Finite Group Databasebeta

← #13159414

Subgroups — #13159414

Order 1728.

1340 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 210 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×D3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C32×#892C211yesmaximal, characteristic, complemented
3C2×#88786C211yesmaximal, characteristic, complemented
4#11722179C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592700—31nomaximal
9#10590592—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6292—21no—
15D3×C4⋊(C3⋊D3)—22no—
16D3×C4⋊(C3⋊D3)C414yescomplemented
17C33×C4⋊C4C2211yescharacteristic, complemented
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C32×(C2×D3)⋊C4—21no—
21C32×C4⋊Dic3C2211yescharacteristic, complemented
22C32×C4×Dic3—21no—
23#88786—22no—
24C22×C4⋊(C3⋊D3)—31no—
25C2×D3×D12—34no—
26C32×#117—31no—
27C2×#3748—31no—
28C3×#892D314yescomplemented
29C2×#3711—34no—
30#63636—31no—
31#63636—31no—
32#63636—31no—
33#63636—34no—
34#63636—64no—
35#63636D311yescharacteristic, complemented
36C3×#809—34no—
37#63564—34no—
38#63562—34no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C3×C3⋊D3—21no—
41C22×C32×D3D411yescharacteristic
42C2×D3×C3⋊D3—24no—
43C2×D3×C3⋊D3—42no—
44C2×D3×C3⋊D3—42no—
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))C2×C412yes—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)C2×C412yes—
51C2×C32×Dic3—21no—
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3C2×C412yes—
54C32×C4×D3D412yes—
55#6292—24no—
56C3⋊(C3⋊(C3⋊C8))—21no—
57C3×C3⋊(C3⋊C8)—21no—
58#5001—94no—
59#4770—91no—
60C23×C3⋊D3—61no—
61C22×D32—64no—
62C22×C32×C4—31no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—34no—
65C2×C4⋊(C3⋊D3)—34no—
66C2×C4⋊(C3⋊D3)—62no—
67C2×C4⋊(C3⋊D3)—61no—
68C2×C4⋊(C3⋊D3)—64no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×C3×D12—34no—
71C2×C3×C4×D3C2×D314yes—
72C2×#626—64no—
73D3×D12—316no—
74D3×D12—68no—
75C32×C4⋊C4—24no—
76C32×C4⋊C4—31no—
77C32×C4⋊C4C2×D314yescomplemented
78C32×C4⋊C4C2×D311yescharacteristic, complemented
79C32×C22⋊C4—61no—
80C32×C42—61no—
81#3748—62no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—34no—
84C2×C3⋊(C3⋊C8)—64no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
86C3×(C2×D3)⋊C4—24no—
87C3×C4⋊Dic3C2×D314yescomplemented
88C3×C4×Dic3—24no—
89C2×C3×C3⋊C8—34no—
90#3711—68no—
91C22×C33C2×D411yescharacteristic
92C2×C3⋊(C3⋊D3)—22no—
93C2×C3⋊(C3⋊D3)—41no—
94C2×C3⋊(C3⋊D3)—41no—
95C2×C3×C3⋊D3—22no—
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3—41no—
98C2×C32×D3—21no—
99C2×C32×D3—21no—
100C2×C32×D3C22⋊C412yes—
101D3×C3⋊D3—48no—
102C33×C4—41no—
103C33×C4C22×C411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C32×Dic3—41no—
106C32×Dic3C22⋊C412yes—
107C2×D3×D4—91no—
108C22×D12—94no—
109C3×#117—34no—
110C2×#208—94no—
111C2×ΓL(1,𝔽25)—91no—
112#892C3⋊D311yescharacteristic, complemented
113C3×#102—91no—
114#832—91no—
115#821—91no—
116#809—34no—
117#809—94no—
118#809—94no—
119#809—94no—
120#809—91no—
121#809—184no—
122C23×C32—31no—
123C22×C3⋊D3—21no—
124C22×C3⋊D3—64no—
125C22×C3⋊D3—61no—
126C22×C3⋊D3—64no—
127C22×C3⋊D3—124no—
128C22×C3⋊D3—122no—
129C22×C3⋊D3—122no—
130C22×C3×D3—64no—
131C22×C3×D3#6714yes—
132C2×D32—616no—
133C2×D32—128no—
134C2×D32—128no—
135C2×C32×C4—21no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—31no—
139C2×C32×C4—32no—
140C2×C32×C4—32no—
141C2×C32×C4—44no—
142C2×C32×C4—61no—
143C2×C32×C4C22×D314yes—
144C2×C32×C4C22×D311yescharacteristic
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)—32no—
147C4⋊(C3⋊D3)—38no—
148C4⋊(C3⋊D3)—68no—
149C4⋊(C3⋊D3)—64no—
150C4⋊(C3⋊D3)—61no—
151C4⋊(C3⋊D3)—64no—
152C4⋊(C3⋊D3)—124no—
153C4⋊(C3⋊D3)C4×D312yes—
154C2×C3×Dic3—24no—
155C2×C3×Dic3#6714yes—
156C3×D12—38no—
157C3×D12—64no—
