Finite Group Databasebeta

← #13160031

Subgroups — #13160031

Order 1728.

1340 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 178 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×C3⋊D3×D12C211yesmaximal, characteristic
2C2×C3×#3743C211yesmaximal, characteristic, complemented
3#11723404C211yesmaximal, characteristic, complemented
4#11722191C211yesmaximal, characteristic, complemented
5C32×#821C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592469—31nomaximal
9#10590939—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6291—21no—
15C3⋊D3×D12—22no—
16C3⋊D3×D12C414yescomplemented
17C2×C33×C8C2211yescharacteristic, complemented
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C3×#3743—22no—
21C32×C4⋊Dic3C2211yescharacteristic, complemented
22#88803—21no—
23Dic3×C3⋊Dic3—21no—
24C2×D4×C3⋊D3—31no—
25C2×D3×D12—34no—
26C2×#3743D311yescharacteristic, complemented
27#64101—31no—
28C2×C3×ΓL(1,𝔽25)—34no—
29#63841—34no—
30C32×#102—31no—
31#63659—31no—
32#63648—31no—
33#63648—34no—
34#63648—64no—
35#63636—31no—
36C3×#821D314yescomplemented
37#63564—34no—
38#63563—34no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C3×C3⋊D3D411yescharacteristic
41C22×C32×D3—21no—
42C2×D3×C3⋊D3—24no—
43C2×D3×C3⋊D3—42no—
44C2×D3×C3⋊D3—42no—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—21no—
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))C2×C412yes—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4×C3⋊D3C2×C412yes—
51C3×C4×C3⋊D3D412yes—
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12C2×C412yes—
55#6291—24no—
56C33×C8—21no—
57C3×C3⋊(C3⋊C8)—21no—
58#5131—91no—
59#4770—94no—
60C23×C3⋊D3—61no—
61C22×D32—64no—
62C2×C32×D4—31no—
63C2×#638—61no—
64D4×C3⋊D3—34no—
65D4×C3⋊D3—62no—
66C2×C4⋊(C3⋊D3)—31no—
67C2×C4⋊(C3⋊D3)—34no—
68C2×C4⋊(C3⋊D3)—64no—
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×C3×D12C2×D314yes—
71C2×C3×C4×D3—34no—
72C2×#626—64no—
73D3×D12—316no—
74D3×D12—68no—
75C2×C32×C8—24no—
76C2×C32×C8C2×D314yescomplemented
77C2×C32×C8C2×D311yescharacteristic, complemented
78C32×C4⋊C4—31no—
79#3752—61no—
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—34no—
82C4⋊(C3⋊Dic3)—64no—
83C4×C3⋊Dic3—61no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
85#3743—22no—
86C3×C4⋊Dic3C2×D314yescomplemented
87C2×C3×C3⋊C8—34no—
88C3×ΓL(1,𝔽25)—68no—
89(C2×D3)⋊Dic3—64no—
90Dic32—64no—
91C22×C33C2×D411yescharacteristic
92C2×C3⋊(C3⋊D3)—22no—
93C2×C3⋊(C3⋊D3)—41no—
94C2×C3⋊(C3⋊D3)—41no—
95C2×C3×C3⋊D3—21no—
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3C22⋊C412yes—
98C2×C32×D3—22no—
99C2×C32×D3—41no—
100C2×C32×D3—41no—
101D3×C3⋊D3—48no—
102C33×C4C22×C411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3⋊(C3⋊Dic3)—41no—
105C3×C3⋊Dic3C22⋊C412yes—
106C32×Dic3—41no—
107C2×D3×D4—94no—
108C22×D12—91no—
109C2×C3×M16—91no—
110#926—91no—
111C2×ΓL(1,𝔽25)—34no—
112#892—94no—
113C3×#102—34no—
114#832—94no—
115#821—91no—
116#821—94no—
117#821—91no—
118#821—91no—
119#821—184no—
120#821C3⋊D311yescharacteristic, complemented
121#809—94no—
122C23×C32—61no—
123C22×C3⋊D3—61no—
124C22×C3⋊D3—64no—
125C22×C3⋊D3—64no—
126C22×C3⋊D3—124no—
127C22×C3⋊D3—122no—
128C22×C3⋊D3—122no—
129C22×C3⋊D3D1211yescharacteristic
130C22×C3×D3—24no—
131C22×C3×D3—34no—
132C2×D32—616no—
133C2×D32—128no—
134C2×D32—128no—
135C32×D4—32no—
136C32×D4—61no—
137C2×C32×C4—24no—
