Finite Group Databasebeta

← #13159452

Subgroups — #13159452

Order 1728.

1140 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1140 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×#89021C211yesmaximal, characteristic, complemented
2C32×#895C211yesmaximal, characteristic, complemented
3C2×#88794C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5#11722180C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592736—31nomaximal
9#10590613—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12—21no—
13C2×C32×D12C2211yescharacteristic
14C2×C32×C4×D3C2211yescharacteristic
15C2×#6290—21no—
16C2×Dic3×C3⋊D3—21no—
17#89021—22no—
18#89021—22no—
19C33×C4⋊C4C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C32×(C2×D3)⋊C4—21no—
23#88794—22no—
24#88789—22no—
25C2×#3828—31no—
26C2×#3795—34no—
27C32×#121—31no—
28C2×#3754—31no—
29C2×#3753—31no—
30C3×#895D314yescomplemented
31C2×#3716—34no—
32C2×#3713—34no—
33#63657—31no—
34#63637—31no—
35#63637—34no—
36#63637—64no—
37#63636D311yescharacteristic, complemented
38C3×#809—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
45C2×C3⋊(C3⋊Dic3)—21no—
46#6308—21no—
47#6308D412yescomplemented
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)D412yescomplemented
50C2×C32×Dic3D411yescharacteristic
51C32×D12—21no—
52C32×D12—21no—
53C32×D12—42no—
54C32×C4×D3—21no—
55C32×C4×D3—21no—
56#6290—24no—
57Dic3×C3⋊D3—24no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5037—94no—
61#4781—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)C2×D311yescharacteristic
69C2×C4×C3⋊D3—61no—
70C2×#634—31no—
71C2×#634—34no—
72C2×#634—64no—
73C2×C3×D12—24no—
74C2×C3×D12—34no—
75C2×C3×D12C2×D314yes—
76C2×C3×C4×D3C2×D314yes—
77C2×#625—64no—
78C2×D3×Dic3—64no—
79#3795—68no—
80#3795—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#3716—68no—
94#3713—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×#237—91no—
111C2×#235—94no—
112C3×#121—34no—
113C2×#214—94no—
114C2×#213—94no—
115C2×#213—91no—
116C2×#204—91no—
117#895C3⋊D311yescharacteristic, complemented
118C3×#102—91no—
119#830—94no—
120(C2×D3)⋊C8—91no—
121#810—94no—
122#810—91no—
123#810—184no—
124#809—34no—
125C23×C32—31no—
126C23×C32—61no—
127C22×C3⋊D3—21no—
128C22×C3×D3—24no—
129C22×C3×D3—64no—
130C22×C3×D3#6714yes—
131C32×D4—61no—
132C32×D4—61no—
133C32×D4—122no—
134C2×C32×C4—21no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—31no—
138C2×C32×C4—44no—
139C2×C32×C4—61no—
140C2×C32×C4—61no—
141C2×C32×C4C22×D314yes—
142C2×C32×C4C22×D311yescharacteristic
143#638—64no—
144C2×C3⋊Dic3—61no—
145C2×C3⋊Dic3—64no—
146C2×C3⋊Dic3—124no—
147C4⋊(C3⋊D3)—21no—
148C4⋊(C3⋊D3)D1212yescomplemented
149C4×C3⋊D3—64no—
150#634—32no—
151#634—38no—
152#634—68no—
153#634—61no—
154#634—64no—
155#634—124no—
156C2×C3×Dic3#6714yes—
157C3×D12—24no—
158C3×D12—24no—
159C3×D12—38no—
160C3×D12—48no—
161C3×D12—64no—
162C3×C4×D3—24no—
163C3×C4×D3—24no—
164#625—616no—
165D3×Dic3—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×D4—34no—
179C2×C3×D4—64no—
180C2×C3×D4—91no—
181C22×C3×C4—34no—
182C2×#67—184no—
183C2×#67—181no—
184C22×Dic3—181no—
185#237—182no—
186#237—182no—
187#235—188no—
188#235—188no—
189C2×D12—21no—
190C2×D12—34no—
191C2×D12C2×C3⋊D311yescharacteristic
192C2×C4×D3—184no—
193C2×C4×D3C2×C3⋊D311yescharacteristic
194C2×Dic6—94no—
195C2×Dic6—91no—
196C2×Dic6—184no—
197C2×C3×C8—91no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4D3214yes—
200C3×C4⋊C4C2×C3⋊D311yescharacteristic
201C3×C22⋊C4—64no—
202#214—188no—
203#213—182no—
204#213—188no—
205(C2×D3)⋊C4—21no—
206C2×C3⋊C8—34no—
207C2×C3⋊C8—94no—
208C2×C3⋊C8—91no—
209C2×C3⋊C8—184no—
210#204—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—22no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3—41no—
223C2×C3×D3—24no—
224C2×C3×D3—24no—
225C2×C3×D3—24no—
226C2×C3×D3—44no—
227C2×C3×D3—44no—
228C2×C3×D3—44no—
229C2×C3×D3—68no—
230C2×C3×D3—124no—
231C2×C3×D3—124no—
232C32×C4—24no—
233C32×C4—24no—
234C32×C4—41no—
235C32×C4—44no—
236C32×C4—61no—
237C32×C4—84no—
238C32×C4C2×D1211yescharacteristic
239C32×C4D3×D414yes—
240C32×C4D3×D414yes—
