Finite Group Databasebeta

← #11779548

Subgroups — #11779548

Order 1152.

509 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 96 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×#5551C211yesmaximal, characteristic
2C3×#5008C211yesmaximal, characteristic, complemented
3#10590290C211yesmaximal, characteristic, complemented
4#10590266C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#10589092C211yesmaximal, characteristic, complemented
7#10589080C211yesmaximal, characteristic, complemented
8#71911—31nomaximal
9#71140—31nomaximal
10C2×C3×#235—21no—
11C32×C4×D4C2211yescharacteristic
12C3×#938—21no—
13C3×#924—21no—
14C3×D3×C4⋊C4—21no—
15C3×#883—22no—
16C3×C4×D12C2211yescharacteristic
17C3×#869C2211yescharacteristic
18C3×#869C2211yescharacteristic
19C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C4×C3⋊(C3⋊C8)C2211yescharacteristic
21C3×#832—21no—
22C3×#808—21no—
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24#63574—21no—
25#63565—22no—
26#63562—21no—
27C3×#547—31no—
28#5551D311yescharacteristic
29C3×#476—31no—
30#5093—31no—
31#5084—31no—
32#5079—31no—
33#5008—31no—
34#5008D311yescharacteristic, complemented
35#4547—31no—
36D4⋊(C3⋊C8)—31no—
37D4⋊(C3⋊C8)—31no—
38D4⋊(C3⋊C8)—31no—
39D4⋊(C3⋊C8)—61no—
40#4486—31no—
41#4483—31no—
42C2×C32×D4D411yescharacteristic
43C22×C32×C4—21no—
44C2×C3×#67—22no—
45C3×#235—42no—
46C3×#235—42no—
47C2×C3×D12D411yescharacteristic
48C2×C3×C4×D3—22no—
49C2×C3×C4×D3—21no—
50C2×C3×Dic6—21no—
51C2×C3×Dic6—21no—
52C32×C4⋊C4D411yescharacteristic
53C32×C22⋊C4—21no—
54C32×C42C2311yescharacteristic
55C2×C3⋊(C3⋊C8)—21no—
56C2×C3⋊(C3⋊C8)—21no—
57C3×#217—22no—
58C3×(C2×D3)⋊C4—21no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3D412yes—
63C3×C4⋊Dic3D411yescharacteristic
64C3×Dic3⋊C4—22no—
65C2×C3×C3⋊C8—21no—
66#1635—91noSylow
67C2×C3×#41—61no—
68C2×#235—21no—
69C3×C4⋊Q8—31no—
70C3×C4⋊Q8—31no—
71C3×#123—62no—
72C3×C22⋊Q8—61no—
73C3×#121—61no—
74C3×C4×D4—21no—
75C3×C4×D4—31no—
76C3×C4×D4C2×D311yescharacteristic
77C3×C4×D4C2×D311yescharacteristic
78C2×C3×C4⋊C4—61no—
79#938—21no—
80#924—21no—
81D3×C4⋊C4—21no—
82#883—22no—
83C4×D12C2×D311yescharacteristic
84#869C2×D311yescharacteristic
85#869C2×D311yescharacteristic
86C3×#106—61no—
87C3×C4⋊C8—31no—
88C3×#102—61no—
89#835—62no—
90#832—21no—
91#832—61no—
92#810—62no—
93#809—61no—
94#808—21no—
95#808—61no—
96#808—61no—
97C4⋊(C3⋊C8)—31no—
98C4⋊(C3⋊C8)—31no—
99C4⋊(C3⋊C8)—61no—
100C4⋊(C3⋊C8)C2×D311yescharacteristic
101C4×C3⋊C8—31no—
102C4×C3⋊C8—31no—
103C4×C3⋊C8—61no—
104C23×C32—21no—
105C22×C3×D3—21no—
106C32×D4—21no—
107C32×D4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C3×#67—44no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—41no—
118C3×D12—21no—
119C3×D12SD1612yes—
120C3×C4×D3—44no—
121C3×C4×D3—41no—
122C3×C4×D3—41no—
123C3×Dic6—41no—
124C3×Dic6—41no—
125C3×Dic6—42no—
126C3⋊(C3⋊C8)—41no—
127C3⋊(C3⋊C8)—41no—
128C3×C3⋊C8—41no—
129#547—31no—
130#487—91no—
131#476—31no—
132#476—91no—
133#332—91no—
134D4⋊C8—91no—
135D4⋊C8—91no—
136C3×#41—122no—
