Finite Group Databasebeta

← #13159454

Subgroups — #13159454

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×#89022C211yesmaximal, characteristic, complemented
2C32×#895C211yesmaximal, characteristic, complemented
3C2×#88792C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C3×#63637C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592736—31nomaximal
9#10590615—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12—21no—
13C2×C32×D12C2211yescharacteristic
14C2×C32×C4×D3C2211yescharacteristic
15C2×#6291—21no—
16C2×#6289—21no—
17#89022—22no—
18#89022—22no—
19C33×C4⋊C4C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C32×(C2×D3)⋊C4—21no—
23#88792—22no—
24#88790—22no—
25C2×#3828—31no—
26C2×#3799—34no—
27C32×#121—31no—
28C2×#3754—31no—
29C2×#3753—31no—
30C3×#895D314yescomplemented
31C2×#3715—34no—
32C2×#3714—34no—
33#63657—31no—
34#63637D311yescharacteristic, complemented
35#63636—31no—
36#63636—34no—
37#63636—64no—
38C3×#810—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C32×D3—21no—
42C22×C32×D3D411yescharacteristic
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))D412yescomplemented
47C2×C3×C3⋊Dic3—21no—
48C3×#634—21no—
49C3×#634D412yescomplemented
50C2×C32×Dic3D411yescharacteristic
51C32×D12—21no—
52C32×D12—21no—
53C32×D12—42no—
54C32×C4×D3—21no—
55C32×C4×D3—21no—
56#6291—24no—
57#6289—24no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5037—94no—
61#4810—91no—
62C2×C32×D4—31no—
63C2×C32×D4—61no—
64C22×C32×C4—31no—
65C2×#638—61no—
66#3828—62no—
67#3828—62no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—34no—
70C2×C4⋊(C3⋊D3)—64no—
71C2×C4×C3⋊D3—61no—
72C2×#634C2×D311yescharacteristic
73C2×C3×D12—24no—
74C2×C3×D12C2×D314yes—
75C2×C3×C4×D3C2×D314yes—
76C2×C3×Dic6—34no—
77C2×#626—64no—
78C2×#624—64no—
79#3799—68no—
80#3799—68no—
81C32×C4⋊C4—24no—
82C32×C4⋊C4C2×D314yes—
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C22⋊C4—61no—
85#3754—62no—
86#3753—62no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—34no—
89C2×C3⋊(C3⋊C8)—64no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C3×(C2×D3)⋊C4—24no—
92C2×C3×C3⋊C8—34no—
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×C32×D3—21no—
100C2×C32×D3—21no—
101C2×C32×D3—21no—
102C2×C32×D3—41no—
103C2×C32×D3—41no—
104C2×C32×D3—41no—
105C33×C4—41no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3×C3⋊Dic3—22no—
109C32×Dic3—21no—
110C2×#239—91no—
111C2×#235—94no—
112C3×#121—34no—
113C2×#215—91no—
114C2×#214—94no—
115C2×#213—94no—
116C2×D24—91no—
117#895C3⋊D311yescharacteristic, complemented
118C3×#103—91no—
119#830—94no—
120(C2×D3)⋊C8—91no—
121#810—34no—
122#809—94no—
123#809—91no—
124#809—184no—
125C23×C32—31no—
126C23×C32—61no—
127C22×C3⋊D3—61no—
128C22×C3⋊D3—64no—
129C22×C3⋊D3—124no—
130C22×C3×D3—24no—
131C22×C3×D3#6714yes—
132C32×D4—61no—
133C32×D4—61no—
134C32×D4—122no—
135C2×C32×C4—21no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—31no—
139C2×C32×C4—44no—
140C2×C32×C4—61no—
141C2×C32×C4—61no—
142C2×C32×C4C22×D314yes—
