Finite Group Databasebeta

← #13160094

Subgroups — #13160094

Order 1728.

1172 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1172 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×#89018C211yesmaximal, characteristic, complemented
2C2×C32×#204C211yesmaximal, characteristic, complemented
3#11723522C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5#11722191C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7#11722038C211yesmaximal, characteristic, complemented
8#10592871—31nomaximal
9#10590823—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6292—21no—
15C2×#6290—21no—
16C2×#6289—21no—
17#89018—22no—
18#89018—22no—
19C2×C33×C8C2211yescharacteristic
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C32×#204—22no—
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88790—22no—
25C2×#3832—31no—
26C2×#3799—34no—
27C2×C32×SD16—31no—
28#64147—31no—
29C2×#3753—31no—
30C2×C3×#204D314yescomplemented
31#63908—34no—
32C2×#3714—34no—
33#63662—31no—
34#63648—31no—
35#63648—34no—
36#63648—64no—
37#63647D311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—34no—
39#63566—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))D412yescomplemented
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C2×C32×Dic3—21no—
51C32×D12—21no—
52C32×D12—21no—
53C32×Dic6—21no—
54C32×Dic6D412yescomplemented
55#6292—24no—
56#6290—42no—
57#6289—24no—
58C33×C8—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5172—94no—
61#4734—91no—
62C2×C32×Q8—31no—
63C2×C32×D4—31no—
64C2×#638—61no—
65#3832—62no—
66#3832—62no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—34no—
69C2×C4⋊(C3⋊D3)—61no—
70C2×C4⋊(C3⋊D3)—64no—
71C2×C4×C3⋊D3—61no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×C3×D12C2×D314yes—
74C2×C3×C4×D3—34no—
75C2×C3×Dic6C2×D314yes—
76C2×#626—64no—
77C2×#625—64no—
78C2×#624—64no—
79#3799—68no—
80#3799—68no—
81C32×SD16—62no—
82C2×C32×C8—24no—
83C2×C32×C8C2×D314yes—
84C2×C32×C8C2×D311yescharacteristic
85#3757—61no—
86#3753—62no—
87C4⋊(C3⋊Dic3)—31no—
88C4⋊(C3⋊Dic3)—34no—
89C4⋊(C3⋊Dic3)—64no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C2×C3×C3⋊C8—34no—
92C3×#204—28no—
93(C2×D3)⋊Dic3—64no—
94#3714—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3⋊(C3⋊D3)—22no—
97C2×C3⋊(C3⋊D3)—41no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C2×C32×D3—41no—
103C2×C32×D3—41no—
104C2×C32×D3—41no—
105C33×C4C2×D411yescharacteristic
106C33×C4C2×D411yescharacteristic
107C3⋊(C3⋊Dic3)—41no—
108C3×C3⋊Dic3—21no—
109C32×Dic3—22no—
110C2×#239—94no—
111C2×#235—91no—
112C2×C3×SD16—34no—
113#938—94no—
114C2×#213—94no—
115#930—91no—
116C2×D24—91no—
117C2×#204C3⋊D311yescharacteristic, complemented
118C3×C22⋊C8—91no—
119#835—94no—
120#821—94no—
121#821—91no—
122#821—184no—
123(C2×D3)⋊C8—34no—
124#816—91no—
125C23×C32—61no—
126C22×C3⋊D3—61no—
127C22×C3⋊D3—64no—
128C22×C3⋊D3—124no—
129C22×C3⋊D3D1211yescharacteristic
130C22×C3×D3—24no—
131C22×C3×D3—34no—
132C32×Q8—32no—
133C32×Q8—61no—
134C32×D4—61no—
135C32×D4—61no—
136C2×C32×C4—24no—
137C2×C32×C4—61no—
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D311yescharacteristic