138C2×C32×C4—61no—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C22×D311yescharacteristic
141#638—64no—
142C2×C3⋊Dic3—61no—
143C2×C3⋊Dic3—64no—
144C2×C3⋊Dic3—124no—
145C2×C3⋊Dic3D1211yescharacteristic
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⋊D3C4×D312yes—
153C4×C3⋊D3#6712yes—
154C2×C3×Dic3—24no—
155C2×C3×Dic3—34no—
156C3×D12—24no—
157C3×D12C4×D318yes—
158C3×C4×D3—38no—
159C3×C4×D3—38no—
160#626—616no—
161C32×C8—21no—
162C32×C8—24no—
163C32×C8—44no—
164C3⋊(C3⋊C8)—21no—
165C3×C3⋊C8—64no—
166#418—271noSylow
167C2×C33C2×C22⋊C411yescharacteristic
168C2×C33AGL(1,ℤ/8)11yescharacteristic
169C2×C33AGL(1,ℤ/8)11yescharacteristic
170C3⋊(C3⋊D3)—42no—
171C3×C3⋊D3—22no—
172C32×D3—42no—
173C23×D3—181no—
174C23×D3—184no—
175C2×C3×D4—34no—
176C22×C3×C4—91no—
177C2×#67—184no—
178D3×D4—916no—
179D3×D4—188no—
180C2×D12—94no—
181C2×D12—91no—
182C2×D12—94no—
183C2×D12—184no—
184C2×D12—182no—
185C2×D12—181no—
186C2×D12C2×C3⋊D311yescharacteristic
187C2×C4×D3—34no—
188C3×M16—182no—
189C2×C3×C8—24no—
190C2×C3×C8—91no—
191C2×C3×C8D3214yescomplemented
192C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
193C3×C4⋊C4—34no—
194#217—181no—
195(C2×D3)⋊C4—184no—
196C4⋊Dic3—94no—
197C4⋊Dic3—91no—
198C4⋊Dic3—184no—
199C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
200C4×Dic3—181no—
201C4×Dic3—184no—
202C2×C3⋊C8—34no—
203ΓL(1,𝔽25)—68no—
204C22×C32—24no—
205C22×C32—62no—
206C22×C32—121no—
207C22×C32—121no—
208C22×C32C2×D1211yescharacteristic
209C22×C32D3×D414yes—
210C2×C3⋊D3—21no—
211C2×C3⋊D3—21no—
212C2×C3⋊D3—62no—
213C2×C3⋊D3—68no—
214C2×C3⋊D3—128no—
215C2×C3⋊D3—128no—
216C2×C3⋊D3—121no—
217C2×C3⋊D3—121no—
218C2×C3⋊D3—124no—
219C2×C3⋊D3—124no—
220C2×C3⋊D3—244no—
221C2×C3⋊D3—244no—
222C2×C3⋊D3(C2×D3)⋊C412yes—
223C2×C3×D3—28no—
224C2×C3×D3—38no—
225C2×C3×D3—44no—
226C2×C3×D3—44no—
227C2×C3×D3—64no—
228C2×C3×D3—64no—
229D32—1232no—
230C32×C4—24no—
231C32×C4—24no—
232C32×C4—121no—
233C32×C4C2×C4×D314yes—
234C32×C4C2×C4×D311yescharacteristic
235C32×C4D3×D414yes—
236C32×C4C2×#6711yescharacteristic
237C3⋊Dic3—121no—
238C3⋊Dic3—124no—
239C3⋊Dic3—244no—
240C3⋊Dic3(C2×D3)⋊C412yes—
241C3×Dic3—38no—
242C3×Dic3—44no—
243C22×D4—271no—
244C2×M16—91no—
245#117—271no—
246#102—31no—
247#102—271no—
248#102—271no—
249#102—271no—
250C33#41811yescharacteristic, complemented, Sylow
251C23×C3—64no—
252C23×C3—91no—
253C22×D3—21no—
254C22×D3—34no—
255C22×D3—184no—
256C22×D3—1816no—
257C22×D3—184no—
258C22×D3—181no—
259C22×D3—364no—
260C22×D3—362no—
261C22×D3—362no—
262C22×D3—368no—
263C22×D3—368no—
264C3×D4—38no—
265C3×D4—64no—
266C2×C3×C4—24no—
267C2×C3×C4—64no—
268C2×C3×C4—91no—
269C2×C3×C4—92no—
270C2×C3×C4—92no—
271C2×C3×C4C2×D3214yes—
272C2×C3×C4C22×C3⋊D311yescharacteristic
273#67—1816no—
274C2×Dic3—21no—
275C2×Dic3—34no—
276C2×Dic3—184no—
277C2×Dic3—181no—
278C2×Dic3—364no—
279D12—21no—
280D12—98no—
281D12—92no—
282D12—188no—
283D12—184no—
284D12—184no—
285D12—181no—
286D12—364no—
287D12C4×C3⋊D312yes—
288C4×D3—38no—
289C4×D3—38no—
290C3×C8—24no—
291C3×C8—21no—
292C3×C8—44no—
293C3×C8—181no—
294C3⋊C8—64no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—24no—
298C2×C32—122no—
299C2×C32D3×C22⋊C414yes—
300C2×C32#90811yescharacteristic
301C2×C32#90811yescharacteristic
302C2×C32#91114yes—
303C2×C32#91414yes—
304C2×C32C2×(C2×D3)⋊C411yescharacteristic
