Finite Group Databasebeta

← #11780438

Subgroups — #11780438

Order 1152.

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

normal subgroups only

This group has 512 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×#5080C211yesmaximal, characteristic, complemented
3#10590363C211yesmaximal, characteristic, complemented
4#10590269C211yesmaximal, characteristic, complemented
5#10589413C211yesmaximal, characteristic, complemented
6#10589081C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic, complemented
8#71913—31nomaximal
9#71343—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
19C2×#3715—21no—
20C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×#808—22no—
23C3×C4×C3⋊C8C2211yescharacteristic
24#63576—21no—
25#63565—21no—
26#63562—21no—
27C3×#547—31no—
28#5726D311yescharacteristic
29C3×#498—31no—
30#5087—31no—
31#5083—31no—
32#5080D311yescharacteristic, complemented
33#5019—31no—
34#5007—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#3715—42no—
67#1654—91noSylow
68C2×C3×#41—61no—
69C2×#239—21no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—31no—
72C3×C4⋊Q8C2×D311yescharacteristic
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×#123—62no—
75C3×C22⋊Q8—61no—
76C3×#121—61no—
77C3×C4×D4—31no—
78C3×C4×D4—31no—
79C2×C3×C4⋊C4—61no—
80#946—21no—
81C2×#215—61no—
82C2×#214—61no—
83#895—21no—
84#894—22no—
85D3×C4⋊C4—21no—
86C4×D12C2×D311yescharacteristic
87C4×D12C2×D311yescharacteristic
88#869C2×D311yescharacteristic
89C3×#106—62no—
90C3×C4×C8—31no—
91#835—61no—
92#832—61no—
93#810—61no—
94#809—61no—
95#808—22no—
96#807—61no—
97#807—61no—
98C4⋊(C3⋊C8)—31no—
99C4⋊(C3⋊C8)—31no—
100C4⋊(C3⋊C8)—31no—
101C4⋊(C3⋊C8)—31no—
102C4⋊(C3⋊C8)—61no—
103C4⋊(C3⋊C8)—61no—
104C4×C3⋊C8C2×D311yescharacteristic
105C22×C3×D3—21no—
106C22×C3×D3—21no—
107C32×Q8—41no—
108C32×Q8—41no—
109C2×C32×C4—22no—
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—41no—
116C3×D12—21no—
117C3×D12—21no—
118C3×D12—21no—
119C3×D12—44no—
120C3×D12SD1612yescomplemented
121C3×C4×D3—44no—
122C3×C4×D3—41no—
123C3×C4×D3—41no—
124C3×C4×D3—41no—
125C3×C4×D3—41no—
126C3×Dic6—41no—
127C3×Dic6—41no—
128C3⋊(C3⋊C8)—41no—
129C3⋊(C3⋊C8)—41no—
130C3×C3⋊C8—22no—
131#547—31no—
132#498—31no—
133#474—91no—
134#461—91no—
135#332—91no—
136D4⋊C8—91no—
137D4⋊C8—91no—
138C3×#41—122no—
139C3×#41—122no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8—41no—
143C2×C3×Q8—61no—
144C2×C3×D4—31no—
145C2×C3×D4—31no—
146C2×C3×D4—62no—
147C22×C3×C4—62no—
148C22×C3×C4—61no—
149C22×C3×C4—61no—
150#239—42no—
151#239—42no—
152C2×D12—22no—
153C2×D12#6711yescharacteristic
154C2×D12#6711yescharacteristic
155C2×C4×D3—22no—
156C2×C4×D3—21no—
157C2×C4×D3—21no—
158C2×Dic6—21no—
159C2×C3×C8—61no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—41no—
166C3×C4⋊C4—62no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4—61no—
169C3×C4⋊C4#6712yes—
170C3×C4⋊C4#6712yes—
171C3×C22⋊C4—62no—
172C3×C22⋊C4—61no—
173C3×C22⋊C4—61no—
174C3×C42—21no—
175C3×C42C22×D311yescharacteristic
176C3×C42C22×D311yescharacteristic
177#215—122no—
178#214—122no—
179(C2×D3)⋊C4—22no—
180(C2×D3)⋊C4—21no—
181(C2×D3)⋊C4—21no—
182C4⋊Dic3—21no—
183C4⋊Dic3—21no—
184C4⋊Dic3#6711yescharacteristic
185C4⋊Dic3#6711yescharacteristic
186Dic3⋊C4—22no—
187C2×C3⋊C8—21no—
188C2×C3⋊C8—61no—
189C2×C3⋊C8—61no—
190C2×C3⋊C8—61no—
191C2×C3⋊C8—61no—
192C2×C3⋊C8—121no—
193C2×C3⋊C8—121no—
194C22×C32#12011yescharacteristic
195C2×C3×D3—22no—
196C2×C3×D3—41no—
197C2×C3×D3—41no—
198C2×C3×D3—41no—
199C2×C3×D3—41no—
200C2×C3×D3—41no—
201C32×C4—21no—
202C32×C4—42no—
203C32×C4C2×SD1611yescharacteristic
204C32×C4#13511yescharacteristic
205C32×C4AGL(1,ℤ/8)11yescharacteristic
206C32×C4AGL(1,ℤ/8)11yescharacteristic
207C3×Dic3—41no—
208C3×Dic3—41no—
209C3×Dic3—81no—
210C2×#41—61no—
211C2×SD16—181no—
212C4⋊Q8—31no—
213C4⋊Q8D3211yescharacteristic
214#123—62no—
215C22⋊Q8—61no—
216#121—61no—
217C4×D4—31no—
218C4×D4—31no—
219C2×C4⋊C4—61no—
220#107—181no—
221#106—62no—
222C4⋊C8—91no—
223C4⋊C8—91no—
224#103—181no—
225#102—181no—
226C4×C8—31no—
227C23×C3—61no—
228C23×C3—61no—
229C22×D3—21no—
230C22×D3—21no—