241C32×C4D3×D411yescharacteristic
242C3⋊Dic3—62no—
243C3⋊Dic3—68no—
244C3⋊Dic3—128no—
245C3×Dic3—24no—
246C2×#41—271no—
247C2×SD16—271no—
248C2×D8—271no—
249#121—31no—
250#103—271no—
251#102—91no—
252C22⋊C8—271no—
253C33#44911yescharacteristic, complemented, Sylow
254C23×C3—34no—
255C23×C3—64no—
256C23×C3—181no—
257C22×D3—21no—
258C22×D3—64no—
259C22×D3#63811yescharacteristic
260C3×D4—64no—
261C3×D4—64no—
262C3×D4—92no—
263C3×D4—128no—
264C3×D4—181no—
265C2×C3×C4—24no—
266C2×C3×C4—21no—
267C2×C3×C4—24no—
268C2×C3×C4—34no—
269C2×C3×C4—44no—
270C2×C3×C4—64no—
271C2×C3×C4—64no—
272C2×C3×C4C2×D3214yes—
273C2×C3×C4C22×C3⋊D311yescharacteristic
274#67—184no—
275#67—1816no—
276C2×Dic3—184no—
277C2×Dic3—184no—
278C2×Dic3—181no—
279C2×Dic3—364no—
280C2×Dic3#63811yescharacteristic
281D12—21no—
282D12—21no—
283D12—38no—
284D12—42no—
285D12—64no—
286C4×D3—21no—
287C4×D3—21no—
288C4×D3—1816no—
289Dic6—98no—
290Dic6—92no—
291Dic6—188no—
292Dic6—184no—
293Dic6—181no—
294Dic6—364no—
295C3×C8—181no—
296C3⋊C8—64no—
297C3⋊C8—184no—
298C3⋊C8—181no—
299C3⋊C8—364no—
300C2×C32—24no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—121no—
304C2×C32—241no—
305C2×C32#88211yescharacteristic
306C2×C32#91111yescharacteristic
307C2×C32#91611yescharacteristic
308C2×C32#93214yes—
309C2×C32#93714yes—
310C2×C32#95014yes—
311C3⋊D3—42no—
312C3×D3—44no—
313C3×D3—84no—
314C3×D3—128no—
315#41—542no—
316#41—542no—
317C2×Q8—271no—
318C2×D4—31no—
319C2×D4—61no—
320C2×D4—91no—
321C2×D4—541no—
322C22×C4—31no—
323C22×C4—541no—
324SD16—542no—
325D8—542no—
326C2×C8—91no—
327C2×C8—271no—
328C4⋊C4D3×C3⋊D311yescharacteristic
329C22⋊C4—61no—
330C22×C3—24no—
331C22×C3—64no—
332C22×C3—64no—
333C22×C3—64no—
334C22×C3—124no—
335C22×C3—124no—
336C22×C3—124no—
337C22×C3—182no—
338C22×C3—361no—
339C22×C3—361no—
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—68no—
349C2×D3—124no—
350C2×D3—124no—
351C3×C4—24no—
352C3×C4—24no—
353C3×C4—41no—
354C3×C4—44no—
355C3×C4—64no—
356C3×C4—84no—
357C3×C4D3×D1214yes—
358C3×C4#380214yes—
359C3×C4D4×C3⋊D311yescharacteristic
360C3×C4D4×C3⋊D311yescharacteristic
361Dic3—21no—
362Dic3—182no—
363Dic3—188no—
364Dic3—368no—
365C32—24no—
366C32#478111yescharacteristic, complemented
367C32#503714yescomplemented
368C23—31no—
369C23—61no—
370C23—181no—
371Q8—272no—
372Q8—541no—
373D4—61no—
374D4—61no—
375D4—92no—
376D4—122no—
377D4—181no—
378D4—544no—
379C2×C4—21no—
380C2×C4—31no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—544no—
384C2×C4—541no—
385C2×C4C2×D3×C3⋊D311yescharacteristic
386C8—181no—
387C8—541no—
388C2×C3—24no—
389C2×C3—24no—
390C2×C3—24no—
391C2×C3—124no—
392C2×C3—244no—
393C2×C3—362no—
394C2×C3#6392214yes—
395C2×C3#6393214yes—
396C2×C3#6394714yes—
397C2×C3#6414111yescharacteristic
398C2×C3#6414611yescharacteristic
399C2×C3#6415911yescharacteristic
400D3—41no—
401D3—81no—
402D3—128no—
403C22—61no—
404C22—61no—
405C22—61no—
406C22—121no—
407C22—121no—
408C22—121no—
409C22—182no—
410C22—361no—
411C22—361no—
412C22D3×#63811yescharacteristic
413C4—41no—
414C4—61no—
415C4—542no—
416C4C3⋊D3×D1211yescharacteristic
417C4#8902611yescharacteristic
418C3—24no—
419C3#1059061314yescomplemented
420C3#1059273611yescharacteristic, complemented
421C2—121no—
422C2—241no—
423C2—362no—
424C2#1172354011yescharacteristic
425C2#1172355511yescharacteristic
426C2#1172358011yescharacteristic
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×#89021C2semidirect
2. C32×#895C2semidirect
3. C2×#88794C2semidirect
4. C2×#88789C2semidirect
5. #11722180C2semidirect
6. C3×#63636C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
30. C3×#895D3semidirect
37. #63636D3semidirect
47. #6308D4semidirect
49. C3×C4⋊(C3⋊D3)D4semidirect
117. #895C3⋊D3semidirect
148. C4⋊(C3⋊D3)D12semidirect
253. C33#449semidirect
366. C32#4781semidirect
367. C32#5037semidirect
419. C3#10590613semidirect
420. C3#10592736semidirect

Maximal subgroups

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

subgroupindex
C2×#890212
C32×#8952
C2×#887942
C2×#887892
#117221802
C3×#636362
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927363
#105906133