Finite Group Databasebeta

← #13159449

Subgroups — #13159449

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
2C3×#64104C211yesmaximal, characteristic, complemented
3C2×#88793C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592509—31nomaximal
9#10590615—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6292—21no—
16C2×#6289—21no—
17#89018—22no—
18#89018—22no—
19C33×C4⋊C4C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×#3752—21no—
22C2×C32×C3⋊C8C2211yescharacteristic
23#88793—22no—
24#88791—22no—
25C2×#3832—31no—
26C2×#3799—34no—
27C2×#3755—31no—
28C2×#3745—31no—
29#64104D311yescharacteristic, complemented
30C3×#895—34no—
31C2×#3715—34no—
32C2×#3714—34no—
33C32×#103—31no—
34#63647—31no—
35#63636—31no—
36#63636—34no—
37#63636—64no—
38C3×#810D314yescomplemented
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))D412yescomplemented
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—42no—
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C2×C32×Dic3—21no—
54C32×Dic6—21no—
55C32×Dic6D412yescomplemented
56#6292—24no—
57#6289—24no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5037—91no—
61#4810—94no—
62C2×C32×Q8—31no—
63#3832—62no—
64#3832—62no—
65C2×C4⋊(C3⋊D3)—21no—
66C2×C4⋊(C3⋊D3)—31no—
67C2×C4⋊(C3⋊D3)—34no—
68C2×C4⋊(C3⋊D3)—61no—
69C2×C4⋊(C3⋊D3)—64no—
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×C4×C3⋊D3—61no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×C3×D12—34no—
74C2×C3×D12—64no—
75C2×C3×C4×D3—34no—
76C2×C3×Dic6C2×D314yes—
77C2×#626—64no—
78C2×#624—64no—
79#3799—68no—
80#3799—68no—
81C2×C32×C8—31no—
82C32×C4⋊C4—24no—
83C32×C4⋊C4C2×D314yes—
84C32×C4⋊C4C2×D311yescharacteristic
85#3755—62no—
86#3752—21no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—34no—
89C2×C3⋊(C3⋊C8)—64no—
90#3745—62no—
91C3×(C2×D3)⋊C4—64no—
92C2×C3×C3⋊C8C2×D314yes—
93#3715—68no—
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×C3×C3⋊D3—41no—
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3—41no—
105C33×C4—41no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3×C3⋊Dic3—21no—
109C32×Dic3—22no—
110C2×#239—94no—
111C2×#235—91no—
112C3×#121—91no—
113C2×#215—94no—
114C2×#214—91no—
115C2×#213—91no—
116C2×D24—94no—
117#895—34no—
118C3×#103—34no—
119#830—91no—
120(C2×D3)⋊C8—94no—
121#810C3⋊D311yescharacteristic, complemented
122#809—94no—
123#809—91no—
124#809—184no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—61no—
127C22×C3⋊D3—64no—
128C22×C3⋊D3—124no—
129C22×C3⋊D3#6711yescharacteristic
130C22×C3×D3—34no—
131C22×C3×D3—64no—
132C32×Q8—32no—
133C32×Q8—61no—
134C2×C32×C4—21no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—44no—
138C2×C32×C4—61no—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3#6711yescharacteristic
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—32no—
145C4⋊(C3⋊D3)—38no—
146C4⋊(C3⋊D3)—42no—
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—64no—
