Finite Group Databasebeta

← #13161015

Subgroups — #13161015

Order 1728.

1216 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1216 conjugacy classes of subgroups — too many to draw. Only its 168 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
1D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic, complemented
3C2×C32×#213C211yesmaximal, characteristic, complemented
4#11723529C211yesmaximal, characteristic, complemented
5#11722050C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592849—31nomaximal
9#10591486—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15C2×#6290—21no—
16#88864—21no—
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C3×#3753—22no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×#213—22no—
22C2×C32×C3⋊C8C2211yescharacteristic
23C2×C32×D8—31no—
24#64149—31no—
25D4×C3⋊Dic3—31no—
26D4×C3⋊Dic3—34no—
27D4×C3⋊Dic3—64no—
28C2×#3753D311yescharacteristic, complemented
29C2×C3×#213—34no—
30C2×C3×#213D314yescomplemented
31#63914—34no—
32#63659—31no—
33#63648—31no—
34#63641—31no—
35#63572—34no—
36#63563—34no—
37#63563—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C22×C32×D3—21no—
41C33×D4—21no—
42C33×D4D412yescomplemented
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊Dic3)—42no—
46C2×C3⋊(C3⋊Dic3)D411yescharacteristic
47C2×C3⋊(C3⋊Dic3)D411yescharacteristic
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C32×D12—21no—
51C32×D12—21no—
52#6290—42no—
53#6290—42no—
54C3×C3⋊(C3⋊C8)—21no—
55C32×C3⋊C8—21no—
56#5150—94no—
57#5150—91no—
58C2×C32×D4—24no—
59C2×C32×D4—31no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C2×D311yescharacteristic
62C2×#638—61no—
63C2×#638—61no—
64C22×C3⋊Dic3—61no—
65C22×C3⋊Dic3—64no—
66C22×C3⋊Dic3—68no—
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×C3×D12—34no—
69C2×C3×D12C2×D314yes—
70C2×#625—64no—
71C2×#625—64no—
72C32×D8—62no—
73C2×C32×C8—31no—
74#3757—61no—
75#3757—64no—
76#3757—68no—
77#3753—22no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—34no—
80C4⋊(C3⋊Dic3)—64no—
81C4×C3⋊Dic3—31no—
82C4×C3⋊Dic3—34no—
83C4×C3⋊Dic3—64no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C3×#213—28no—
86C3×#213—68no—
87C2×C3×C3⋊C8—34no—
88C2×C3×C3⋊C8C2×D314yes—
89C22×C33—22no—
90C22×C33—41no—
91C22×C33—41no—
92C22×C33C2×D411yescharacteristic
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3—41no—
96C2×C32×D3—41no—
97C2×C32×D3—41no—
98C2×C32×D3—41no—
99C33×C4C2×D411yescharacteristic
100C33×C4#4111yescharacteristic
101C3⋊(C3⋊Dic3)—21no—
102C3⋊(C3⋊Dic3)—41no—
103C3⋊(C3⋊Dic3)D812yescomplemented
104C2×C3×D8—34no—
105C2×C3×D8—91no—
106#940—94no—
107#940—91no—
108D4×Dic3—94no—
109D4×Dic3—91no—
110D4×Dic3—184no—
111C2×#213—34no—
112C2×#213C3⋊D311yescharacteristic, complemented
113#832—94no—
114#832—91no—
115#821—94no—
116#821—91no—
117Dic3⋊C8—94no—
118Dic3⋊C8—91no—
119C23×C32—28no—
120C23×C32—24no—
121C23×C32—21no—
122C23×C32—61no—
123C22×C3⋊D3—21no—
124C22×C3×D3—24no—
125C22×C3×D3—64no—
126C32×D4—28no—
127C32×D4—24no—
128C32×D4—21no—
129C32×D4—28no—
130C32×D4—61no—
131C32×D4—61no—
132C32×D4#6718yescomplemented
133C32×D4#6712yescomplemented
