Finite Group Databasebeta

← #11923055

Subgroups — #11923055

Order 1152.

878 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 101 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×Q8×A4C211yesmaximal, characteristic, complemented
2C2×C4×C3⋊S4C211yesmaximal, characteristic
3#10593647C212yesmaximal
4C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
5#10593645C212yesmaximal, complemented
6#85506—33nomaximal
7#84068—31nomaximal
8C22×C3⋊S4C2211yescharacteristic
9C3×Q8×A4—22no—
10C2×C3×C4×A4C2212yes—
11C2×C3×C4×A4C2211yescharacteristic
12C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
13C2×C3⋊SL(2,ℤ/4)C2212yes—
14C4×C3⋊S4—22no—
15#64155—41nomaximal
16C23×C3×Q8D311yescharacteristic, complemented
17C2×Q8×A4D313yescomplemented
18C2×C4×S4—33no—
19C2×#946—31no—
20C2×#946—31no—
21C2×C4×#67—31no—
22(C2×S4)⋊C4—36no—
23C4⋊SL(2,ℤ/4)—33no—
24SL(2,ℤ/4)⋊C4—36no—
25#5231—32no—
26#5230—32no—
27#5229—31no—
28#5229—31no—
29#5213—32no—
30#5212—31no—
31#5210—32no—
32C22×C3×A4C2311yescharacteristic
33C2×C3⋊S4—21no—
34C2×C3⋊S4—21no—
35C2×C3⋊S4Q812yescomplemented
36C2×C32×Q8—41no—
37C2×C4×C3⋊D3—41no—
38C3×C4×A4—22no—
39C3×C4×A4D412yes—
40C3⋊SL(2,ℤ/4)—21no—
41C3⋊SL(2,ℤ/4)—22no—
42#3752—42no—
43C4⋊(C3⋊Dic3)—41no—
44#3750—42no—
45#2409—91noSylow
46C22×S4—33no—
47C22×C3×Q8—22no—
48C22×C3×Q8—31no—
49C22×C3×Q8—31no—
50C22×C3×Q8—32no—
51C22×C3×Q8—62no—
52C23×C3×C4C2×D312yes—
53C23×C3×C4C2×D311yescharacteristic
54C22×#67—31no—
55C22×C4×D3—31no—
56Q8×A4—26no—
57C2×C4×A4C2×D316yes—
58C2×C4×A4C2×D313yes—
59C2×SL(2,ℤ/4)—33no—
60C2×SL(2,ℤ/4)—36no—
61C4×S4—66no—
62C2×#217—31no—
63C2×#217—32no—
64#946—64no—
65#946—61no—
66#946—62no—
67#946—61no—
68#946—123no—
69C4×#67—62no—
70C4×#67—62no—
71C2×(C2×D3)⋊C4—32no—
72C2×(C2×D3)⋊C4—32no—
73C2×(C2×D3)⋊C4—31no—
74C2×C4⋊Dic3—31no—
75C2×C4⋊Dic3—31no—
76C2×Dic3⋊C4—32no—
77C2×Dic3⋊C4—32no—
78C2×Dic3⋊C4—31no—
79C2×C4×Dic3—31no—
80C3⋊#95—32no—
81C3⋊#95—32no—
82C3⋊#95—32no—
83C3⋊#95—32no—
84C3⋊#95—32no—
85C22×C3⋊D3—41no—
86C2×C3×A4C2×D411yescharacteristic
87C2×C3×A4C2×Q811yescharacteristic
88C2×C3×A4#4111yescharacteristic
89C3⋊S4—22no—
90C32×Q8—82no—
91C2×C32×C4—41no—
92C2×C32×C4—42no—
93C2×C3⋊Dic3—41no—
94C2×C3⋊Dic3—42no—
95C4×C3⋊D3—82no—
96C23×Q8C3⋊D311yescharacteristic, complemented
97C2×C22⋊Q8—91no—
98C2×C22⋊Q8—91no—
99C2×C4×D4—91no—
100#395—92no—
101#393—92no—
102#391—91no—
103#391—91no—
104#386—92no—
105#385—92no—
106#380—91no—
107C24×C3C22×D311yescharacteristic
108C23×D3—31no—
109C22×A4C22×D313yes—
110C2×S4—36no—
111C2×S4—63no—
112C2×S4—63no—
113C2×C3×Q8—34no—
114C2×C3×Q8—32no—
115C2×C3×Q8—31no—
116C2×C3×Q8—43no—
117C2×C3×Q8—61no—
118C2×C3×Q8—62no—
119C2×C3×Q8—62no—
120C2×C3×Q8—62no—
121C2×C3×Q8—64no—
122C2×C3×Q8—64no—
123C2×C3×Q8S411yescharacteristic, complemented
124C22×C3×C4—22no—
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—32no—
128C22×C3×C4—32no—
129C22×C3×C4—32no—
130C22×C3×C4—62no—
131C22×C3×C4#6712yes—
132C2×#67—32no—
133C2×#67—32no—
134C2×#67—61no—
135C2×#67—61no—
136C2×#67—61no—
137C2×#67—61no—
138C22×Dic3—31no—
139C22×Dic3—31no—
140C22×Dic3—32no—
141C22×Dic3—32no—
142C22×Dic3—31no—
