Finite Group Databasebeta

← #13160103

Subgroups — #13160103

Order 1728.

1276 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 158 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×#89022C211yesmaximal, characteristic, complemented
2C2×#88852C211yesmaximal, characteristic, complemented
3#11723516C211yesmaximal, characteristic, complemented
4C2×#88797C211yesmaximal, characteristic, complemented
5C3×#63643C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic, complemented
8#10592433—31nomaximal
9#10590967—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6291—21no—
15C2×#6291—21no—
16C2×#6289—21no—
17#89022—22no—
18#89022—22no—
19C2×C33×C8C2211yescharacteristic
20#88852—22no—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88804—21no—
24#88797—22no—
25C2×#3828—31no—
26C2×#3799—34no—
27#64139—31no—
28C2×#3745—31no—
29C2×#3745—34no—
30C2×#3745—64no—
31C2×#3744—31no—
32#63909—34no—
33C2×#3717—34no—
34C32×C22⋊C8—31no—
35#63648—31no—
36#63643D311yescharacteristic, complemented
37C3×(C2×D3)⋊C8D314yescomplemented
38C3×#816—34no—
39#63564—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3⋊(C3⋊D3)—21no—
42C22×C32×D3D411yescharacteristic
43C2×C33×C4C2311yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4⋊(C3⋊(C3⋊D3))D412yescomplemented
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C3×#634—21no—
51C3×#634D412yescomplemented
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55#6291—24no—
56#6291—42no—
57#6289—24no—
58C33×C8—21no—
59C32×C3⋊C8—21no—
60#4810—91no—
61#4734—94no—
62C22×C32×C4—31no—
63C2×#638—61no—
64C2×#638—61no—
65#3828—62no—
66#3828—62no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—34no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—34no—
71C2×C4⋊(C3⋊D3)—64no—
72C2×C4⋊(C3⋊D3)—64no—
73C2×C4×C3⋊D3—61no—
74C2×#634C2×D311yescharacteristic
75C2×C3×C4×D3C2×D314yes—
76C2×C3×Dic6—34no—
77C2×#626—64no—
78C2×#626—64no—
79C2×#624—64no—
80#3799—68no—
81#3799—68no—
82C2×C32×C8—24no—
83C2×C32×C8—31no—
84C2×C32×C8C2×D314yes—
85C2×C32×C8C2×D311yescharacteristic
86#3752—61no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88#3745—62no—
89#3745—68no—
90#3745—616no—
91#3744—62no—
92C3×C4⋊Dic3—34no—
93C2×C3×C3⋊C8C2×D314yes—
94#3722—64no—
95#3717—68no—
96C22×C33C2×D411yescharacteristic
97C2×C3⋊(C3⋊D3)—22no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C3⋊(C3⋊D3)—41no—
100C2×C3⋊(C3⋊D3)—41no—
101C2×C3⋊(C3⋊D3)—41no—
102C2×C3⋊(C3⋊D3)—41no—
103C2×C32×D3—21no—
104C2×C32×D3—21no—
105C2×C32×D3—21no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3×C3⋊Dic3—22no—
109C3×C3⋊Dic3—41no—
110C32×Dic3—21no—
111C2×#239—91no—
112C2×#235—94no—
113C2×#215—91no—
114#930—94no—
115C2×D24—94no—
116C2×D24—91no—
117C2×D24—184no—
118C2×#204—94no—
119#895—91no—
120C3×#103—91no—
121C3×C22⋊C8—34no—
122#821—94no—
123(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
124#816—34no—
125#809—91no—
126C23×C32—31no—
127C22×C3⋊D3—61no—
128C22×C3⋊D3—61no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3—64no—
131C22×C3⋊D3—124no—
132C22×C3⋊D3—124no—
133C22×C3×D3D1214yes—
134C2×C32×C4—24no—
135C2×C32×C4—31no—
136C2×C32×C4—61no—
137C2×C32×C4—61no—
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D311yescharacteristic
140#638—64no—
