Finite Group Databasebeta

← #13159447

Subgroups — #13159447

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×#89014C211yesmaximal, characteristic, complemented
2C3×#64104C211yesmaximal, characteristic, complemented
3C2×#88796C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5#11722180C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592480—31nomaximal
9#10590613—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6290—21no—
16C2×D3×C3⋊Dic3—21no—
17#89014—22no—
18#89014—22no—
19C33×C4⋊C4C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×#3752—21no—
22C2×C32×C3⋊C8C2211yescharacteristic
23#88796—22no—
24#88789—22no—
25C2×#3830—31no—
26C2×#3795—34no—
27C2×#3753—31no—
28C2×#3744—31no—
29#64104D311yescharacteristic, complemented
30C3×#895—34no—
31C2×#3716—34no—
32C2×#3713—34no—
33C32×#102—31no—
34#63647—31no—
35#63637—31no—
36#63637—34no—
37#63637—64no—
38C3×#809D314yescomplemented
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46#6308—21no—
47#6308D412yescomplemented
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—42no—
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C32×D12—21no—
55C32×D12D412yescomplemented
56#6290—24no—
57D3×C3⋊Dic3—24no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5037—91no—
61#4781—94no—
62C2×C32×D4—31no—
63C2×#638—61no—
64C22×C3⋊Dic3—61no—
65#3830—62no—
66#3830—62no—
67C2×C4⋊(C3⋊D3)—21no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×#634—31no—
71C2×#634—34no—
72C2×#634—64no—
73C2×C3×D12—34no—
74C2×C3×D12—64no—
75C2×C3×D12C2×D314yes—
76C2×C3×C4×D3—34no—
77C2×#625—64no—
78C2×D3×Dic3—64no—
79#3795—68no—
80#3795—68no—
81C2×C32×C8—31no—
82C32×C4⋊C4—24no—
83C32×C4⋊C4C2×D314yes—
84C32×C4⋊C4C2×D311yescharacteristic
85#3753—62no—
86#3752—21no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—34no—
89C2×C3⋊(C3⋊C8)—64no—
90#3744—62no—
91C3×(C2×D3)⋊C4—64no—
92C2×C3×C3⋊C8C2×D314yes—
93#3716—68no—
94#3713—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—21no—
98C2×C3×C3⋊D3—21no—
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3—41no—
102C2×C32×D3—22no—
103C2×C32×D3—41no—
104C2×C32×D3—41no—
105C33×C4—41no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3⋊(C3⋊Dic3)—22no—
109C3×C3⋊Dic3—21no—
110C2×#237—94no—
111C2×#235—91no—
112C3×#121—91no—
113C2×#214—91no—
114C2×#213—94no—
115C2×#213—91no—
116C2×#204—94no—
117#895—34no—
118C3×#102—34no—
119#830—91no—
120(C2×D3)⋊C8—94no—
121#810—94no—
122#810—91no—
123#810—184no—
124#809C3⋊D311yescharacteristic, complemented
125C23×C32—61no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3#6711yescharacteristic
128C22×C3×D3—24no—
129C22×C3×D3—34no—
130C22×C3×D3—64no—
131C32×D4—32no—
132C32×D4—61no—
133C2×C32×C4—21no—
134C2×C32×C4—24no—
135C2×C32×C4—24no—
136C2×C32×C4—44no—
137C2×C32×C4C22×D314yes—
138C2×C32×C4C22×D311yescharacteristic
139#638—64no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—124no—
144C2×C3⋊Dic3#6711yescharacteristic
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)—42no—
148C4×C3⋊D3—21no—
149C4×C3⋊D3—21no—
150#634—32no—
151#634—38no—
152#634—68no—
153#634—61no—
154#634—64no—
155#634—124no—
156C2×C3×Dic3—34no—
157C3×D12—24no—
158C3×D12—64no—
159C3×D12—64no—
160C3×D12—128no—
161C3×D12D1218yescomplemented
162C3×C4×D3—64no—
163C3×C4×D3—64no—
164#625—616no—
165D3×Dic3—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—41no—
176C3×C3⋊D3—81no—
177C32×D3—42no—
178C2×C3×D4—34no—
179C2×C3×D4—91no—
180C2×C3×D4—181no—
181C22×C3×C4—91no—
182C2×#67—181no—
183C2×#67—184no—
184C22×Dic3—184no—
185#237—188no—
186#237—188no—
187#235—182no—
188#235—182no—
189C2×D12—34no—
190C2×D12—64no—
191C2×D12C2×C3⋊D311yescharacteristic
192C2×C4×D3—34no—
193C2×C4×D3—181no—
194C2×Dic6—94no—
195C2×Dic6—91no—
196C2×Dic6—184no—
197C2×C3×C8—34no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4D3214yes—
200C3×C4⋊C4C2×C3⋊D311yescharacteristic
201C3×C22⋊C4—181no—
202#214—182no—
203#213—188no—
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
210#204—188no—
211C22×C32—24no—
212C22×C32—62no—
213C22×C32—121no—
214C22×C32—121no—
215C22×C32D3×D414yes—
216C22×C32C2×#6711yescharacteristic
217C2×C3⋊D3—21no—
218C2×C3⋊D3—21no—
219C2×C3⋊D3—21no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3—41no—
223C2×C3×D3—28no—
224C2×C3×D3—44no—