143C2×C4×D3—62no—
144C2×C4×D3—61no—
145C2×C4×D3—62no—
146C2×C4×D3—61no—
147C2×C4×D3—123no—
148C4×A4—26no—
149C4×A4#6716yes—
150SL(2,ℤ/4)—66no—
151SL(2,ℤ/4)—63no—
152#217—61no—
153#217—62no—
154#217—62no—
155#217—61no—
156(C2×D3)⋊C4—62no—
157(C2×D3)⋊C4—64no—
158(C2×D3)⋊C4—62no—
159(C2×D3)⋊C4—62no—
160(C2×D3)⋊C4—126no—
161C4⋊Dic3—62no—
162C4⋊Dic3—61no—
163C4⋊Dic3—61no—
164C4⋊Dic3—123no—
165Dic3⋊C4—62no—
166Dic3⋊C4—64no—
167Dic3⋊C4—62no—
168Dic3⋊C4—62no—
169Dic3⋊C4—126no—
170C4×Dic3—62no—
171C22×C32—41no—
172C2×C3⋊D3—42no—
173C2×C3⋊D3—81no—
174C2×C3⋊D3—81no—
175C3×A4C22⋊Q811yescharacteristic, complemented
176C32×C4—42no—
177C32×C4—82no—
178C3⋊Dic3—81no—
179C3⋊Dic3—82no—
180C22×Q8—22no—
181C22×Q8—31no—
182C22×Q8—31no—
183C22×Q8—32no—
184C22×Q8—62no—
185C22×D4—91no—
186C23×C4—91no—
187C23×C4C2×C3⋊D312yes—
188C23×C4C2×C3⋊D311yescharacteristic
189C22⋊Q8—184no—
190C22⋊Q8—181no—
191C22⋊Q8—182no—
192C22⋊Q8—181no—
193C4×D4—182no—
194C4×D4—182no—
195C2×C4⋊C4—91no—
196C2×C4⋊C4—91no—
197C2×C4⋊C4—91no—
198C2×C4⋊C4—92no—
199C2×C4⋊C4—92no—
200C2×C22⋊C4—91no—
201C2×C22⋊C4—92no—
202C2×C22⋊C4—92no—
203C2×C22⋊C4—91no—
204C2×C22⋊C4—92no—
205C2×C42—91no—
206#95—92no—
207#95—92no—
208#95—92no—
209#95—92no—
210#95—92no—
211C23×C3—31no—
212C23×C3—31no—
213C23×C3—31no—
214C23×C3—31no—
215C23×C3D3×Q811yescharacteristic
216C23×C3#23911yescharacteristic
217C23×C3C2×#6711yescharacteristic
218C22×D3—32no—
219C22×D3—61no—
220C22×D3—61no—
221C22×D3—61no—
222C22×D3—61no—
223C22×D3—61no—
224C22×D3—61no—
225C22×D3—123no—
226C2×A4D3×Q813yes—
227C2×A4#23913yes—
228C2×A4C2×#6713yes—
229S4—66no—
230C3×Q8—22no—
231C3×Q8—64no—
232C3×Q8—64no—
233C3×Q8—62no—
234C3×Q8—86no—
235C2×C3×C4—31no—
236C2×C3×C4—32no—
237C2×C3×C4—32no—
238C2×C3×C4—32no—
239C2×C3×C4—34no—
240C2×C3×C4—32no—
241C2×C3×C4—43no—
242C2×C3×C4—46no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—61no—
246C2×C3×C4—62no—
247C2×C3×C4C2×S412yes—
248C2×C3×C4C2×S411yescharacteristic
249#67—62no—
250#67—62no—
251#67—62no—
252#67—62no—
253C2×Dic3—32no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—62no—
259C2×Dic3—61no—
260C2×Dic3—62no—
261C2×Dic3—62no—
262C2×Dic3—61no—
263C2×Dic3—62no—
264C2×Dic3—61no—
265C2×Dic3—62no—
266C2×Dic3—61no—
267C2×Dic3—62no—
268C2×Dic3—123no—
269C2×Dic3—126no—
270C4×D3—122no—
271C4×D3—122no—
272C4×D3—246no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—41no—
276C3⋊D3—82no—
277C24—91no—
278C24C22×C3⋊D311yescharacteristic
279C2×Q8—34no—
280C2×Q8—32no—
281C2×Q8—31no—
282C2×Q8—61no—
283C2×Q8—62no—
284C2×Q8—62no—
285C2×Q8—62no—
286C2×Q8—64no—
287C2×Q8—64no—
288C2×Q8C3⋊S411yescharacteristic, complemented
289C2×D4—92no—
290C2×D4—92no—
291C2×D4—181no—
292C2×D4—181no—
293C2×D4—181no—
294C2×D4—181no—
295C22×C4—22no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—62no—
302C22×C4—91no—
303C22×C4—91no—
304C22×C4—92no—
305C22×C4—92no—
306C22×C4—91no—
307C22×C4—182no—
308C22×C4—181no—
309C22×C4—182no—
310C22×C4—181no—
311C22×C4#63812yes—
312C4⋊C4—182no—
313C4⋊C4—184no—
314C4⋊C4—182no—
315C4⋊C4—181no—
316C4⋊C4—182no—
317C4⋊C4—182no—
318C4⋊C4—181no—
319C22⋊C4—182no—