140#638—122no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—124no—
144C2×C3⋊Dic3D1211yescharacteristic
145C4⋊(C3⋊D3)—32no—
146C4⋊(C3⋊D3)—38no—
147C4⋊(C3⋊D3)—68no—
148C4⋊(C3⋊D3)—64no—
149C4⋊(C3⋊D3)—61no—
150C4⋊(C3⋊D3)—64no—
151C4⋊(C3⋊D3)—124no—
152C4×C3⋊D3—21no—
153C4×C3⋊D3—21no—
154C4×C3⋊D3—64no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—34no—
157C3×D12—24no—
158C3×D12—24no—
159C3×C4×D3—64no—
160C3×C4×D3—64no—
161C3×Dic6—24no—
162C3×Dic6#6718yescomplemented
163#626—616no—
164#625—128no—
165#624—616no—
166C32×C8—21no—
167C32×C8—24no—
168C32×C8—44no—
169C3⋊(C3⋊C8)—21no—
170C3×C3⋊C8—64no—
171#449—271noSylow
172C2×C33#12011yescharacteristic
173C2×C33#13511yescharacteristic
174C2×C33AGL(1,ℤ/8)11yescharacteristic
175C3⋊(C3⋊D3)—42no—
176C3×C3⋊D3—41no—
177C32×D3—81no—
178C2×C3×Q8—34no—
179C2×C3×D4—34no—
180C22×C3×C4—91no—
181C2×#67—181no—
182C2×#67—181no—
183C2×#67—184no—
184#239—188no—
185#239—188no—
186#235—182no—
187#235—182no—
188C2×D12—94no—
189C2×D12—91no—
190C2×D12—184no—
191C2×D12—184no—
192C2×D12C2×C3⋊D311yescharacteristic
193C2×C4×D3—34no—
194C2×C4×D3—181no—
195C2×C4×D3—184no—
196C2×Dic6C2×C3⋊D311yescharacteristic
197C3×SD16—68no—
198C2×C3×C8—24no—
199C2×C3×C8—91no—
200C2×C3×C8D3214yes—
201C2×C3×C8C2×C3⋊D311yescharacteristic
202#217—184no—
203#213—188no—
204(C2×D3)⋊C4—181no—
205C4⋊Dic3—94no—
206C4⋊Dic3—91no—
207C4⋊Dic3—184no—
208C2×C3⋊C8—34no—
209D24—182no—
210#204—22no—
211C22×C32—24no—
212C22×C32—121no—
213C22×C32—121no—
214C22×C32—121no—
215C22×C32C2×D1211yescharacteristic
216C22×C32D3×D414yes—
217C2×C3⋊D3—21no—
218C2×C3⋊D3—21no—
219C2×C3⋊D3—21no—
220C2×C3⋊D3—62no—
221C2×C3⋊D3—68no—
222C2×C3⋊D3—128no—
223C2×C3⋊D3—121no—
224C2×C3⋊D3—121no—
225C2×C3⋊D3—124no—
226C2×C3⋊D3—124no—
227C2×C3⋊D3—244no—
228C2×C3⋊D3—244no—
229C2×C3×D3—44no—
230C2×C3×D3—44no—
231C2×C3×D3—44no—
232C2×C3×D3—64no—
233C2×C3×D3—64no—
234C2×C3×D3—64no—
235C32×C4—24no—
236C32×C4—24no—
237C32×C4—62no—
238C32×C4D3×D414yes—
239C32×C4D3×D411yescharacteristic
240C32×C4D3×D411yescharacteristic
241C32×C4C2×#6714yes—
242C3⋊Dic3—21no—
243C3⋊Dic3—121no—
244C3⋊Dic3—124no—
245C3⋊Dic3—244no—
246C3×Dic3—28no—
247C3×Dic3—64no—
248C2×#41—271no—
249C2×SD16—31no—
250C2×D8—271no—
251#121—271no—
252#103—271no—
253#102—271no—
254C22⋊C8—91no—
255C33#44911yescharacteristic, complemented, Sylow
256C23×C3—64no—
257C23×C3—91no—
258C22×D3—21no—
259C22×D3—34no—
260C22×D3—184no—
261C22×D3—181no—
262C22×D3—364no—
263C3×Q8—38no—
264C3×Q8—64no—
265C3×D4—64no—
266C3×D4—64no—
267C2×C3×C4—24no—
268C2×C3×C4—64no—
269C2×C3×C4—91no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4C2×D3214yes—
273C2×C3×C4C22×C3⋊D311yescharacteristic
274#67—184no—
275#67—362no—
276#67—368no—
277C2×Dic3—21no—
278C2×Dic3—34no—
279C2×Dic3—184no—
280C2×Dic3—181no—
281C2×Dic3—364no—
282D12—21no—
283D12—21no—
284D12—98no—
285D12—92no—
286D12—188no—
287D12—1816no—
288D12—184no—
289D12—181no—
290D12—364no—
291C4×D3—64no—
292C4×D3—64no—
293C4×D3—184no—
294C4×D3—1816no—
295Dic6—21no—
296Dic6#63812yescomplemented
297C3×C8—24no—
298C3×C8—21no—
299C3×C8—44no—
300C3×C8—181no—
301C3⋊C8—64no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32—24no—
