Finite Group Databasebeta

← #11779536

Subgroups — #11779536

Order 1152.

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

normal subgroups only

This group has 500 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×#5726C211yesmaximal, characteristic
2C3×#5093C211yesmaximal, characteristic, complemented
3#10590266C211yesmaximal, characteristic, complemented
4#10590266C211yesmaximal, characteristic, complemented
5#10589413C211yesmaximal, characteristic, complemented
6#10589081C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic, complemented
8#71911—31nomaximal
9#71228—31nomaximal
10C2×C3×#239—21no—
11C32×C4⋊Q8C2211yescharacteristic
12C3×#946—21no—
13C3×#895—21no—
14C3×#894—22no—
15C3×D3×C4⋊C4—21no—
16C3×C4×D12C2211yescharacteristic
17C3×C4×D12C2211yescharacteristic
18C3×#869C2211yescharacteristic
19C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×#810—22no—
22C3×C4×C3⋊C8C2211yescharacteristic
23#63574—21no—
24#63574—21no—
25#63562—21no—
26#63562—21no—
27C3×#547—31no—
28#5726D311yescharacteristic
29C3×#487—31no—
30#5093D311yescharacteristic, complemented
31#5084—31no—
32#5084—31no—
33#5008—31no—
34#5008—31no—
35#4547—31no—
36#4547—31no—
37#4547—61no—
38D4⋊(C3⋊C8)—31no—
39D4⋊(C3⋊C8)—31no—
40#4459—31no—
41#4459—31no—
42C2×C32×Q8—21no—
43C3×#239—42no—
44C3×#239—42no—
45C2×C3×D12—22no—
46C2×C3×D12D411yescharacteristic
47C2×C3×D12D411yescharacteristic
48C2×C3×C4×D3—22no—
49C2×C3×C4×D3—21no—
50C2×C3×C4×D3—21no—
51C2×C3×Dic6—21no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4D412yes—
54C32×C42C2311yescharacteristic
55C2×C3⋊(C3⋊C8)—21no—
56C2×C3⋊(C3⋊C8)—21no—
57C3×(C2×D3)⋊C4—22no—
58C3×(C2×D3)⋊C4—21no—
59C3×(C2×D3)⋊C4—21no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3D411yescharacteristic
63C3×C4⋊Dic3D411yescharacteristic
64C3×Dic3⋊C4—22no—
65C2×C3×C3⋊C8—21no—
66#1635—91noSylow
67C2×C3×#41—61no—
68C2×#239—21no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—31no—
71C3×C4⋊Q8C2×D311yescharacteristic
72C3×C4⋊Q8C2×D311yescharacteristic
73C3×#123—62no—
74C3×C22⋊Q8—61no—
75C3×#121—61no—
76C3×C4×D4—31no—
77C3×C4×D4—31no—
78C2×C3×C4⋊C4—61no—
79#946—21no—
80#895—21no—
81#894—22no—
82D3×C4⋊C4—21no—
83C4×D12C2×D311yescharacteristic
84C4×D12C2×D311yescharacteristic
85#869C2×D311yescharacteristic
86C3×#103—62no—
87C3×C4×C8—31no—
88#832—61no—
89#832—61no—
90#810—22no—
91#809—61no—
92#809—61no—
93#808—61no—
94#808—61no—
95#808—61no—
96#808—61no—
97C4⋊(C3⋊C8)—31no—
98C4⋊(C3⋊C8)—31no—
99C4⋊(C3⋊C8)—31no—
100C4⋊(C3⋊C8)—31no—
101C4⋊(C3⋊C8)—61no—
102C4⋊(C3⋊C8)—61no—
103C4×C3⋊C8C2×D311yescharacteristic
104C22×C3×D3—21no—
105C22×C3×D3—21no—
106C32×Q8—41no—
107C32×Q8—41no—
108C2×C32×C4—22no—
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—41no—
115C3×D12—21no—
116C3×D12—21no—
117C3×D12—21no—
118C3×D12—44no—
119C3×D12SD1612yes—
120C3×C4×D3—44no—
121C3×C4×D3—41no—
122C3×C4×D3—41no—
123C3×C4×D3—41no—
124C3×C4×D3—41no—
125C3×Dic6—42no—
126C3⋊(C3⋊C8)—41no—