320C22⋊C4—184no—
321C22⋊C4—182no—
322C22⋊C4—181no—
323C22⋊C4—182no—
324C22⋊C4—182no—
325C22⋊C4—181no—
326C22⋊C4—182no—
327C42—182no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—43no—
338C22×C3—61no—
339C22×C3#94611yescharacteristic, complemented
340C22×C3C22×S411yescharacteristic
341C2×D3—62no—
342C2×D3—62no—
343C2×D3—62no—
344C2×D3—126no—
345C2×D3—121no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—121no—
349C2×D3—243no—
350C2×D3—243no—
351A4#94613yescomplemented
352C3×C4—22no—
353C3×C4—32no—
354C3×C4—46no—
355C3×C4—62no—
356C3×C4—86no—
357C3×C4GL(2,ℤ/4)12yes—
358Dic3—62no—
359Dic3—122no—
360Dic3—121no—
361Dic3—122no—
362Dic3—121no—
363Dic3—243no—
364Dic3—246no—
365C32—41noSylow
366C23—31no—
367C23—31no—
368C23—31no—
369C23—31no—
370C23—92no—
371C23—181no—
372C23—181no—
373C23—181no—
374C23—181no—
375C23—181no—
376C23—181no—
377C23Q8×C3⋊D311yescharacteristic
378C23#383211yescharacteristic
379C23C2×#63811yescharacteristic
380Q8—22no—
381Q8—64no—
382Q8—64no—
383Q8—62no—
384D4—182no—
385D4—182no—
386D4—182no—
387D4—182no—
388C2×C4—31no—
389C2×C4—32no—
390C2×C4—32no—
391C2×C4—32no—
392C2×C4—34no—
393C2×C4—32no—
394C2×C4—62no—
395C2×C4—62no—
396C2×C4—61no—
397C2×C4—62no—
398C2×C4—92no—
399C2×C4—181no—
400C2×C4—181no—
401C2×C4—181no—
402C2×C4—182no—
403C2×C4—181no—
404C2×C4—182no—
405C2×C4—182no—
406C2×C4—181no—
407C2×C4—181no—
408C2×C4—182no—
409C2×C4—181no—
410C2×C4—182no—
411C2×C4—182no—
412C2×C4—181no—
413C2×C4—362no—
414C2×C4—362no—
415C2×C4C2×C3⋊S412yes—
416C2×C4C2×C3⋊S411yescharacteristic
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—43no—
422C2×C3—43no—
423C2×C3—43no—
424C2×C3Q8×S411yescharacteristic
425C2×C3#591911yescharacteristic
426C2×C3C2×GL(2,ℤ/4)11yescharacteristic
427D3—122no—
428D3—246no—
429C22—31no—
430C22—31no—
431C22—31no—
432C22—31no—
433C22—31no—
434C22—31no—
435C22—31no—
436C22—31no—
437C22—31no—
438C22—61no—
439C22—182no—
440C22—182no—
441C22—182no—
442C22—361no—
443C22—361no—
444C22—361no—
445C22—361no—
446C22#6415511yescharacteristic, complemented
447C22C22×C3⋊S411yescharacteristic
448C4—22no—
449C4—32no—
450C4—62no—
451C4—182no—
452C4—362no—
453C4—361no—
454C4—362no—
455C4—361no—
456C4#6426912yes—
457C3—43no—
458C3#8550611yescharacteristic, complemented
459C2—31no—
460C2—31no—
461C2—31no—
462C2—31no—
463C2—362no—
464C2Q8×C3⋊S411yescharacteristic
465C2#1059658211yescharacteristic
466C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC22×C3×A4C23
FittingC23×C3×Q8D3
socleC24×C3C22×D3
prime radicalC22×C3⋊S4C22
hypercenterC2×Q8C3⋊S4
hypocenterC3×A4C22⋊Q8
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×A4C2semidirect
4. C4⋊(C3⋊SL(2,ℤ/4))C2semidirect
5. #10593645C2semidirect
16. C23×C3×Q8D3semidirect
17. C2×Q8×A4D3semidirect
35. C2×C3⋊S4Q8semidirect
96. C23×Q8C3⋊D3semidirect
123. C2×C3×Q8S4semidirect
175. C3×A4C22⋊Q8semidirect
288. C2×Q8C3⋊S4semidirect
339. C22×C3#946semidirect
351. A4#946semidirect
446. C22#64155semidirect
458. C3#85506semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×A42
C2×C4×C3⋊S42
#105936472
C4⋊(C3⋊SL(2,ℤ/4))2
#105936452
#855063
#840683
#641554