Finite Group Databasebeta

← #11921860

Subgroups — #11921860

Order 1152.

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

normal subgroups only

This group has 562 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
1C3×#5912C211yesmaximal, characteristic
2A4×#208C211yesmaximal, characteristic, complemented
3C2×C3⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4#10593175C212yesmaximal, complemented
5C3⋊C8×S4C212yesmaximal, complemented
6#85442—31nomaximal
7#82918—31nomaximal
8C2×C3×C4×A4C2211yescharacteristic
9C3×GL(2,ℤ/4)C412yes—
10C3×C4⋊S4C411yescharacteristic
11C3×C4×S4C2212yes—
12C3×A4⋊Q8C411yescharacteristic
13#63815—41nomaximal
14A4×C3⋊C8C2212yes—
15C3⋊(A4⋊C8)C2211yescharacteristic
16C3⋊(A4⋊C8)C2211yescharacteristic
17#5912D311yescharacteristic
18C3×#525—31no—
19C2×#948—31no—
20C22×#208D311yescharacteristic, complemented
21A4×M16—31no—
22C2×A4⋊C8—31no—
23C2×A4⋊C8—61no—
24#5400—32no—
25C8×S4—32no—
26#5234—31no—
27C2×#830—31no—
28#5012—32no—
29#5011—32no—
30#5011—32no—
31D4×C3⋊C8—32no—
32#4999—31no—
33#4997—31no—
34C22×C3×A4C2×C411yescharacteristic
35C2×C3×S4C2×C412yes—
36C3×#235—41no—
37C3×C4×A4C2×C411yescharacteristic
38C3×C4×A4C2311yescharacteristic
39C3×SL(2,ℤ/4)C2×C412yes—
40C2×C3⋊(C3⋊C8)—41no—
41C3×#208—41no—
42#3711—42no—
43D3×C3⋊C8—42no—
44#2933—91noSylow
45C2×C3×#41—31no—
46C23×C3×C4C2×D311yescharacteristic
47C2×C4×A4—21no—
48C2×C4×A4C2×D311yescharacteristic
49GL(2,ℤ/4)Dic312yes—
50C4⋊S4Dic311yescharacteristic
51C4×S4C2×D312yes—
52A4⋊Q8Dic311yescharacteristic
53C3×#123—32no—
54C3×C22⋊Q8—31no—
55C3×C22⋊Q8—31no—
56C3×#121—31no—
57C3×#121—31no—
58C3×#120—32no—
59C3×C4×D4—32no—
60C3×C4×D4—32no—
61C3×#117—31no—
62#948—61no—
63#948—62no—
64#948—61no—
65C2×#208—31no—
66C2×#208—31no—
67C2×#208—32no—
68C2×#208—32no—
69C22×C3⋊C8—31no—
70C22×C3⋊C8—31no—
71C22×C3⋊C8—32no—
72C22×C3⋊C8C2×D312yes—
73#907—121no—
74C8×A4—32no—
75A4⋊C8—31no—
76A4⋊C8—31no—
77A4⋊C8—61no—
78A4⋊C8—61no—
79#830—31no—
80#830—31no—
81#830—32no—
82#830—32no—
83#830—31no—
84#830—31no—
85C4⋊(C3⋊C8)—32no—
86C4⋊(C3⋊C8)—32no—
87#803—32no—
88C4×C3⋊C8—32no—
89C2×C3×A4—21no—
90C2×C3×A4C22×C411yescharacteristic
91C3×S4—22no—
92C2×C32×C4—41no—
93C3×#67—42no—
94C3×D12—41no—
95C3×C4×D3—42no—
96C3×Dic6—41no—
97C3⋊(C3⋊C8)—41no—
98C3⋊(C3⋊C8)—41no—
99C3×C3⋊C8—42no—
100C2×#131—91no—
101C22×M16—31no—
102#525—31no—
103#436—92no—
104#435—92no—
105#435—92no—
106C8×D4—92no—
107#424—91no—
108#408—91no—
109C2×C22⋊C8—91no—
110#405—91no—
111C24×C3C4×D311yescharacteristic
112C22×A4—21no—
113C22×A4C2×Dic311yescharacteristic
114C2×S4C2×Dic312yes—
115C3×#41—61no—
116C3×#41—61no—
117C3×#41—61no—
118C3×#41—61no—
119C2×C3×Q8—31no—
120C2×C3×D4—32no—
121C2×C3×D4—31no—
122C2×C3×D4—32no—
123C2×C3×D4—32no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—32no—
