Finite Group Databasebeta

← #11921913

Subgroups — #11921913

Order 1152.

728 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 73 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×C4×C3⋊S4C211yesmaximal, characteristic
2C2×C3×A4⋊Q8C211yesmaximal, characteristic, complemented
3A4×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#10593242C212yesmaximal
5Dic3⋊SL(2,ℤ/4)C212yesmaximal, complemented
6#85312—31nomaximal
7#82815—31nomaximal
8C22×C3⋊S4C2211yescharacteristic
9C2×C3×C4×A4C2211yescharacteristic
10C2×Dic3×A4C2212yes—
11C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
12C4×C3⋊S4—22no—
13C2×C3×SL(2,ℤ/4)C2212yes—
14C3×A4⋊Q8—22no—
15#63871—41nomaximal
16C2×C4×S4—31no—
17C2×C4×S4—61no—
18C2×A4⋊Q8D311yescharacteristic, complemented
19C2×C3×C22⋊Q8—31no—
20C2×#946—31no—
21C2×C4×#67—31no—
22C22×C4⋊Dic3D311yescharacteristic, complemented
23A4×C4⋊C4—31no—
24(C2×S4)⋊C4—32no—
25C4⋊SL(2,ℤ/4)—32no—
26#5229—31no—
27#4994—32no—
28#4987—31no—
29#4982—32no—
30#4972—31no—
31#4958—32no—
32#4942—32no—
33C22×C3×A4C2311yescharacteristic
34C2×C3⋊S4—21no—
35C2×C3⋊S4—21no—
36C2×C3⋊S4Q812yescomplemented
37C2×C4×C3⋊D3—41no—
38C2×C3×Dic6—41no—
39C3×C4×A4D412yes—
40Dic3×A4—22no—
41C3⋊SL(2,ℤ/4)—21no—
42C3×SL(2,ℤ/4)—22no—
43C3×C4⋊Dic3—41no—
44Dic3⋊Dic3—42no—
45#3722—42no—
46#2417—91noSylow
47C22×S4—31no—
48C22×S4—61no—
49C22×C3×Q8—31no—
50C23×C3×C4C2×D311yescharacteristic
51C22×#67—31no—
52C23×Dic3C2×D312yes—
53C22×C4×D3—31no—
54C2×C4×A4—21no—
55C2×C4×A4—32no—
56C2×C4×A4C2×D311yescharacteristic
57C2×SL(2,ℤ/4)—31no—
58C2×SL(2,ℤ/4)—61no—
59C2×SL(2,ℤ/4)C2×D312yes—
60C4×S4—64no—
61C4×S4—62no—
62A4⋊Q8—22no—
63C3×C22⋊Q8—62no—
64C3×C22⋊Q8—62no—
65C2×C3×C4⋊C4—31no—
66C2×C3×C4⋊C4—32no—
67C2×C3×C22⋊C4—32no—
68C2×#217—31no—
69#946—61no—
70#946—62no—
71#946—61no—
72C4×#67—62no—
73C4×#67—62no—
74C2×(C2×D3)⋊C4—32no—
75C2×(C2×D3)⋊C4—31no—
76C2×(C2×D3)⋊C4—32no—
77C2×C4⋊Dic3—31no—
78C2×C4⋊Dic3—31no—
79C2×C4⋊Dic3—32no—
80C2×Dic3⋊C4—32no—
81C2×Dic3⋊C4—31no—
82C2×Dic3⋊C4—32no—
83C2×C4×Dic3—31no—
84#897—121no—
85C3⋊#95—32no—
86C3⋊#95—32no—
87C3⋊#95—32no—
88C3⋊#95—32no—
89C22×C3⋊D3—41no—
90C2×C3×A4C2×D411yescharacteristic
91C2×C3×A4C2×Q811yescharacteristic
92C2×C3×A4#4111yescharacteristic
93C3⋊S4—22no—
94C2×C32×C4—41no—
95C2×C3⋊Dic3—41no—
96C4×C3⋊D3—82no—
97C2×C3×Dic3—42no—
98C2×C3×Dic3—42no—
99C3×Dic6—82no—
100C2×C22⋊Q8—31no—
101C2×C22⋊Q8—91no—
102C2×C4×D4—91no—
103C22×C4⋊C4—31no—
104#398—92no—
105#396—92no—
106#391—91no—
107#389—91no—
108#386—92no—
109#380—92no—
110#380—91no—
111C24×C3C22×D311yescharacteristic
112C23×D3—31no—
113C22×A4—21no—
114C22×A4C22×D311yescharacteristic
115C2×S4—32no—
116C2×S4—62no—
117C2×S4—62no—
118C2×S4—62no—
119C2×S4—61no—
120C2×S4—61no—
121C2×C3×Q8—61no—
122C2×C3×Q8—62no—
123C2×C3×Q8—62no—
124C2×C3×Q8—61no—
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—32no—
128C22×C3×C4—32no—
129C22×C3×C4—32no—
130C22×C3×C4D1212yes—
131C2×#67—32no—
132C2×#67—32no—
133C2×#67—61no—
134C2×#67—61no—
135C2×#67—61no—
136C2×#67—61no—
137C22×Dic3—22no—
138C22×Dic3—31no—
139C22×Dic3—32no—
140C22×Dic3—32no—
141C22×Dic3—31no—
142C22×Dic3—31no—
143C22×Dic3—62no—
144C2×C4×D3—62no—