305C2×C32—241no—
306C2×C32#88211yescharacteristic
307C2×C32#90411yescharacteristic
308C2×C32#90811yescharacteristic
309C2×C32#91414yes—
310C2×C32#91614yes—
311C2×C32#93714yes—
312C3⋊D3—41no—
313C3⋊D3—122no—
314C3⋊D3—128no—
315C3⋊D3—248no—
316C3×D3—84no—
317C3×D3—124no—
318#41—542no—
319#41—542no—
320C2×Q8—31no—
321C2×D4—31no—
322C2×D4—271no—
323C2×D4—541no—
324C2×D4—541no—
325C22×C4—91no—
326C22×C4—541no—
327SD16—62no—
328D8—542no—
329C2×C8—91no—
330C2×C8D3×C3⋊D311yescharacteristic
331C4⋊C4—271no—
332C22⋊C4—541no—
333C22×C3—24no—
334C22×C3—124no—
335C22×C3—124no—
336C22×C3—124no—
337C22×C3—181no—
338C22×C3—181no—
339C22×C3—181no—
340C22×C3D3×D1214yes—
341C22×C3D4×C3⋊D311yescharacteristic
342C2×D3—41no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—64no—
346C2×D3—64no—
347C2×D3—64no—
348C2×D3—182no—
349C2×D3—188no—
350C2×D3—368no—
351C2×D3—364no—
352C2×D3—364no—
353C2×D3—361no—
354C2×D3—361no—
355C2×D3—724no—
356C2×D3—724no—
357C3×C4—24no—
358C3×C4—24no—
359C3×C4—68no—
360C3×C4—181no—
361C3×C4#380214yes—
362C3×C4D3×#6714yes—
363C3×C4D4×C3⋊D311yescharacteristic
364C3×C4C2×#63811yescharacteristic
365Dic3—22no—
366Dic3—64no—
367Dic3—364no—
368Dic3—361no—
369Dic3—724no—
370C32—24no—
371C32#473411yescharacteristic, complemented
372C32#517214yescomplemented
373C23—61no—
374C23—91no—
375C23—541no—
376Q8—32no—
377Q8—61no—
378D4—61no—
379D4—61no—
380D4—272no—
381D4—544no—
382D4—541no—
383D4—1082no—
384C2×C4—61no—
385C2×C4—91no—
386C2×C4—181no—
387C2×C4—181no—
388C2×C4—544no—
389C2×C4—541no—
390C2×C4C2×D3×C3⋊D311yescharacteristic
391C8—21no—
392C8—181no—
393C2×C3—24no—
394C2×C3—24no—
395C2×C3—24no—
396C2×C3—244no—
397C2×C3—361no—
398C2×C3#6379414yes—
399C2×C3#6380614yes—
400C2×C3#6392214yes—
401C2×C3#6412311yescharacteristic
402C2×C3#6412511yescharacteristic
403C2×C3#6414611yescharacteristic
404D3—81no—
405D3—124no—
406D3—368no—
407D3—362no—
408D3—728no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22—181no—
413C22—181no—
414C22—181no—
415C22—542no—
416C22—1081no—
417C22—1081no—
418C22C3⋊D3×D1211yescharacteristic
419C4—62no—
420C4—181no—
421C4—1081no—
422C4#8902611yescharacteristic
423C4D3×#63811yescharacteristic
424C3—24no—
425C3#1059082314yescomplemented
426C3#1059287111yescharacteristic, complemented
427C2—241no—
428C2—361no—
429C2—1082no—
430C2#1172331911yescharacteristic
431C2#1172333811yescharacteristic
432C2#1172354011yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
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×#89018C2semidirect
2. C2×C32×#204C2semidirect
3. #11723522C2semidirect
4. C2×#88790C2semidirect
5. #11722191C2semidirect
6. C3×#63647C2semidirect
7. #11722038C2semidirect
30. C2×C3×#204D3semidirect
37. #63647D3semidirect
46. C4⋊(C3⋊(C3⋊D3))D4semidirect
54. C32×Dic6D4semidirect
117. C2×#204C3⋊D3semidirect
162. C3×Dic6#67semidirect
255. C33#449semidirect
296. Dic6#638semidirect
371. C32#4734semidirect
372. C32#5172semidirect
425. C3#10590823semidirect
426. C3#10592871semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
C2×C32×#2042
#117235222
C2×#887902
#117221912
C3×#636472
#117220382
#105928713
#105908233