137C3×#41—122no—
138C2×C3×Q8—61no—
139C2×C3×Q8—61no—
140C2×C3×D4—21no—
141C2×C3×D4—31no—
142C2×C3×D4—62no—
143C2×C3×D4#6711yescharacteristic
144C2×C3×D4#6711yescharacteristic
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—41no—
148C22×C3×C4—62no—
149C22×C3×C4—61no—
150C2×#67—22no—
151#235—42no—
152#235—42no—
153C2×D12#6711yescharacteristic
154C2×C4×D3—22no—
155C2×C4×D3—21no—
156C2×Dic6—21no—
157C2×Dic6—21no—
158C2×C3×C8—61no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—32no—
162C3×C4⋊C4—62no—
163C3×C4⋊C4—61no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4#6711yescharacteristic
167C3×C4⋊C4#6711yescharacteristic
168C3×C22⋊C4—21no—
169C3×C22⋊C4—21no—
170C3×C22⋊C4—41no—
171C3×C22⋊C4—62no—
172C3×C22⋊C4—61no—
173C3×C42—21no—
174C3×C42C22×D311yescharacteristic
175C3×C42C22×D311yescharacteristic
176#217—22no—
177(C2×D3)⋊C4—21no—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180C4⋊Dic3—21no—
181C4⋊Dic3#6711yescharacteristic
182C4⋊Dic3#6712yes—
183Dic3⋊C4—22no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8—61no—
186C2×C3⋊C8—61no—
187C2×C3⋊C8—61no—
188C2×C3⋊C8—61no—
189C2×C3⋊C8—121no—
190C2×C3⋊C8—121no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32#12011yescharacteristic
195C2×C3×D3—22no—
196C2×C3×D3—41no—
197C2×C3×D3—41no—
198C32×C4—21no—
199C32×C4—41no—
200C32×C4C2×SD1611yescharacteristic
201C32×C4#13511yescharacteristic
202C32×C4AGL(1,ℤ/8)11yescharacteristic
203C32×C4#13711yescharacteristic
204C3×Dic3—42no—
205C3×Dic3—41no—
206C3×Dic3—81no—
207C2×#41—61no—
208C4⋊Q8—31no—
209C4⋊Q8—31no—
210#123—62no—
211C22⋊Q8—61no—
212#121—61no—
213C4×D4—31no—
214C4×D4D3211yescharacteristic
215C2×C4⋊C4—61no—
216#106—61no—
217#106—181no—
218C4⋊C8—31no—
219C4⋊C8—91no—
220#103—182no—
221#102—61no—
222#102—181no—
223C4×C8—91no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—41no—
227C23×C3—61no—
228C22×D3—21no—
229C3×Q8—122no—
230C3×Q8—121no—
231C3×Q8—121no—
232C3×D4—22no—
233C3×D4—21no—
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—21no—
237C3×D4—32no—
238C3×D4—41no—
239C3×D4—61no—
240C3×D4—124no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—61no—
252C2×C3×C4—62no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4—124no—
256C2×C3×C4—121no—
257C2×C3×C4—121no—
258C2×C3×C4—121no—
259C2×C3×C4D3×D411yescharacteristic
260C2×C3×C4D3×D411yescharacteristic
261C2×C3×C4C2×#6711yescharacteristic
262C2×C3×C4C2×#6711yescharacteristic
263C2×C3×C4C2×#6711yescharacteristic
264C2×C3×C4C2×#6711yescharacteristic
265#67—44no—
266C2×Dic3—21no—
267C2×Dic3—22no—
268C2×Dic3—41no—
269D12—21no—
270D12#21412yes—
271C4×D3—44no—
272C4×D3—41no—
273C4×D3—41no—
274Dic6—41no—
275Dic6—41no—
276Dic6—42no—
277C3×C8—121no—
278C3⋊C8—41no—
279C3⋊C8—121no—
280C3⋊C8—121no—
281C3⋊C8—122no—
282C3⋊C8—121no—
283C3⋊C8—121no—
284C3⋊C8—241no—
285C2×C32—81no—
286C2×C32#45011yescharacteristic
287C2×C32#45211yescharacteristic
288C2×C32#45411yescharacteristic
289C3×D3—42no—
290#41—122no—
291#41—122no—
292C2×Q8—61no—
293C2×Q8—61no—
294C2×D4—31no—