305C3⋊D3—22no—
306C3⋊D3—122no—
307C3⋊D3—128no—
308C3⋊D3—248no—
309C3×D3—48no—
310C3×D3—68no—
311C24—541no—
312C2×D4—31no—
313C2×D4—274no—
314C2×D4—271no—
315C2×D4—542no—
316C2×D4—541no—
317C22×C4—91no—
318M16—182no—
319C2×C8—91no—
320C2×C8D3×C3⋊D311yescharacteristic, complemented
321C4⋊C4—31no—
322C4⋊C4—271no—
323C22⋊C4—541no—
324C42—541no—
325C22×C3—24no—
326C22×C3—68no—
327C22×C3—92no—
328C22×C3—124no—
329C22×C3—124no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3D3×D1214yes—
333C22×C3D4×C3⋊D311yescharacteristic
334C2×D3—22no—
335C2×D3—38no—
336C2×D3—41no—
337C2×D3—41no—
338C2×D3—64no—
339C2×D3—64no—
340C2×D3—182no—
341C2×D3—188no—
342C2×D3—368no—
343C2×D3—368no—
344C2×D3—3632no—
345C2×D3—364no—
346C2×D3—364no—
347C2×D3—361no—
348C2×D3—361no—
349C2×D3—724no—
350C2×D3—724no—
351C3×C4—24no—
352C3×C4—24no—
353C3×C4—92no—
354C3×C4—124no—
355C3×C4C4×D3214yes—
356C3×C4D3×#6714yes—
357C3×C4C2×C4×C3⋊D311yescharacteristic
358C3×C4D4×C3⋊D311yescharacteristic
359Dic3—38no—
360Dic3—41no—
361Dic3—364no—
362Dic3—361no—
363Dic3—724no—
364C32—24no—
365C32#477014yescomplemented
366C32#513111yescharacteristic, complemented
367C23—61no—
368C23—91no—
369C23—544no—
370C23—541no—
371C23—1082no—
372C23—1082no—
373D4—32no—
374D4—61no—
375D4—272no—
376D4—544no—
377D4—541no—
378C2×C4—61no—
379C2×C4—91no—
380C2×C4—92no—
381C2×C4—92no—
382C2×C4—541no—
383C2×C4C2×D3×C3⋊D311yescharacteristic
384C8—21no—
385C8—181no—
386C2×C3—24no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—128no—
390C2×C3—182no—
391C2×C3#6379414yes—
392C2×C3#6379514yes—
393C2×C3D3×(C2×D3)⋊C414yes—
394C2×C3C22⋊C4×C3⋊D311yescharacteristic
395C2×C3#6412011yescharacteristic
396C2×C3#6412311yescharacteristic
397D3—42no—
398D3—68no—
399D3—362no—
400D3—368no—
401D3—728no—
402C22—62no—
403C22—92no—
404C22—121no—
405C22—121no—
406C22—181no—
407C22—181no—
408C22—542no—
409C22—1088no—
410C22—1081no—
411C22—1081no—
412C22C3⋊D3×D1211yescharacteristic
413C4—92no—
414C4—121no—
415C4—1081no—
416C4C4×D3×C3⋊D311yescharacteristic
417C4C3⋊D3×#6711yescharacteristic
418C3—24no—
419C3#1059093914yescomplemented
420C3#1059246911yescharacteristic, complemented
421C2—122no—
422C2—182no—
423C2—1082no—
424C2#1172331911yescharacteristic
425C2#1172332011yescharacteristic
426C2C3⋊D3×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3⋊D3×D12C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#418
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3743C2semidirect
3. #11723404C2semidirect
4. #11722191C2semidirect
5. C32×#821C2semidirect
6. #11722035C2semidirect
7. #11722032C2semidirect
16. C3⋊D3×D12C4semidirect
17. C2×C33×C8C22semidirect
18. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
19. C2×C3×C3⋊(C3⋊C8)C22semidirect
21. C32×C4⋊Dic3C22semidirect
26. C2×#3743D3semidirect
36. C3×#821D3semidirect
76. C2×C32×C8C2×D3semidirect
77. C2×C32×C8C2×D3semidirect
84. C2×C3⋊(C3⋊C8)C2×D3semidirect
86. C3×C4⋊Dic3C2×D3semidirect
120. #821C3⋊D3semidirect
191. C2×C3×C8D32semidirect
192. C2×C3×C8C2×C3⋊D3semidirect
199. C4⋊Dic3C2×C3⋊D3semidirect
250. C33#418semidirect
320. C2×C8D3×C3⋊D3semidirect
365. C32#4770semidirect
366. C32#5131semidirect
419. C3#10590939semidirect
420. C3#10592469semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
C2×C3×#37432
#117234042
#117221912
C32×#8212
#117220352
#117220322
#105924693
#105909393