143C2×C32×C4C22×D311yescharacteristic
144#638—64no—
145C2×C3⋊Dic3—21no—
146C4⋊(C3⋊D3)—32no—
147C4⋊(C3⋊D3)—38no—
148C4⋊(C3⋊D3)—68no—
149C4⋊(C3⋊D3)—61no—
150C4⋊(C3⋊D3)—64no—
151C4⋊(C3⋊D3)—124no—
152C4×C3⋊D3—64no—
153#634—21no—
154#634D1212yescomplemented
155C2×C3×Dic3—64no—
156C2×C3×Dic3#6714yes—
157C3×D12—24no—
158C3×D12—24no—
159C3×D12—48no—
160C3×C4×D3—24no—
161C3×C4×D3—24no—
162C3×Dic6—38no—
163C3×Dic6—64no—
164#626—616no—
165#624—616no—
166C3⋊(C3⋊C8)—21no—
167C3⋊(C3⋊C8)—61no—
168C3⋊(C3⋊C8)—64no—
169C3⋊(C3⋊C8)—124no—
170C3×C3⋊C8—64no—
171#449—271noSylow
172C2×C33#12011yescharacteristic
173C2×C33#13511yescharacteristic
174C2×C33AGL(1,ℤ/8)11yescharacteristic
175C3⋊(C3⋊D3)—42no—
176C32×D3—41no—
177C32×D3—81no—
178C2×C3×Q8—91no—
179C2×C3×D4—34no—
180C2×C3×D4—64no—
181C22×C3×C4—34no—
182C2×#67—184no—
183#239—182no—
184#239—182no—
185#235—188no—
186#235—188no—
187C2×D12—21no—
188C2×D12—94no—
189C2×D12—91no—
190C2×D12—184no—
191C2×D12—181no—
192C2×D12C2×C3⋊D311yescharacteristic
193C2×C4×D3—184no—
194C2×C4×D3—181no—
195C2×C4×D3C2×C3⋊D311yescharacteristic
196C2×Dic6—34no—
197C2×C3×C8—91no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4D3214yes—
200C3×C4⋊C4C2×C3⋊D311yescharacteristic
201C3×C22⋊C4—64no—
202#215—182no—
203#214—188no—
204#213—188no—
205(C2×D3)⋊C4—21no—
206C2×C3⋊C8—34no—
207C2×C3⋊C8—94no—
208C2×C3⋊C8—91no—
209C2×C3⋊C8—184no—
210D24—182no—
211C22×C32—24no—
212C22×C32—61no—
213C22×C32—61no—
214C22×C32—61no—
215C22×C32—121no—
216C22×C32—121no—
217C22×C32—121no—
218C22×C32D3×D411yescharacteristic
219C22×C32C2×#6714yes—
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—24no—
230C2×C3×D3—24no—
231C2×C3×D3—24no—
232C2×C3×D3—44no—
233C2×C3×D3—44no—
234C2×C3×D3—44no—
235C32×C4—24no—
236C32×C4—24no—
237C32×C4—41no—
238C32×C4—44no—
239C32×C4—61no—
240C32×C4—84no—
241C32×C4C2×D1211yescharacteristic
242C32×C4D3×D414yes—
243C32×C4D3×D414yes—
244C32×C4D3×D411yescharacteristic
245C3⋊Dic3—22no—
246C3×Dic3—24no—
247C3×Dic3—68no—
248C2×#41—271no—
249C2×SD16—271no—
250C2×D8—271no—
251#121—31no—
252#103—91no—
253#102—271no—
254C22⋊C8—271no—
255C33#44911yescharacteristic, complemented, Sylow
256C23×C3—34no—
257C23×C3—64no—
258C22×D3—21no—
259C22×D3—184no—
260C22×D3—181no—
261C22×D3—364no—
262C22×D3#63811yescharacteristic
263C3×Q8—92no—
264C3×Q8—181no—
265C3×D4—64no—
266C3×D4—64no—
267C3×D4—128no—
268C2×C3×C4—24no—
269C2×C3×C4—21no—
270C2×C3×C4—24no—
271C2×C3×C4—34no—
272C2×C3×C4—44no—
273C2×C3×C4—64no—
274C2×C3×C4—64no—
275C2×C3×C4—181no—
276C2×C3×C4C2×D3214yes—
277C2×C3×C4C22×C3⋊D311yescharacteristic
278#67—1816no—
279C2×Dic3—64no—
280C2×Dic3#63811yescharacteristic
281D12—21no—
282D12—21no—
283D12—42no—
284D12—98no—
285D12—92no—
286D12—188no—
287D12—184no—
288D12—184no—
289D12—181no—
290D12—364no—
291C4×D3—21no—
292C4×D3—21no—
293C4×D3—184no—
294C4×D3—1816no—
295Dic6—38no—
296Dic6—64no—
297C3×C8—181no—
298C3⋊C8—64no—
299C3⋊C8—184no—
300C3⋊C8—181no—
301C3⋊C8—364no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32—24no—