145C2×C4×D3—61no—
146C2×C4×D3—62no—
147C2×C4×D3—61no—
148C2×C4×D3—121no—
149C2×C4×D3—241no—
150C2×Dic6—41no—
151C4×A4—22no—
152C4×A4—62no—
153C4×A4#6712yes—
154SL(2,ℤ/4)—22no—
155SL(2,ℤ/4)—62no—
156SL(2,ℤ/4)—61no—
157C3×C4⋊C4—62no—
158C3×C4⋊C4—64no—
159C3×C4⋊C4—121no—
160C3×C22⋊C4—62no—
161C3×C22⋊C4—62no—
162#217—61no—
163#217—61no—
164(C2×D3)⋊C4—62no—
165(C2×D3)⋊C4—62no—
166(C2×D3)⋊C4—62no—
167(C2×D3)⋊C4—122no—
168(C2×D3)⋊C4—122no—
169C4⋊Dic3—31no—
170C4⋊Dic3—32no—
171C4⋊Dic3—61no—
172C4⋊Dic3—122no—
173C4⋊Dic3S411yescharacteristic, complemented
174Dic3⋊C4—62no—
175Dic3⋊C4—62no—
176Dic3⋊C4—62no—
177Dic3⋊C4—122no—
178C4×Dic3—62no—
179C22×C32—41no—
180C2×C3⋊D3—42no—
181C2×C3⋊D3—81no—
182C2×C3⋊D3—81no—
183C3×A4C22⋊Q811yescharacteristic, complemented
184C32×C4—42no—
185C3⋊Dic3—81no—
186C3×Dic3—82no—
187C3×Dic3—82no—
188C22×Q8—31no—
189C22×D4—91no—
190C23×C4—32no—
191C23×C4—91no—
192C23×C4D3211yescharacteristic
193C22⋊Q8—62no—
194C22⋊Q8—62no—
195C22⋊Q8—181no—
196C22⋊Q8—182no—
197C22⋊Q8—181no—
198C4×D4—182no—
199C4×D4—182no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—32no—
202C2×C4⋊C4—91no—
203C2×C4⋊C4—91no—
204C2×C4⋊C4—92no—
205C2×C4⋊C4—91no—
206C2×C4⋊C4—92no—
207C2×C4⋊C4—92no—
208C2×C22⋊C4—32no—
209C2×C22⋊C4—91no—
210C2×C22⋊C4—92no—
211C2×C22⋊C4—91no—
212C2×C22⋊C4—92no—
213C2×C42—91no—
214#95—92no—
215#95—92no—
216#95—92no—
217#95—92no—
218C23×C3—31no—
219C23×C3—31no—
220C23×C3—31no—
221C23×C3—31no—
222C23×C3C2×D1211yescharacteristic
223C23×C3#23711yescharacteristic
224C23×C3D3×Q811yescharacteristic
225C22×D3—32no—
226C22×D3—61no—
227C22×D3—61no—
228C22×D3—61no—
229C22×D3—61no—
230C22×D3—61no—
231C22×D3—61no—
232C22×D3—121no—
233C22×D3—241no—
234C2×A4—21no—
235C2×A4—21no—
236C2×A4—21no—
237C2×A4D3×Q811yescharacteristic
238C2×A4#23911yescharacteristic
239C2×A4C2×#6711yescharacteristic
240S4—64no—
241S4—62no—
242C3×Q8—122no—
243C3×Q8—122no—
244C2×C3×C4—31no—
245C2×C3×C4—32no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—41no—
249C2×C3×C4—61no—
250C2×C3×C4—62no—
251C2×C3×C4—62no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—81no—
257C2×C3×C4—122no—
258C2×C3×C4C2×S411yescharacteristic
259#67—62no—
260#67—62no—
261#67—62no—
262#67—62no—
263C2×Dic3—32no—
264C2×Dic3—32no—
265C2×Dic3—42no—
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—121no—
279C2×Dic3—241no—
280C2×Dic3C2×S412yes—
281C4×D3—122no—
282C4×D3—122no—
283C4×D3—242no—
284C4×D3—244no—
285Dic6—82no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32—41no—
289C3⋊D3—82no—
290C24—91no—
291C24C2×D3211yescharacteristic
292C2×Q8—62no—
293C2×Q8—61no—
294C2×Q8—62no—
295C2×Q8—61no—
296C2×D4—92no—
297C2×D4—92no—
298C2×D4—181no—
299C2×D4—181no—
300C2×D4—181no—
301C2×D4—181no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—32no—
305C22×C4—32no—
306C22×C4—32no—
307C22×C4—62no—
308C22×C4—92no—
309C22×C4—91no—
310C22×C4—92no—
311C22×C4—91no—
312C22×C4—91no—
313C22×C4—182no—
314C22×C4—181no—
315C22×C4—182no—
316C22×C4—182no—
317C22×C4—181no—
318C22×C4#62612yes—
319C4⋊C4—31no—
320C4⋊C4—62no—
321C4⋊C4—64no—
322C4⋊C4—92no—
323C4⋊C4—91no—
324C4⋊C4—182no—
325C4⋊C4—181no—
326C4⋊C4—182no—
327C4⋊C4—182no—