127C3⋊(C3⋊C8)—41no—
128C3×C3⋊C8—41no—
129#547—31no—
130#487—31no—
131#476—91no—
132#476—91no—
133#332—91no—
134D4⋊C8—91no—
135D4⋊C8—91no—
136C3×#41—122no—
137C3×#41—122no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—41no—
141C2×C3×Q8—61no—
142C2×C3×D4—31no—
143C2×C3×D4—31no—
144C2×C3×D4—62no—
145C22×C3×C4—62no—
146C22×C3×C4—61no—
147C22×C3×C4—61no—
148#239—42no—
149#239—42no—
150C2×D12—22no—
151C2×D12#6711yescharacteristic
152C2×D12#6711yescharacteristic
153C2×C4×D3—22no—
154C2×C4×D3—21no—
155C2×C4×D3—21no—
156C2×Dic6—21no—
157C2×C3×C8—61no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4—62no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4—61no—
167C3×C4⋊C4#6712yes—
168C3×C4⋊C4#6712yes—
169C3×C22⋊C4—62no—
170C3×C22⋊C4—61no—
171C3×C22⋊C4—61no—
172C3×C42—21no—
173C3×C42C22×D311yescharacteristic
174C3×C42C22×D311yescharacteristic
175(C2×D3)⋊C4—22no—
176(C2×D3)⋊C4—21no—
177(C2×D3)⋊C4—21no—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180C4⋊Dic3#6711yescharacteristic
181C4⋊Dic3#6711yescharacteristic
182Dic3⋊C4—22no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8—61no—
185C2×C3⋊C8—61no—
186C2×C3⋊C8—61no—
187C2×C3⋊C8—61no—
188C2×C3⋊C8—121no—
189C2×C3⋊C8—121no—
190C22×C32#12011yescharacteristic
191C2×C3×D3—22no—
192C2×C3×D3—41no—
193C2×C3×D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—41no—
196C2×C3×D3—41no—
197C32×C4—21no—
198C32×C4—42no—
199C32×C4C2×SD1611yescharacteristic
200C32×C4#13511yescharacteristic
201C32×C4AGL(1,ℤ/8)11yescharacteristic
202C32×C4#13711yescharacteristic
203C3×Dic3—41no—
204C3×Dic3—41no—
205C3×Dic3—81no—
206C2×#41—61no—
207C4⋊Q8—31no—
208C4⋊Q8D3211yescharacteristic
209#123—62no—
210C22⋊Q8—61no—
211#121—61no—
212C4×D4—31no—
213C4×D4—31no—
214C2×C4⋊C4—61no—
215#106—181no—
216#106—181no—
217C4⋊C8—91no—
218C4⋊C8—91no—
219#103—62no—
220#102—181no—
221#102—181no—
222C4×C8—31no—
223C23×C3—61no—
224C23×C3—61no—
225C22×D3—21no—
226C22×D3—21no—
227C3×Q8—41no—
228C3×Q8—41no—
229C3×Q8—41no—
230C3×Q8—41no—
231C3×Q8—81no—
232C3×Q8—81no—
233C3×Q8—122no—
234C3×D4—32no—
235C3×D4—61no—
236C3×D4—61no—
237C3×D4—61no—
238C3×D4—124no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—42no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—124no—
248C2×C3×C4—121no—
249C2×C3×C4—121no—
250C2×C3×C4—121no—
251C2×C3×C4—121no—
252C2×C3×C4—121no—
253C2×C3×C4D3×D411yescharacteristic
254C2×C3×C4D3×D411yescharacteristic
255C2×C3×C4C2×#6711yescharacteristic
256C2×C3×C4C2×#6711yescharacteristic
257C2×C3×C4C2×#6711yescharacteristic
258C2×C3×C4C2×#6711yescharacteristic
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—41no—
262D12—21no—
263D12—21no—
264D12—21no—
265D12—44no—
266D12#21412yes—
267C4×D3—44no—
268C4×D3—41no—
269C4×D3—41no—
270C4×D3—41no—
271C4×D3—41no—
272Dic6—42no—
273C3×C8—121no—
274C3⋊C8—41no—
275C3⋊C8—121no—
276C3⋊C8—121no—
277C3⋊C8—121no—
278C3⋊C8—121no—
279C3⋊C8—241no—
280C3⋊C8—241no—
281C2×C32#45011yescharacteristic