305C2×C32—121no—
306C2×C32—241no—
307C2×C32#88211yescharacteristic
308C2×C32#91411yescharacteristic
309C2×C32#91911yescharacteristic
310C2×C32#93214yes—
311C2×C32#93714yes—
312C2×C32#95014yes—
313C3⋊D3—122no—
314C3⋊D3—128no—
315C3⋊D3—248no—
316C3×D3—44no—
317C3×D3—84no—
318#41—542no—
319#41—542no—
320C2×Q8—91no—
321C2×D4—31no—
322C2×D4—61no—
323C2×D4—271no—
324C2×D4—541no—
325C22×C4—31no—
326C22×C4—541no—
327SD16—542no—
328D8—542no—
329C2×C8—91no—
330C2×C8—271no—
331C4⋊C4D3×C3⋊D311yescharacteristic
332C22⋊C4—61no—
333C22×C3—24no—
334C22×C3—64no—
335C22×C3—64no—
336C22×C3—64no—
337C22×C3—124no—
338C22×C3—124no—
339C22×C3—124no—
340C22×C3D3×#6714yes—
341C22×C3C2×#63811yescharacteristic
342C2×D3—21no—
343C2×D3—21no—
344C2×D3—21no—
345C2×D3—41no—
346C2×D3—41no—
347C2×D3—41no—
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—64no—
362C3×C4—84no—
363C3×C4—182no—
364C3×C4D3×D1214yes—
365C3×C4#380214yes—
366C3×C4D4×C3⋊D311yescharacteristic
367C3×C4D4×C3⋊D311yescharacteristic
368Dic3—21no—
369Dic3—68no—
370C32—24no—
371C32#481011yescharacteristic, complemented
372C32#503714yescomplemented
373C23—31no—
374C23—61no—
375C23—541no—
376Q8—92no—
377Q8—181no—
378D4—61no—
379D4—61no—
380D4—122no—
381D4—272no—
382D4—544no—
383D4—541no—
384C2×C4—21no—
385C2×C4—31no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—181no—
389C2×C4—544no—
390C2×C4C2×D3×C3⋊D311yescharacteristic
391C8—181no—
392C8—541no—
393C2×C3—24no—
394C2×C3—24no—
395C2×C3—24no—
396C2×C3—124no—
397C2×C3—244no—
398C2×C3#6392214yes—
399C2×C3#6393414yes—
400C2×C3#6394914yes—
401C2×C3#6414111yescharacteristic
402C2×C3#6414611yescharacteristic
403C2×C3#6415911yescharacteristic
404D3—41no—
405D3—81no—
406D3—368no—
407D3—362no—
408D3—728no—
409C22—61no—
410C22—61no—
411C22—61no—
412C22—121no—
413C22—121no—
414C22—121no—
415C22—542no—
416C22—1081no—
417C22—1081no—
418C22D3×#63811yescharacteristic
419C4—41no—
420C4—61no—
421C4—182no—
422C4C3⋊D3×D1211yescharacteristic
423C4#8902611yescharacteristic
424C3—24no—
425C3#1059061514yescomplemented
426C3#1059273611yescharacteristic, complemented
427C2—121no—
428C2—241no—
429C2—1082no—
430C2#1172354011yescharacteristic
431C2#1172355711yescharacteristic
432C2#1172358211yescharacteristic
subgroupG/H
centerC22D3×#638
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×#89022C2semidirect
2. C32×#895C2semidirect
3. C2×#88792C2semidirect
4. C2×#88790C2semidirect
5. #11722179C2semidirect
6. C3×#63637C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
30. C3×#895D3semidirect
34. #63637D3semidirect
46. C4⋊(C3⋊(C3⋊D3))D4semidirect
49. C3×#634D4semidirect
117. #895C3⋊D3semidirect
154. #634D12semidirect
255. C33#449semidirect
371. C32#4810semidirect
372. C32#5037semidirect
425. C3#10590615semidirect
426. C3#10592736semidirect

Maximal subgroups

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

subgroupindex
C2×#890222
C32×#8952
C2×#887922
C2×#887902
#117221792
C3×#636372
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927363
#105906153