158C3×D12—64no—
159C3×D12—128no—
160C3×C4×D3—64no—
161C3×C4×D3—64no—
162C3×Dic6—24no—
163C3×Dic6D1218yescomplemented
164#626—616no—
165#624—616no—
166C32×C8—61no—
167C3⋊(C3⋊C8)—61no—
168C3⋊(C3⋊C8)—64no—
169C3⋊(C3⋊C8)—124no—
170C3×C3⋊C8—24no—
171#449—271noSylow
172C2×C33#12011yescharacteristic
173C2×C33#13511yescharacteristic
174C2×C33AGL(1,ℤ/8)11yescharacteristic
175C3⋊(C3⋊D3)—42no—
176C3×C3⋊D3—41no—
177C3×C3⋊D3—81no—
178C2×C3×Q8—34no—
179C2×C3×D4—91no—
180C2×C3×D4—181no—
181C22×C3×C4—91no—
182C2×#67—181no—
183#239—188no—
184#239—188no—
185#235—182no—
186#235—182no—
187C2×D12—34no—
188C2×D12—64no—
189C2×D12—94no—
190C2×D12—91no—
191C2×D12—184no—
192C2×D12—184no—
193C2×C4×D3—34no—
194C2×C4×D3—181no—
195C2×C4×D3—184no—
196C2×Dic6C2×C3⋊D311yescharacteristic
197C2×C3×C8—34no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4D3214yes—
200C3×C4⋊C4C2×C3⋊D311yescharacteristic
201C3×C22⋊C4—181no—
202#215—188no—
203#214—182no—
204#213—182no—
205(C2×D3)⋊C4—64no—
206C2×C3⋊C8—94no—
207C2×C3⋊C8—91no—
208C2×C3⋊C8—184no—
209C2×C3⋊C8C2×C3⋊D311yescharacteristic
210D24—188no—
211C22×C32—24no—
212C22×C32D3×D414yes—
213C22×C32C2×#6711yescharacteristic
214C2×C3⋊D3—21no—
215C2×C3⋊D3—21no—
216C2×C3⋊D3—21no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—41no—
219C2×C3⋊D3—41no—
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—64no—
230C2×C3×D3—64no—
231C2×C3×D3—64no—
232C2×C3×D3—124no—
233C2×C3×D3—124no—
234C2×C3×D3—124no—
235C32×C4—24no—
236C32×C4—24no—
237C32×C4—41no—
238C32×C4—44no—
239C32×C4—62no—
240C32×C4—84no—
241C32×C4C2×D1214yes—
242C32×C4D3×D414yes—
243C32×C4D3×D411yescharacteristic
244C32×C4D3×D411yescharacteristic
245C3⋊Dic3—21no—
246C3×Dic3—28no—
247C3×Dic3—64no—
248C2×#41—271no—
249C2×SD16—271no—
250C2×D8—271no—
251#121—91no—
252#103—31no—
253#102—271no—
254C22⋊C8—271no—
255C33#44911yescharacteristic, complemented, Sylow
256C23×C3—91no—
257C23×C3—181no—
258C22×D3—34no—
259C22×D3—64no—
260C22×D3—184no—
261C22×D3—181no—
262C22×D3—364no—
263C3×Q8—38no—
264C3×Q8—64no—
265C3×D4—181no—
266C3×D4—181no—
267C3×D4—362no—
268C2×C3×C4—24no—
269C2×C3×C4—21no—
270C2×C3×C4—24no—
271C2×C3×C4—44no—
272C2×C3×C4—64no—
273C2×C3×C4—91no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4C2×D3214yes—
277C2×C3×C4C22×C3⋊D311yescharacteristic
278#67—184no—
279C2×Dic3—21no—
280C2×Dic3—34no—
281D12—64no—
282D12—64no—
283D12—98no—
284D12—92no—
285D12—128no—
286D12—188no—
287D12—1816no—
288D12—184no—
289D12—181no—
290D12—364no—
291C4×D3—64no—
292C4×D3—64no—
293C4×D3—184no—
294C4×D3—1816no—
295Dic6—21no—
296Dic6C4⋊(C3⋊D3)12yescomplemented
297C3×C8—64no—
298C3⋊C8—21no—
299C3⋊C8—184no—
300C3⋊C8—181no—
301C3⋊C8—364no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32—24no—
305C2×C32#88214yes—
306C2×C32#91414yes—
307C2×C32#91914yes—
308C2×C32#93211yescharacteristic
309C2×C32#93711yescharacteristic
310C2×C32#95011yescharacteristic
311C3⋊D3—41no—