134C2×C32×C4—24no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C22×D311yescharacteristic
137#638—122no—
138#638—122no—
139C2×C3⋊Dic3—31no—
140C2×C3⋊Dic3—31no—
141C2×C3⋊Dic3—34no—
142C2×C3⋊Dic3—34no—
143C2×C3⋊Dic3—64no—
144C2×C3⋊Dic3—64no—
145C2×C3⋊Dic3—68no—
146C2×C3⋊Dic3—64no—
147C2×C3⋊Dic3—61no—
148C2×C3⋊Dic3—122no—
149C2×C3⋊Dic3—128no—
150C2×C3⋊Dic3—1216no—
151C4⋊(C3⋊D3)—21no—
152C4⋊(C3⋊D3)—21no—
153C3×D12—24no—
154C3×D12—24no—
155C3×D12—64no—
156C3×D12—64no—
157#625—128no—
158#625—128no—
159C32×C8—61no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—24no—
162C3×C3⋊C8—64no—
163#459—271noSylow
164C2×C33—42no—
165C2×C33#12111yescharacteristic
166C2×C33C2×D811yescharacteristic
167C2×C33AGL(1,ℤ/8)11yescharacteristic
168C3×C3⋊D3—81no—
169C32×D3—81no—
170C2×C3×D4—24no—
171C2×C3×D4—34no—
172C2×C3×D4—91no—
173C2×C3×D4D3214yes—
174C2×C3×D4C2×C3⋊D311yescharacteristic
175C2×#67—181no—
176C2×#67—181no—
177C2×#67—184no—
178C2×#67—184no—
179C22×Dic3—184no—
180C22×Dic3—181no—
181C22×Dic3—188no—
182C2×D12—34no—
183C2×D12C2×C3⋊D311yescharacteristic
184C3×D8—68no—
185C3×D8—182no—
186C2×C3×C8—34no—
187C2×C3×C8—91no—
188#217—184no—
189#217—181no—
190#217—188no—
191#213—22no—
192#213—68no—
193C4⋊Dic3—94no—
194C4⋊Dic3—91no—
195C4⋊Dic3—184no—
196C4×Dic3—94no—
197C4×Dic3—91no—
198C4×Dic3—184no—
199C2×C3⋊C8—34no—
200C2×C3⋊C8C2×C3⋊D311yescharacteristic
201C22×C32—216no—
202C22×C32—28no—
203C22×C32—22no—
204C22×C32—24no—
205C22×C32—48no—
206C22×C32—48no—
207C22×C32—44no—
208C22×C32—41no—
209C22×C32—44no—
210C22×C32—41no—
211C22×C32—121no—
212C22×C32—121no—
213C22×C32—121no—
214C22×C32D3×D414yes—
215C22×C32D3×D411yescharacteristic
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—41no—
219C2×C3×D3—44no—
220C2×C3×D3—44no—
221C2×C3×D3—44no—
222C2×C3×D3—124no—
223C2×C3×D3—124no—
224C2×C3×D3—124no—
225C32×C4—24no—
226C32×C4—24no—
227C32×C4#23714yes—
228C32×C4#23711yescharacteristic
229C32×C4C2×#6714yes—
230C32×C4C2×#6711yescharacteristic
231C3⋊Dic3—32no—
232C3⋊Dic3—38no—
233C3⋊Dic3—68no—
234C3⋊Dic3—61no—
235C3⋊Dic3—64no—
236C3⋊Dic3—124no—
237C3⋊Dic3—128no—
238C3⋊Dic3—124no—
239C3⋊Dic3—121no—
240C2×D8—31no—
241C2×D8—91no—
242#127—271no—
243C4×D4—271no—
244C4⋊C8—271no—
245#102—271no—
246#102—271no—
247C33#45911yescharacteristic, complemented, Sylow
248C23×C3—28no—
249C23×C3—24no—
250C23×C3—21no—
251C23×C3—64no—
252C23×C3—181no—
253C22×D3—21no—
254C22×D3—64no—
255C3×D4—28no—
256C3×D4—24no—
257C3×D4—21no—
258C3×D4—28no—
259C3×D4—64no—
260C3×D4—64no—
261C3×D4—181no—
262C3×D4—181no—
263C3×D4#62518yescomplemented
264C3×D4#63812yescomplemented
265C2×C3×C4—24no—
266C2×C3×C4C2×D3214yes—
267C2×C3×C4C22×C3⋊D311yescharacteristic
268#67—362no—
269#67—368no—
270#67—362no—
271#67—368no—
272C2×Dic3—94no—
273C2×Dic3—91no—
274C2×Dic3—94no—
275C2×Dic3—91no—
276C2×Dic3—184no—
277C2×Dic3—184no—
278C2×Dic3—188no—
279C2×Dic3—184no—
280C2×Dic3—181no—
281C2×Dic3—362no—
282C2×Dic3—368no—
283C2×Dic3—3616no—
284D12—21no—
285D12—21no—
286D12—64no—
287D12—64no—
288C3×C8—64no—
289C3×C8—181no—
290C3⋊C8—21no—
291C3⋊C8—64no—
292C2×C32—24no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—416no—
296C2×C32—48no—
297C2×C32—42no—