129C22×C3×C4C4×D311yescharacteristic
130C22×C3×C4C22×D311yescharacteristic
131#235—41no—
132C4×A4—21no—
133C4×A4—21no—
134C4×A4C2×Dic311yescharacteristic
135C4×A4C22×D311yescharacteristic
136SL(2,ℤ/4)C2×Dic312yes—
137C3×M16—121no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—32no—
141C3×C4⋊C4—32no—
142C3×C22⋊C4—32no—
143C3×C22⋊C4—32no—
144C3×C22⋊C4—32no—
145C3×C22⋊C4—32no—
146C3×C22⋊C4—32no—
147C3×C42—32no—
148#208—31no—
149#208—32no—
150#208—61no—
151#208—62no—
152#208—62no—
153#208S411yescharacteristic, complemented
154C2×C3⋊C8—32no—
155C2×C3⋊C8—32no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8—31no—
158C2×C3⋊C8—31no—
159C2×C3⋊C8—31no—
160C2×C3⋊C8—62no—
161C2×C3⋊C8—61no—
162C2×C3⋊C8—61no—
163C2×C3⋊C8—61no—
164C2×C3⋊C8—61no—
165C2×C3⋊C8—62no—
166C2×C3⋊C8—121no—
167C2×C3⋊C8—241no—
168ΓL(1,𝔽25)—122no—
169D3×C8—122no—
170C22×C32—41no—
171C2×C3×D3—42no—
172C3×A4#13111yescharacteristic, complemented
173C32×C4—41no—
174C32×C4—41no—
175C3×Dic3—42no—
176C2×#41—31no—
177C23×C4D3211yescharacteristic
178#131—181no—
179#131—182no—
180#131—181no—
181C2×M16—91no—
182C2×M16—91no—
183C2×M16—92no—
184C2×M16—92no—
185C22×C8—32no—
186C22×C8—91no—
187C22×C8—92no—
188C22×C8—91no—
189#123—32no—
190C22⋊Q8—31no—
191C22⋊Q8—31no—
192#121—31no—
193#121—31no—
194#120—32no—
195C4×D4—32no—
196C4×D4—32no—
197#117—31no—
198C4⋊C8—92no—
199C4⋊C8—92no—
200C22⋊C8—91no—
201C22⋊C8—91no—
202C22⋊C8—92no—
203C22⋊C8—92no—
204C22⋊C8—91no—
205C22⋊C8—91no—
206#97—92no—
207C4×C8—92no—
208C23×C3—21no—
209C23×C3—31no—
210C23×C3—31no—
211C23×C3—32no—
212C23×C3—61no—
213C23×C3C2×C4×D311yescharacteristic
214C2×A4—22no—
215C2×A4—21no—
216C2×A4—21no—
217C2×A4C22×Dic311yescharacteristic
218S4—22no—
219C3×Q8—61no—
220C3×Q8—61no—
221C3×D4—61no—
222C3×D4—62no—
223C3×D4—62no—
224C3×D4—62no—
225C3×D4—62no—
226C3×D4—61no—
227C3×D4—122no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—32no—
236C2×C3×C4—32no—
237C2×C3×C4—32no—
238C2×C3×C4—41no—
239C2×C3×C4—61no—
240C2×C3×C4—62no—
241C2×C3×C4—62no—
242C2×C3×C4—81no—
243C2×C3×C4C2×S411yescharacteristic
244#67—42no—
245D12—41no—
246C4×D3—42no—
247Dic6—41no—
248C3×C8—122no—
249C3⋊C8—32no—
250C3⋊C8—61no—
251C3⋊C8—61no—
252C3⋊C8—61no—
253C3⋊C8—61no—
254C3⋊C8—121no—
255C3⋊C8—121no—
256C3⋊C8—241no—
257C3⋊C8—241no—
258C3⋊C8C2×S412yes—
259C2×C32—41no—
260C2×C32—81no—
261C3×D3—82no—
262C24D3×Dic311yescharacteristic
263#41—61no—
264#41—61no—
265#41—61no—
266#41—61no—
267C2×Q8—31no—
268C2×D4—32no—
269C2×D4—31no—
270C2×D4—32no—
271C2×D4—32no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—31no—
276C22×C4—32no—
277C22×C4D3×Dic311yescharacteristic
278C22×C4C2×D3211yescharacteristic
279M16—31no—
280M16—92no—
281M16—91no—
282M16—182no—
283M16—181no—
284M16—182no—
285C2×C8—92no—
286C2×C8—92no—