141#638—122no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3—21no—
144C4⋊(C3⋊D3)—32no—
145C4⋊(C3⋊D3)—38no—
146C4⋊(C3⋊D3)—68no—
147C4⋊(C3⋊D3)—68no—
148C4⋊(C3⋊D3)—61no—
149C4⋊(C3⋊D3)—61no—
150C4⋊(C3⋊D3)—64no—
151C4⋊(C3⋊D3)—61no—
152C4⋊(C3⋊D3)—64no—
153C4⋊(C3⋊D3)—64no—
154C4⋊(C3⋊D3)—124no—
155C4⋊(C3⋊D3)—124no—
156C4×C3⋊D3—64no—
157#634—21no—
158#634D1212yescomplemented
159C2×C3×Dic3—64no—
160C2×C3×Dic3—64no—
161C2×C3×Dic3D1214yes—
162C3×C4×D3—24no—
163C3×C4×D3—24no—
164C3×Dic6—38no—
165C3×Dic6—64no—
166#626—616no—
167#626—128no—
168#624—616no—
169C32×C8—28no—
170C32×C8—24no—
171C32×C8—21no—
172C32×C8—61no—
173C3×C3⋊C8—24no—
174#449—271noSylow
175C2×C33#12011yescharacteristic
176C2×C33#13511yescharacteristic
177C2×C33AGL(1,ℤ/8)11yescharacteristic
178C3⋊(C3⋊D3)—42no—
179C3⋊(C3⋊D3)—81no—
180C32×D3—41no—
181C2×C3×Q8—91no—
182C22×C3×C4—34no—
183C2×#67—184no—
184C2×#67—184no—
185#239—182no—
186#239—182no—
187#235—188no—
188#235—188no—
189C2×D12—94no—
190C2×D12—91no—
191C2×D12—94no—
192C2×D12—91no—
193C2×D12—184no—
194C2×D12—184no—
195C2×D12—181no—
196C2×D12—181no—
197C2×C4×D3—184no—
198C2×C4×D3—181no—
199C2×C4×D3C2×C3⋊D311yescharacteristic
200C2×Dic6—34no—
201C2×C3×C8—24no—
202C2×C3×C8—34no—
203C2×C3×C8D3214yes—
204C2×C3×C8C2×C3⋊D311yescharacteristic
205C3×C4⋊C4—91no—
206#215—182no—
207(C2×D3)⋊C4—184no—
208(C2×D3)⋊C4—181no—
209C4⋊Dic3—34no—
210C2×C3⋊C8C2×C3⋊D311yescharacteristic
211D24—188no—
212D24—182no—
213D24—1816no—
214#204—188no—
215C22×C32—24no—
216C22×C32—61no—
217C22×C32—61no—
218C22×C32—61no—
219C22×C32C2×D1214yes—
220C22×C32D3×D411yescharacteristic
221C2×C3⋊D3—62no—
222C2×C3⋊D3—68no—
223C2×C3⋊D3—128no—
224C2×C3⋊D3—121no—
225C2×C3⋊D3—121no—
226C2×C3⋊D3—128no—
227C2×C3⋊D3—124no—
228C2×C3⋊D3—121no—
229C2×C3⋊D3—121no—
230C2×C3⋊D3—121no—
231C2×C3⋊D3—124no—
232C2×C3⋊D3—124no—
233C2×C3⋊D3—124no—
234C2×C3⋊D3—124no—
235C2×C3⋊D3—244no—
236C2×C3⋊D3—244no—
237C2×C3⋊D3—244no—
238C2×C3⋊D3—244no—
239C2×C3×D3—24no—
240C2×C3×D3—24no—
241C2×C3×D3—24no—
242C32×C4—24no—
243C32×C4—24no—
244C32×C4—61no—
245C32×C4C2×D1211yescharacteristic
246C32×C4D3×D414yes—
247C32×C4D3×D414yes—
248C32×C4D3×D411yescharacteristic
249C3⋊Dic3—22no—
250C3⋊Dic3—41no—
251C3×Dic3—24no—
252C3×Dic3—68no—
253C3×Dic3—124no—
254C2×#41—271no—
255C2×SD16—271no—
256C2×D8—271no—
257#121—271no—
258#103—91no—
259#102—271no—
260C22⋊C8—31no—
261C33#44911yescharacteristic, complemented, Sylow
262C23×C3—34no—
263C22×D3—184no—
264C22×D3—181no—
265C22×D3—184no—
266C22×D3—181no—
267C22×D3—364no—
268C22×D3—364no—
269C22×D3C4⋊(C3⋊D3)11yescharacteristic
270C3×Q8—92no—
271C3×Q8—181no—
272C2×C3×C4—24no—
273C2×C3×C4—34no—
274C2×C3×C4—64no—
275C2×C3×C4—64no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4C2×D3214yes—
279C2×C3×C4C22×C3⋊D311yescharacteristic
280#67—1816no—
281#67—368no—
282C2×Dic3—64no—
283C2×Dic3—64no—
284C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
285D12—98no—
286D12—92no—
287D12—188no—
288D12—188no—
289D12—184no—
290D12—184no—
291D12—184no—
292D12—184no—
293D12—181no—
294D12—181no—
295D12—181no—
296D12—364no—
297D12—364no—
298D12—362no—
299C4×D3—21no—
300C4×D3—21no—
301C4×D3—184no—
302C4×D3—1816no—
303Dic6—38no—
304Dic6—64no—
305C3×C8—28no—
306C3×C8—24no—
307C3×C8—21no—
308C3×C8—64no—