298C2×C32—241no—
299C2×C32D3×D814yes—
300C2×C32D3×D811yescharacteristic
301C2×C32#91114yes—
302C2×C32#91111yescharacteristic
303C2×C32#93914yes—
304C2×C32#93911yescharacteristic
305C3⋊D3—81no—
306C3×D3—84no—
307C3×D3—244no—
308C2×D4—31no—
309C2×D4—91no—
310C2×D4—541no—
311C2×D4—541no—
312C2×D4D3×C3⋊D311yescharacteristic
313C22×C4—541no—
314D8—62no—
315D8—182no—
316C2×C8—31no—
317C2×C8—91no—
318C4⋊C4—271no—
319C22⋊C4—541no—
320C42—271no—
321C22×C3—216no—
322C22×C3—22no—
323C22×C3—28no—
324C22×C3—24no—
325C22×C3—48no—
326C22×C3—48no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—44no—
330C22×C3—44no—
331C22×C3—124no—
332C22×C3—124no—
333C22×C3—124no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3—361no—
337C22×C3#381114yes—
338C22×C3D4×C3⋊D311yescharacteristic
339C2×D3—41no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—124no—
343C2×D3—124no—
344C2×D3—124no—
345C3×C4—24no—
346C3×C4—24no—
347C3×C4#380514yes—
348C3×C4C2×#62514yes—
349C3×C4#383011yescharacteristic
350C3×C4C2×#63811yescharacteristic
351Dic3—92no—
352Dic3—98no—
353Dic3—188no—
354Dic3—181no—
355Dic3—184no—
356Dic3—364no—
357Dic3—368no—
358Dic3—361no—
359Dic3—364no—
360C32—24no—
361C32#515014yescomplemented
362C32#515011yescharacteristic, complemented
363C23—21no—
364C23—61no—
365C23—181no—
366D4—21no—
367D4—61no—
368D4—61no—
369D4—181no—
370D4—181no—
371D4—1082no—
372D4—1082no—
373D4#629012yescomplemented
374C2×C4—271no—
375C2×C4—271no—
376C2×C4—541no—
377C2×C4—1082no—
378C2×C4C2×D3×C3⋊D311yescharacteristic
379C8—61no—
380C8—181no—
381C2×C3—24no—
382C2×C3—24no—
383C2×C3—24no—
384C2×C3—416no—
385C2×C3—42no—
386C2×C3—48no—
387C2×C3—244no—
388C2×C3—721no—
389C2×C3#6392614yes—
390C2×C3#6392714yes—
391C2×C3#6398014yes—
392C2×C3D8×C3⋊D311yescharacteristic
393C2×C3#6412011yescharacteristic
394C2×C3#6414811yescharacteristic
395D3—81no—
396D3—244no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22—361no—
404C22—361no—
405C22—361no—
406C22#8903911yescharacteristic
407C4—272no—
408C4—541no—
409C4—1081no—
410C4#8902911yescharacteristic
411C4C2×#629011yescharacteristic
412C3—24no—
413C3#1059148614yescomplemented
414C3#1059284911yescharacteristic, complemented
415C2—42no—
416C2—241no—
417C2—721no—
418C2#1172354711yescharacteristic
419C2#1172354811yescharacteristic
420C2#1172363711yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#459
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. D4×C3⋊(C3⋊Dic3)C2semidirect
2. C2×C3×#3753C2semidirect
3. C2×C32×#213C2semidirect
4. #11723529C2semidirect
5. #11722050C2semidirect
6. #11722034C2semidirect
7. #11722032C2semidirect
28. C2×#3753D3semidirect
30. C2×C3×#213D3semidirect
42. C33×D4D4semidirect
103. C3⋊(C3⋊Dic3)D8semidirect
112. C2×#213C3⋊D3semidirect
132. C32×D4#67semidirect
133. C32×D4#67semidirect
247. C33#459semidirect
263. C3×D4#625semidirect
264. C3×D4#638semidirect
361. C32#5150semidirect
362. C32#5150semidirect
373. D4#6290semidirect
413. C3#10591486semidirect
414. C3#10592849semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊(C3⋊Dic3)2
C2×C3×#37532
C2×C32×#2132
#117235292
#117220502
#117220342
#117220322
#105928493
#105914863