225C2×C3×D3—44no—
226C2×C3×D3—64no—
227C2×C3×D3—64no—
228C2×C3×D3—64no—
229C2×C3×D3—124no—
230C2×C3×D3—124no—
231C2×C3×D3—124no—
232C32×C4—24no—
233C32×C4—24no—
234C32×C4—41no—
235C32×C4—44no—
236C32×C4—84no—
237C32×C4C2×D1214yes—
238C32×C4D3×D414yes—
239C32×C4D3×D411yescharacteristic
240C32×C4D3×D411yescharacteristic
241C3⋊Dic3—21no—
242C3⋊Dic3—62no—
243C3⋊Dic3—68no—
244C3⋊Dic3—128no—
245C3×Dic3—64no—
246C2×#41—271no—
247C2×SD16—271no—
248C2×D8—271no—
249#121—91no—
250#103—271no—
251#102—31no—
252C22⋊C8—271no—
253C33#44911yescharacteristic, complemented, Sylow
254C23×C3—64no—
255C23×C3—91no—
256C23×C3—181no—
257C22×D3—21no—
258C22×D3—34no—
259C22×D3—64no—
260C3×D4—38no—
261C3×D4—64no—
262C3×D4—181no—
263C3×D4—181no—
264C3×D4—362no—
265C2×C3×C4—24no—
266C2×C3×C4—21no—
267C2×C3×C4—24no—
268C2×C3×C4—44no—
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—1816no—
276C2×Dic3—34no—
277C2×Dic3—1816no—
278C2×Dic3—184no—
279C2×Dic3—181no—
280C2×Dic3—364no—
281D12—21no—
282D12—64no—
283D12—64no—
284D12—128no—
285D12C4⋊(C3⋊D3)12yescomplemented
286C4×D3—64no—
287C4×D3—64no—
288C4×D3—184no—
289Dic6—98no—
290Dic6—92no—
291Dic6—188no—
292Dic6—184no—
293Dic6—181no—
294Dic6—364no—
295C3×C8—64no—
296C3⋊C8—21no—
297C3⋊C8—184no—
298C3⋊C8—181no—
299C3⋊C8—364no—
300C2×C32—24no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—122no—
304C2×C32#88214yes—
305C2×C32#91114yes—
306C2×C32#91614yes—
307C2×C32#93211yescharacteristic
308C2×C32#93711yescharacteristic
309C2×C32#95011yescharacteristic
310C3⋊D3—41no—
311C3⋊D3—81no—
312C3×D3—48no—
313C3×D3—124no—
314C3×D3—244no—
315#41—542no—
316#41—542no—
317C2×Q8—271no—
318C2×D4—31no—
319C2×D4—91no—
320C2×D4—181no—
321C2×D4—541no—
322C22×C4—91no—
323C22×C4—541no—
324SD16—542no—
325D8—542no—
326C2×C8—31no—
327C2×C8—271no—
328C4⋊C4D3×C3⋊D311yescharacteristic
329C22⋊C4—181no—
330C22×C3—24no—
331C22×C3—68no—
332C22×C3—124no—
333C22×C3—124no—
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—22no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—64no—
346C2×D3—64no—
347C2×D3—64no—
348C2×D3—124no—
349C2×D3—124no—
350C2×D3—124no—
351C3×C4—24no—
352C3×C4—24no—
353C3×C4—41no—
354C3×C4—44no—
355C3×C4—84no—
356C3×C4—181no—
357C3×C4D3×D1214yes—
358C3×C4#380214yes—
359C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
360C3×C4D4×C3⋊D311yescharacteristic
361Dic3—64no—
362Dic3—182no—
363Dic3—188no—
364Dic3—368no—
365C32—24no—
366C32#478114yescomplemented
367C32#503711yescharacteristic, complemented
368C23—61no—
369C23—91no—
370C23—181no—
371Q8—272no—
372Q8—541no—
373D4—32no—
374D4—61no—
375D4—181no—
376D4—181no—
377D4—362no—
378D4—544no—
379C2×C4—21no—
380C2×C4—91no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4—544no—
384C2×C4—541no—
385C2×C4C2×D3×C3⋊D311yescharacteristic
386C8—61no—
387C8—541no—
388C2×C3—24no—
389C2×C3—24no—
390C2×C3—24no—
391C2×C3—128no—
392C2×C3—361no—
393C2×C3—721no—
394C2×C3#6392214yes—
395C2×C3#6393214yes—
396C2×C3#6394714yes—
397C2×C3#6409111yescharacteristic
398C2×C3#6412011yescharacteristic
399C2×C3#6412511yescharacteristic
400D3—42no—
401D3—124no—
402D3—244no—
403C22—62no—
404C22—121no—
405C22—121no—
406C22—181no—
407C22—181no—
408C22—181no—
409C22—361no—
410C22—361no—
411C22—361no—
412C22C3⋊D3×#6711yescharacteristic
413C4—41no—
414C4—181no—
415C4—542no—
416C4D3×C4⋊(C3⋊D3)11yescharacteristic
417C4#8902611yescharacteristic
418C3—24no—
419C3#1059061314yescomplemented
420C3#1059248011yescharacteristic, complemented
421C2—122no—
422C2—361no—
423C2—721no—
424C2#1172354111yescharacteristic
425C2#1172356011yescharacteristic
426C2#1172358311yescharacteristic
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×#89014C2semidirect
2. C3×#64104C2semidirect
3. C2×#88796C2semidirect
4. C2×#88789C2semidirect
5. #11722180C2semidirect
6. C32×#809C2semidirect
7. #11722023C2semidirect
29. #64104D3semidirect
38. C3×#809D3semidirect
47. #6308D4semidirect
55. C32×D12D4semidirect
124. #809C3⋊D3semidirect
161. C3×D12D12semidirect
253. C33#449semidirect
285. D12C4⋊(C3⋊D3)semidirect
366. C32#4781semidirect
367. C32#5037semidirect
419. C3#10590613semidirect
420. C3#10592480semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
C3×#641042
C2×#887962
C2×#887892
#117221802
C32×#8092
#117220232
#105924803
#105906133