231C3×Q8—41no—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×Q8—81no—
236C3×Q8—81no—
237C3×Q8—121no—
238C3×Q8—121no—
239C3×D4—32no—
240C3×D4—61no—
241C3×D4—61no—
242C3×D4—61no—
243C3×D4—124no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—42no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—124no—
253C2×C3×C4—121no—
254C2×C3×C4—121no—
255C2×C3×C4—121no—
256C2×C3×C4—121no—
257C2×C3×C4—121no—
258C2×C3×C4D3×D411yescharacteristic
259C2×C3×C4D3×D411yescharacteristic
260C2×C3×C4C2×#6711yescharacteristic
261C2×C3×C4C2×#6711yescharacteristic
262C2×C3×C4C2×#6711yescharacteristic
263C2×C3×C4C2×#6711yescharacteristic
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—41no—
267D12—21no—
268D12—21no—
269D12—21no—
270D12—44no—
271D12#21412yescomplemented
272C4×D3—44no—
273C4×D3—41no—
274C4×D3—41no—
275C4×D3—41no—
276C4×D3—41no—
277Dic6—41no—
278Dic6—41no—
279C3×C8—62no—
280C3⋊C8—22no—
281C3⋊C8—121no—
282C3⋊C8—121no—
283C3⋊C8—121no—
284C3⋊C8—121no—
285C3⋊C8—241no—
286C3⋊C8—241no—
287C2×C32#44911yescharacteristic
288C2×C32#45011yescharacteristic
289C2×C32#45611yescharacteristic
290C3×D3—42no—
291C3×D3—81no—
292#41—122no—
293#41—122no—
294C2×Q8—21no—
295C2×Q8—61no—
296C2×D4—31no—
297C2×D4—31no—
298C2×D4—62no—
299C22×C4—62no—
300C22×C4—61no—
301C22×C4—61no—
302SD16—362no—
303C2×C8—61no—
304C2×C8—181no—
305C2×C8—181no—
306C4⋊C4—21no—
307C4⋊C4—31no—
308C4⋊C4—31no—
309C4⋊C4—62no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4#62512yes—
313C22⋊C4—62no—
314C22⋊C4—61no—
315C22⋊C4—61no—
316C42C2×D3211yescharacteristic
317C22×C3—21no—
318C22×C3—62no—
319C22×C3—121no—
320C22×C3—121no—
321C22×C3—121no—
322C22×C3—121no—
323C22×C3—121no—
324C22×C3#93711yescharacteristic
325C22×C3#95311yescharacteristic
326C2×D3—22no—
327C2×D3—41no—
328C2×D3—41no—
329C2×D3—41no—
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—42no—
340C3×C4—42no—
341C3×C4—82no—
342C3×C4—121no—
343C3×C4—121no—
344C3×C4—241no—
345C3×C4D3×SD1611yescharacteristic
346C3×C4#91411yescharacteristic
347C3×C4#91411yescharacteristic
348C3×C4#91611yescharacteristic
349C3×C4#93211yescharacteristic
350C3×C4C2×#21411yescharacteristic
351C3×C4#94911yescharacteristic
352C3×C4#95011yescharacteristic
353Dic3—41no—
354Dic3—41no—
355Dic3—81no—
356C32#165411yescharacteristic, complemented, Sylow
357C23—61no—
358C23—61no—
359Q8—41no—
360Q8—41no—
361Q8—121no—
362Q8—121no—
363D4—32no—
364D4—61no—
365D4—61no—
366D4—61no—
367D4—124no—
368C2×C4—22no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—124no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4C2×#62511yescharacteristic
378C2×C4D3×#6711yescharacteristic
379C2×C4D3×#6711yescharacteristic
380C8—62no—
381C8—361no—
382C8—361no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—122no—
387C2×C3—241no—
388C2×C3#509511yescharacteristic
389C2×C3#516911yescharacteristic
390C2×C3#517211yescharacteristic
391C2×C3#521811yescharacteristic
392C2×C3#523811yescharacteristic
393C2×C3#524611yescharacteristic
394D3—42no—
395D3—81no—
396C22—62no—
397C22—121no—
398C22—121no—
399C22—121no—
400C22—121no—
401C22—121no—
402C22#6397611yescharacteristic
403C4—21no—
404C4—42no—
405C4—121no—
406C4—121no—
407C4—241no—
408C4D3×#21411yescharacteristic
409C4#6393411yescharacteristic
410C4#6393911yescharacteristic
411C4#6394711yescharacteristic
412C3—21no—
413C3#7134311yescharacteristic, complemented
414C3#7191311yescharacteristic, complemented
415C2—122no—
416C2—241no—
417C2#1059145811yescharacteristic
418C2#1059158611yescharacteristic
419C2#1059164811yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radicalC3×#5726C2
hypercenterC4⋊Q8D32
hypocenterC32#1654
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×#5080C2semidirect
3. #10590363C2semidirect
4. #10590269C2semidirect
5. #10589413C2semidirect
6. #10589081C2semidirect
7. #10589081C2semidirect
32. #5080D3semidirect
120. C3×D12SD16semidirect
271. D12#214semidirect
356. C32#1654semidirect
413. C3#71343semidirect
414. C3#71913semidirect

Maximal subgroups

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

subgroupindex
C3×#57262
C3×#50802
#105903632
#105902692
#105894132
#105890812
#719133
#713433