312C3⋊D3—81no—
313C3⋊D3—122no—
314C3⋊D3—128no—
315C3⋊D3—248no—
316C3×D3—124no—
317C3×D3—244no—
318#41—542no—
319#41—542no—
320C2×Q8—31no—
321C2×D4—91no—
322C2×D4—181no—
323C2×D4—271no—
324C2×D4—541no—
325C22×C4—91no—
326C22×C4—541no—
327SD16—542no—
328D8—542no—
329C2×C8—31no—
330C2×C8—271no—
331C4⋊C4D3×C3⋊D311yescharacteristic
332C22⋊C4—181no—
333C22×C3—24no—
334C22×C3—181no—
335C22×C3—181no—
336C22×C3—181no—
337C22×C3—361no—
338C22×C3—361no—
339C22×C3—361no—
340C22×C3D3×#6714yes—
341C22×C3D4×C3⋊D311yescharacteristic
342C2×D3—64no—
343C2×D3—64no—
344C2×D3—64no—
345C2×D3—124no—
346C2×D3—124no—
347C2×D3—124no—
348C2×D3—182no—
349C2×D3—188no—
350C2×D3—368no—
351C2×D3—361no—
352C2×D3—361no—
353C2×D3—364no—
354C2×D3—364no—
355C2×D3—724no—
356C2×D3—724no—
357C3×C4—24no—
358C3×C4—24no—
359C3×C4—41no—
360C3×C4—44no—
361C3×C4—68no—
362C3×C4—84no—
363C3×C4—181no—
364C3×C4D3×D1214yes—
365C3×C4#380214yes—
366C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
367C3×C4D4×C3⋊D311yescharacteristic
368Dic3—22no—
369Dic3—64no—
370C32—24no—
371C32#481014yescomplemented
372C32#503711yescharacteristic, complemented
373C23—91no—
374C23—181no—
375C23—541no—
376Q8—32no—
377Q8—61no—
378D4—181no—
379D4—181no—
380D4—272no—
381D4—362no—
382D4—544no—
383D4—541no—
384C2×C4—21no—
385C2×C4—61no—
386C2×C4—91no—
387C2×C4—181no—
388C2×C4—181no—
389C2×C4—544no—
390C2×C4C2×D3×C3⋊D311yescharacteristic
391C8—61no—
392C8—541no—
393C2×C3—24no—
394C2×C3—24no—
395C2×C3—24no—
396C2×C3—361no—
397C2×C3—721no—
398C2×C3#6392214yes—
399C2×C3#6393414yes—
400C2×C3#6394914yes—
401C2×C3#6409111yescharacteristic
402C2×C3#6412311yescharacteristic
403C2×C3#6412811yescharacteristic
404D3—124no—
405D3—244no—
406D3—368no—
407D3—362no—
408D3—728no—
409C22—181no—
410C22—181no—
411C22—181no—
412C22—361no—
413C22—361no—
414C22—361no—
415C22—542no—
416C22—1081no—
417C22—1081no—
418C22C3⋊D3×#6711yescharacteristic
419C4—41no—
420C4—62no—
421C4—181no—
422C4D3×C4⋊(C3⋊D3)11yescharacteristic
423C4#8902611yescharacteristic
424C3—24no—
425C3#1059061514yescomplemented
426C3#1059250911yescharacteristic, complemented
427C2—361no—
428C2—721no—
429C2—1082no—
430C2#1172354111yescharacteristic
431C2#1172356211yescharacteristic
432C2#1172358511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#89018C2semidirect
2. C3×#64104C2semidirect
3. C2×#88793C2semidirect
4. C2×#88791C2semidirect
5. #11722179C2semidirect
6. C32×#810C2semidirect
7. #11722023C2semidirect
29. #64104D3semidirect
38. C3×#810D3semidirect
46. C4⋊(C3⋊(C3⋊D3))D4semidirect
55. C32×Dic6D4semidirect
121. #810C3⋊D3semidirect
163. C3×Dic6D12semidirect
255. C33#449semidirect
296. Dic6C4⋊(C3⋊D3)semidirect
371. C32#4810semidirect
372. C32#5037semidirect
425. C3#10590615semidirect
426. C3#10592509semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
C3×#641042
C2×#887932
C2×#887912
#117221792
C32×#8102
#117220232
#105925093
#105906153