287C2×C8—91no—
288C2×C8—91no—
289C2×C8—91no—
290C2×C8—91no—
291C2×C8—182no—
292C2×C8—182no—
293C2×C8—181no—
294C2×C8—181no—
295C2×C8—181no—
296C2×C8—181no—
297C4⋊C4—31no—
298C4⋊C4—31no—
299C4⋊C4—32no—
300C4⋊C4—32no—
301C22⋊C4—32no—
302C22⋊C4—32no—
303C22⋊C4—32no—
304C22⋊C4—32no—
305C22⋊C4—32no—
306C42—32no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—31no—
310C22×C3—41no—
311C22×C3—61no—
312C22×C3—61no—
313C22×C3—61no—
314C22×C3—61no—
315C22×C3—62no—
316C22×C3—62no—
317C22×C3—62no—
318C22×C3—81no—
319C22×C3#90711yescharacteristic, complemented
320C22×C3C4×S411yescharacteristic
321C2×D3—42no—
322A4—21no—
323A4#94811yescharacteristic, complemented
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—41no—
327C3×C4—41no—
328C3×C4—62no—
329C3×C4—62no—
330C3×C4—62no—
331C3×C4—81no—
332C3×C4—81no—
333C3×C4C4×S411yescharacteristic
334C3×C4C22×S411yescharacteristic
335Dic3—42no—
336C32—41noSylow
337C23—21no—
338C23—31no—
339C23—31no—
340C23—32no—
341C23—61no—
342C23C2×D3×Dic311yescharacteristic
343Q8—61no—
344Q8—61no—
345D4—61no—
346D4—62no—
347D4—62no—
348D4—62no—
349D4—62no—
350D4—61no—
351D4—122no—
352C2×C4—31no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—32no—
360C2×C4—32no—
361C2×C4—32no—
362C2×C4—61no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4D3×S411yescharacteristic
366C8—32no—
367C8—92no—
368C8—181no—
369C8—181no—
370C8—181no—
371C8—181no—
372C2×C3—21no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—41no—
376C2×C3—61no—
377C2×C3—81no—
378C2×C3—82no—
379C2×C3—81no—
380C2×C3—122no—
381C2×C3C2×C4×S411yescharacteristic
382D3—82no—
383C22—31no—
384C22—31no—
385C22—31no—
386C22—61no—
387C22—61no—
388C22—61no—
389C22—61no—
390C22—62no—
391C22—62no—
392C22—62no—
393C22#6381511yescharacteristic, complemented
394C22Dic3×S411yescharacteristic
395C4—31no—
396C4—31no—
397C4—62no—
398C4—62no—
399C4—62no—
400C4Dic3×S411yescharacteristic
401C4C2×D3×S411yescharacteristic
402C3—41no—
403C3—81no—
404C3#8544211yescharacteristic, complemented
405C2—21no—
406C2—31no—
407C2—31no—
408C2—61no—
409C2—122no—
410C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×S4
commutatorC2×C3×A4C22×C4
FittingC23×C3×C4C2×D3
socleC23×C3C2×C4×D3
prime radicalC3×#5912C2
cosocleC3×C4×A4C23
hypercenterC2×C4D3×S4
hypocenterC3×A4#131
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. A4×#208C2semidirect
3. C2×C3⋊(A4⋊C8)C2semidirect
4. #10593175C2semidirect
5. C3⋊C8×S4C2semidirect
20. C22×#208D3semidirect
153. #208S4semidirect
172. C3×A4#131semidirect
319. C22×C3#907semidirect
323. A4#948semidirect
393. C22#63815semidirect
404. C3#85442semidirect

Maximal subgroups

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

subgroupindex
C3×#59122
A4×#2082
C2×C3⋊(A4⋊C8)2
#105931752
C3⋊C8×S42
#854423
#829183
#638154