⋊C4—182no—
322C4⋊C4—362no—
323C22⋊C4—62no—
324C22⋊C4—61no—
325C22⋊C4—61no—
326C22⋊C4—182no—
327C22⋊C4—182no—
328C22⋊C4—182no—
329C22⋊C4—182no—
330C22⋊C4—362no—
331C42—62no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—31no—
341C22×C3—41no—
342C22×C3—61no—
343C22×C3—62no—
344C22×C3—62no—
345C22×C3—62no—
346C22×C3—81no—
347C22×C3—121no—
348C22×C3—121no—
349C22×C3—121no—
350C22×C3—121no—
351C22×C3#89711yescharacteristic, complemented
352C22×C3C22×S411yescharacteristic
353C2×D3—42no—
354C2×D3—81no—
355C2×D3—81no—
356A4—21no—
357A4#92411yescharacteristic, complemented
358C3×C4—32no—
359C3×C4—42no—
360C3×C4—62no—
361C3×C4—82no—
362C3×C4—121no—
363C3×C4—121no—
364C3×C4—242no—
365C3×C4C4⋊S412yes—
366Dic3—22no—
367Dic3—62no—
368Dic3—81no—
369Dic3—122no—
370Dic3—122no—
371Dic3—242no—
372Dic3—244no—
373C32—41noSylow
374C23—31no—
375C23—31no—
376C23—31no—
377C23—31no—
378C23—32no—
379C23—61no—
380C23—61no—
381C23—61no—
382C23—61no—
383C23—61no—
384C23—61no—
385C23D3×Dic611yescharacteristic
386C23#379511yescharacteristic
387C23C2×#62611yescharacteristic
388Q8—362no—
389Q8—362no—
390D4—62no—
391D4—62no—
392D4—62no—
393D4—62no—
394C2×C4—31no—
395C2×C4—32no—
396C2×C4—32no—
397C2×C4—32no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—92no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—182no—
413C2×C4—182no—
414C2×C4—182no—
415C2×C4—182no—
416C2×C4—182no—
417C2×C4—182no—
418C2×C4—182no—
419C2×C4—182no—
420C2×C4—182no—
421C2×C4—182no—
422C2×C4D3×S411yescharacteristic
423C2×C3—31no—
424C2×C3—31no—
425C2×C3—31no—
426C2×C3—31no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—81no—
431C2×C3—81no—
432C2×C3—81no—
433C2×C3—122no—
434C2×C3C2×C4⋊S411yescharacteristic
435C2×C3#591411yescharacteristic
436C2×C3Q8×S411yescharacteristic
437D3—82no—
438C22—31no—
439C22—31no—
440C22—31no—
441C22—31no—
442C22—31no—
443C22—31no—
444C22—31no—
445C22—31no—
446C22—31no—
447C22—61no—
448C22—62no—
449C22—62no—
450C22—62no—
451C22—121no—
452C22—121no—
453C22—121no—
454C22—121no—
455C22#6385711yescharacteristic, complemented
456C22C2×D3×S411yescharacteristic
457C4—32no—
458C4—62no—
459C4—62no—
460C4—121no—
461C4—121no—
462C4—182no—
463C4—362no—
464C4—362no—
465C4#6420712yes—
466C3—41no—
467C3—81no—
468C3#8531211yescharacteristic, complemented
469C2—31no—
470C2—31no—
471C2—31no—
472C2—31no—
473C2—122no—
474C2Dic6×S411yescharacteristic
475C2#1059642411yescharacteristic
476C2C2×#6420711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC22×C3×A4C23
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
hypercenterC2×C4D3×S4
hypocenterC3×A4C22⋊Q8
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#64259C2semidirect
3. A4×C4⋊Dic3C2semidirect
5. Dic3⋊SL(2,ℤ/4)C2semidirect
20. C22×C4⋊Dic3D3semidirect
36. C2×C3×S4Q8semidirect
116. C2×S4Dic6semidirect
170. C4⋊Dic3S4semidirect
179. C3×A4C22⋊Q8semidirect
351. C22×C3#897semidirect
357. A4#924semidirect
455. C22#63857semidirect
468. C3#85312semidirect

Maximal subgroups

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

subgroupindex
C2×#642592
C2×C3×C4×S42
A4×C4⋊Dic32
(C2×S4)⋊Dic32
Dic3⋊SL(2,ℤ/4)2
#853123
#823533
#638574