309C3⋊C8—21no—
310C2×C32—24no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—121no—
314C2×C32#88214yes—
315C2×C32#88211yescharacteristic
316C2×C32#90414yes—
317C2×C32#90814yes—
318C2×C32#91411yescharacteristic
319C2×C32#91911yescharacteristic
320C3⋊D3—122no—
321C3⋊D3—128no—
322C3⋊D3—248no—
323C3⋊D3—248no—
324C3⋊D3—244no—
325C3⋊D3—241no—
326C3×D3—44no—
327#41—542no—
328#41—542no—
329C2×Q8—91no—
330C2×D4—271no—
331C2×D4—271no—
332C2×D4—541no—
333C2×D4—541no—
334C22×C4—31no—
335C22×C4—541no—
336SD16—542no—
337D8—542no—
338C2×C8—31no—
339C2×C8D3×C3⋊D311yescharacteristic
340C4⋊C4—91no—
341C22⋊C4—541no—
342C22×C3—24no—
343C22×C3—64no—
344C22×C3—64no—
345C22×C3—64no—
346C22×C3D3×D1214yes—
347C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
348C2×D3—21no—
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—182no—
352C2×D3—188no—
353C2×D3—368no—
354C2×D3—364no—
355C2×D3—364no—
356C2×D3—361no—
357C2×D3—361no—
358C2×D3—368no—
359C2×D3—364no—
360C2×D3—364no—
361C2×D3—361no—
362C2×D3—361no—
363C2×D3—361no—
364C2×D3—364no—
365C2×D3—724no—
366C2×D3—724no—
367C2×D3—724no—
368C2×D3—724no—
369C3×C4—24no—
370C3×C4—24no—
371C3×C4—64no—
372C3×C4—182no—
373C3×C4—361no—
374C3×C4D3×D1214yes—
375C3×C4#381114yes—
376C3×C4D4×C3⋊D311yescharacteristic
377C3×C4D4×C3⋊D311yescharacteristic
378Dic3—21no—
379Dic3—68no—
380Dic3—124no—
381C32—24no—
382C32#473414yescomplemented
383C32#481011yescharacteristic, complemented
384C23—31no—
385C23—541no—
386C23—541no—
387Q8—92no—
388Q8—181no—
389D4—272no—
390D4—544no—
391D4—541no—
392D4—541no—
393D4—541no—
394D4—1082no—
395C2×C4—31no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—544no—
401C2×C4C2×D3×C3⋊D311yescharacteristic
402C8—21no—
403C8—61no—
404C2×C3—24no—
405C2×C3—24no—
406C2×C3—24no—
407C2×C3—124no—
408C2×C3#6379414yes—
409C2×C3#6380814yes—
410C2×C3#6392314yes—
411C2×C3#6409111yescharacteristic
412C2×C3#6411311yescharacteristic
413C2×C3#6411711yescharacteristic
414D3—41no—
415D3—362no—
416D3—368no—
417D3—728no—
418D3—728no—
419D3—721no—
420D3—724no—
421C22—61no—
422C22—61no—
423C22—61no—
424C22—542no—
425C22—1081no—
426C22—1081no—
427C22—1081no—
428C22—1081no—
429C22—1081no—
430C22D3×C4⋊(C3⋊D3)11yescharacteristic
431C4—61no—
432C4—182no—
433C4—361no—
434C4C3⋊D3×D1211yescharacteristic
435C4#8903911yescharacteristic
436C3—24no—
437C3#1059096714yescomplemented
438C3#1059243311yescharacteristic, complemented
439C2—121no—
440C2—1082no—
441C2—2161no—
442C2#1172331511yescharacteristic
443C2#1172333711yescharacteristic
444C2#1172354211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#89022C2semidirect
2. C2×#88852C2semidirect
3. #11723516C2semidirect
4. C2×#88797C2semidirect
5. C3×#63643C2semidirect
6. C32×(C2×D3)⋊C8C2semidirect
7. #11722036C2semidirect
36. #63643D3semidirect
37. C3×(C2×D3)⋊C8D3semidirect
47. C4⋊(C3⋊(C3⋊D3))D4semidirect
51. C3×#634D4semidirect
123. (C2×D3)⋊C8C3⋊D3semidirect
158. #634D12semidirect
261. C33#449semidirect
382. C32#4734semidirect
383. C32#4810semidirect
437. C3#10590967semidirect
438. C3#10592433semidirect

Maximal subgroups

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

subgroupindex
C2×#890222
C2×#888522
#117235162
C2×#887972
C3×#636432
C32×(C2×D3)⋊C82
#117220362
#105924333
#105909673