295C2×D4—62no—
296C2×D4#62511yescharacteristic
297C22×C4—21no—
298C22×C4—62no—
299C22×C4—61no—
300C2×C8—61no—
301C2×C8—181no—
302C2×C8—181no—
303C4⋊C4—32no—
304C4⋊C4—31no—
305C4⋊C4—62no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4#62511yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—62no—
312C22⋊C4—61no—
313C42C2×D3211yescharacteristic
314C22×C3—21no—
315C22×C3—42no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—62no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3—121no—
326C22×C3—121no—
327C22×C3#93711yescharacteristic
328C22×C3#95311yescharacteristic
329C2×D3—22no—
330C2×D3—41no—
331C2×D3—41no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—41no—
339C3×C4—41no—
340C3×C4—41no—
341C3×C4—81no—
342C3×C4—122no—
343C3×C4—121no—
344C3×C4—241no—
345C3×C4D3×SD1611yescharacteristic
346C3×C4#91411yescharacteristic
347C3×C4#93211yescharacteristic
348C3×C4C2×#21411yescharacteristic
349C3×C4#94211yescharacteristic
350C3×C4#95011yescharacteristic
351C3×C4#95011yescharacteristic
352C3×C4#95111yescharacteristic
353Dic3—42no—
354Dic3—41no—
355Dic3—81no—
356C32#163511yescharacteristic, complemented, Sylow
357C23—21no—
358C23—61no—
359Q8—122no—
360Q8—121no—
361Q8—121no—
362D4—21no—
363D4—21no—
364D4—32no—
365D4—61no—
366D4—124no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—61no—
371C2×C4—62no—
372C2×C4—124no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4C2×#62511yescharacteristic
377C2×C4D3×#6711yescharacteristic
378C2×C4D3×#6711yescharacteristic
379C8—121no—
380C8—361no—
381C8—361no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—82no—
386C2×C3—81no—
387C2×C3—81no—
388C2×C3—122no—
389C2×C3#504711yescharacteristic
390C2×C3#516911yescharacteristic
391C2×C3#520111yescharacteristic
392C2×C3#521811yescharacteristic
393C2×C3#523911yescharacteristic
394C2×C3#524511yescharacteristic
395D3—42no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—62no—
400C22—121no—
401C22—121no—
402C22#6397611yescharacteristic
403C4—21no—
404C4—41no—
405C4—121no—
406C4—122no—
407C4—241no—
408C4#6382211yescharacteristic
409C4#6382711yescharacteristic
410C4D3×#21411yescharacteristic
411C4#6393411yescharacteristic
412C3—21no—
413C3#7114011yescharacteristic, complemented
414C3#7191111yescharacteristic, complemented
415C2—81no—
416C2—122no—
417C2#1059117011yescharacteristic
418C2#1059145811yescharacteristic
419C2#1059168411yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×D4C22
socleC22×C32#120
prime radicalC3×#5551C2
hypercenterC4×D4D32
hypocenterC32#1635
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C3×#5008C2semidirect
3. #10590290C2semidirect
4. #10590266C2semidirect
5. D4⋊(C3⋊(C3⋊C8))C2semidirect
6. #10589092C2semidirect
7. #10589080C2semidirect
34. #5008D3semidirect
356. C32#1635semidirect
413. C3#71140semidirect
414. C3#71911semidirect

Maximal subgroups

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

subgroupindex
C3×#55512
C3×#50082
#105902902
#105902662
D4⋊(C3⋊(C3⋊C8))2
#105890922
#105890802
#719113
#711403