282C2×C32#45211yescharacteristic
283C2×C32#45411yescharacteristic
284C3×D3—42no—
285C3×D3—81no—
286#41—122no—
287#41—122no—
288C2×Q8—21no—
289C2×Q8—61no—
290C2×D4—31no—
291C2×D4—31no—
292C2×D4—62no—
293C22×C4—62no—
294C22×C4—61no—
295C22×C4—61no—
296C2×C8—61no—
297C2×C8—181no—
298C2×C8—181no—
299C4⋊C4—21no—
300C4⋊C4—31no—
301C4⋊C4—31no—
302C4⋊C4—62no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4#62512yes—
306C22⋊C4—62no—
307C22⋊C4—61no—
308C22⋊C4—61no—
309C42C2×D3211yescharacteristic
310C22×C3—21no—
311C22×C3—62no—
312C22×C3—121no—
313C22×C3—121no—
314C22×C3—121no—
315C22×C3—121no—
316C22×C3—121no—
317C22×C3#93711yescharacteristic
318C22×C3#95311yescharacteristic
319C2×D3—22no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—41no—
332C3×C4—42no—
333C3×C4—42no—
334C3×C4—82no—
335C3×C4—121no—
336C3×C4—121no—
337C3×C4—241no—
338C3×C4D3×SD1611yescharacteristic
339C3×C4#91411yescharacteristic
340C3×C4#91811yescharacteristic
341C3×C4#91911yescharacteristic
342C3×C4#93211yescharacteristic
343C3×C4C2×#21411yescharacteristic
344C3×C4#95011yescharacteristic
345C3×C4#95111yescharacteristic
346Dic3—41no—
347Dic3—41no—
348Dic3—81no—
349C32#163511yescharacteristic, complemented, Sylow
350C23—61no—
351C23—61no—
352Q8—41no—
353Q8—41no—
354Q8—122no—
355D4—32no—
356D4—61no—
357D4—61no—
358D4—61no—
359D4—124no—
360C2×C4—22no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—124no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4C2×#62511yescharacteristic
370C2×C4D3×#6711yescharacteristic
371C2×C4D3×#6711yescharacteristic
372C8—121no—
373C8—361no—
374C8—361no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—122no—
379C2×C3—241no—
380C2×C3#506211yescharacteristic
381C2×C3#516911yescharacteristic
382C2×C3#518711yescharacteristic
383C2×C3#521811yescharacteristic
384C2×C3#523911yescharacteristic
385C2×C3#524511yescharacteristic
386D3—42no—
387D3—81no—
388C22—62no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22#6397611yescharacteristic
395C4—21no—
396C4—42no—
397C4—121no—
398C4—121no—
399C4—241no—
400C4D3×#21411yescharacteristic
401C4#6393411yescharacteristic
402C4#6394311yescharacteristic
403C4#6394911yescharacteristic
404C3—21no—
405C3#7122811yescharacteristic, complemented
406C3#7191111yescharacteristic, complemented
407C2—122no—
408C2—241no—
409C2#1059145811yescharacteristic
410C2#1059159711yescharacteristic
411C2#1059164511yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radicalC3×#5726C2
hypercenterC4⋊Q8D32
hypocenterC32#1635
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5093C2semidirect
3. #10590266C2semidirect
4. #10590266C2semidirect
5. #10589413C2semidirect
6. #10589081C2semidirect
7. #10589081C2semidirect
30. #5093D3semidirect
349. C32#1635semidirect
405. C3#71228semidirect
406. C3#71911semidirect

Maximal subgroups

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

subgroupindex
C3×#57262
C3×#50932
#105902662
#105894132
#105890812
#719113
#712283