328C4⋊C4—362no—
329C22⋊C4—62no—
330C22⋊C4—62no—
331C22⋊C4—182no—
332C22⋊C4—182no—
333C22⋊C4—182no—
334C22⋊C4—181no—
335C22⋊C4—181no—
336C22⋊C4—362no—
337C42—182no—
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—31no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—31no—
344C22×C3—31no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—41no—
348C22×C3—61no—
349C22×C3—81no—
350C22×C3#89711yescharacteristic, complemented
351C22×C3C22×S411yescharacteristic
352C2×D3—62no—
353C2×D3—62no—
354C2×D3—62no—
355C2×D3—122no—
356C2×D3—121no—
357C2×D3—121no—
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—242no—
361C2×D3—241no—
362C2×D3—241no—
363C2×D3—242no—
364C2×D3—242no—
365A4—21no—
366A4#94611yescharacteristic, complemented
367C3×C4—32no—
368C3×C4—42no—
369C3×C4—82no—
370C3×C4—122no—
371C3×C4—122no—
372C3×C4—242no—
373C3×C4C4⋊S412yes—
374Dic3—22no—
375Dic3—62no—
376Dic3—62no—
377Dic3—82no—
378Dic3—121no—
379Dic3—121no—
380Dic3—241no—
381Dic3—242no—
382C32—41noSylow
383C23—31no—
384C23—31no—
385C23—31no—
386C23—31no—
387C23—92no—
388C23—181no—
389C23—181no—
390C23—181no—
391C23—181no—
392C23—181no—
393C23—181no—
394C23#379611yescharacteristic
395C23#379711yescharacteristic
396C23C2×#62611yescharacteristic
397Q8—122no—
398Q8—122no—
399D4—182no—
400D4—182no—
401D4—182no—
402D4—182no—
403C2×C4—31no—
404C2×C4—32no—
405C2×C4—32no—
406C2×C4—32no—
407C2×C4—32no—
408C2×C4—61no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—92no—
416C2×C4—92no—
417C2×C4—182no—
418C2×C4—182no—
419C2×C4—181no—
420C2×C4—181no—
421C2×C4—182no—
422C2×C4—182no—
423C2×C4—181no—
424C2×C4—181no—
425C2×C4—181no—
426C2×C4—181no—
427C2×C4—181no—
428C2×C4—181no—
429C2×C4—362no—
430C2×C4—362no—
431C2×C4D3×S411yescharacteristic
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—41no—
437C2×C3—41no—
438C2×C3—41no—
439C2×C3—81no—
440C2×C3—81no—
441C2×C3—81no—
442C2×C3C2×C4⋊S411yescharacteristic
443C2×C3#591411yescharacteristic
444C2×C3Q8×S411yescharacteristic
445D3—122no—
446D3—242no—
447D3—244no—
448C22—31no—
449C22—31no—
450C22—31no—
451C22—31no—
452C22—31no—
453C22—31no—
454C22—31no—
455C22—31no—
456C22—31no—
457C22—61no—
458C22—182no—
459C22—182no—
460C22—182no—
461C22—361no—
462C22—361no—
463C22—361no—
464C22—361no—
465C22#6387111yescharacteristic, complemented
466C22C2×D3×S411yescharacteristic
467C4—32no—
468C4—62no—
469C4—122no—
470C4—122no—
471C4—182no—
472C4—182no—
473C4—361no—
474C4—361no—
475C4#6420712yes—
476C3—41no—
477C3—81no—
478C3#8531211yescharacteristic, complemented
479C2—31no—
480C2—31no—
481C2—31no—
482C2—31no—
483C2—362no—
484C2(C3⋊S4)⋊Q811yescharacteristic
485C2#1059642511yescharacteristic
486C2C2×#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
2. C2×C3×A4⋊Q8C2semidirect
3. A4×C4⋊Dic3C2semidirect
5. Dic3⋊SL(2,ℤ/4)C2semidirect
18. C2×A4⋊Q8D3semidirect
22. C22×C4⋊Dic3D3semidirect
36. C2×C3⋊S4Q8semidirect
173. C4⋊Dic3S4semidirect
183. C3×A4C22⋊Q8semidirect
350. C22×C3#897semidirect
366. A4#946semidirect
465. C22#63871semidirect
478. C3#85312semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊S42
C2×C3×A4⋊Q82
A4×C4⋊Dic32
#105932422
Dic3⋊SL(2,ℤ/4)2
#853123
#828153
#638714