158C3×C4×D3C4×D318yes—
159C3×C4×D3D1218yes—
160#626—616no—
161C3⋊(C3⋊C8)—21no—
162C3⋊(C3⋊C8)—61no—
163C3⋊(C3⋊C8)—64no—
164C3⋊(C3⋊C8)—124no—
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—42no—
172C32×D3—22no—
173C23×D3—184no—
174C23×D3—181no—
175C2×C3×D4—91no—
176C22×C3×C4—34no—
177C2×#67—181no—
178D3×D4—94no—
179D3×D4—182no—
180C2×D12—34no—
181C2×D12—916no—
182C2×D12—94no—
183C2×D12—91no—
184C2×D12—184no—
185C2×D12—188no—
186C2×D12—184no—
187C2×C4×D3C2×C3⋊D311yescharacteristic
188C2×C3×C8—91no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—34no—
191C3×C4⋊C4D3214yescomplemented
192C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
193C3×C22⋊C4—64no—
194C3×C42—64no—
195(C2×D3)⋊C4—21no—
196C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
197C4×Dic3—21no—
198#208—188no—
199C2×C3⋊C8—34no—
200C2×C3⋊C8—94no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—184no—
203ΓL(1,𝔽25)—182no—
204C22×C32—24no—
205C22×C32—32no—
206C22×C32—61no—
207C22×C32—61no—
208C22×C32D3×D411yescharacteristic
209C22×C32C2×#6714yes—
210C2×C3⋊D3—22no—
211C2×C3⋊D3—41no—
212C2×C3⋊D3—41no—
213C2×C3⋊D3—62no—
214C2×C3⋊D3—68no—
215C2×C3⋊D3—128no—
216C2×C3⋊D3—128no—
217C2×C3⋊D3—121no—
218C2×C3⋊D3—121no—
219C2×C3⋊D3—124no—
220C2×C3⋊D3—124no—
221C2×C3⋊D3—244no—
222C2×C3⋊D3—244no—
223C2×C3×D3—24no—
224C2×C3×D3—24no—
225C2×C3×D3—68no—
226C2×C3×D3—124no—
227C2×C3×D3—124no—
228C2×C3×D3(C2×D3)⋊C418yes—
229D32—1232no—
230C32×C4—24no—
231C32×C4—24no—
232C32×C4—32no—
233C32×C4—41no—
234C32×C4—44no—
235C32×C4—84no—
236C32×C4—121no—
237C32×C4C2×C4×D314yes—
238C32×C4C2×C4×D311yescharacteristic
239C32×C4C2×D1214yes—
240C32×C4D3×D411yescharacteristic
241C3×Dic3—44no—
242C3×Dic3(C2×D3)⋊C418yes—
243C22×D4—271no—
244C2×M16—271no—
245#117—31no—
246#102—91no—
247#102—271no—
248#102—271no—
249#102—271no—
250C33#41811yescharacteristic, complemented, Sylow
251C23×C3—34no—
252C23×C3—181no—
253C22×D3—64no—
254C22×D3—1816no—
255C22×D3—184no—
256C22×D3—181no—
257C22×D3—184no—
258C22×D3—364no—
259C22×D3—362no—
260C22×D3—362no—
261C22×D3—368no—
262C22×D3—368no—
263C22×D3#63811yescharacteristic
264C3×D4—92no—
265C3×D4—181no—
266C2×C3×C4—24no—
267C2×C3×C4—21no—
268C2×C3×C4—24no—
269C2×C3×C4—34no—
270C2×C3×C4—38no—
271C2×C3×C4—38no—
272C2×C3×C4—44no—
273C2×C3×C4—64no—
274C2×C3×C4C2×D3214yes—
275C2×C3×C4C22×C3⋊D311yescharacteristic
276#67—184no—
277C2×Dic3—21no—
278C2×Dic3#63811yescharacteristic
279D12—38no—
280D12—64no—
281D12—98no—
282D12—92no—
283D12—188no—
284D12—1816no—
285D12—184no—
286D12—181no—
287D12—364no—
288C4×D3C4×C3⋊D312yes—
289C4×D3C4⋊(C3⋊D3)12yes—
290C3×C8—181no—
291C3⋊C8—64no—
292C3⋊C8—184no—
293C3⋊C8—181no—
294C3⋊C8—364no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—24no—
298C2×C32—62no—
299C2×C32D3×C22⋊C411yescharacteristic
300C2×C32#91111yescharacteristic
301C2×C32#91411yescharacteristic
302C2×C32C2×(C2×D3)⋊C414yes—
303C2×C32#94914yes—
304C2×C32#94914yes—
305C3⋊D3—42no—
306C3⋊D3—122no—
307C3⋊D3—128no—
308C3⋊D3—248no—
309C3×D3—28no—
310C3×D3—128no—
311C24—541no—
312C2×D4—91no—
313C2×D4—274no—
314C2×D4—271no—
315C2×D4—542no—
316C2×D4—541no—
317C22×C4—31no—
318M16—542no—
319C2×C8—91no—
320C2×C8—271no—
321C4⋊C4—31no—
322C4⋊C4D3×C3⋊D311yescharacteristic, complemented
323C22⋊C4—61no—
324C42—61no—
325C22×C3—24no—
326C22×C3—38no—
327C22×C3—64no—
328C22×C3—64no—
329C22×C3—182no—
330C22×C3—361no—
331C22×C3—361no—
332C22×C3D3×#6714yes—
333C22×C3C2×#63811yescharacteristic
334C2×D3—21no—
335C2×D3—21no—
336C2×D3—68no—
337C2×D3—124no—
338C2×D3—124no—
339C2×D3—182no—
340C2×D3—188no—
341C2×D3—368no—
342C2×D3—368no—
343C2×D3—3632no—
344C2×D3—361no—
345C2×D3—361no—
346C2×D3—364no—
347C2×D3—364no—
348C2×D3—724no—
349C2×D3—724no—
350C2×D3#375212yes—
351C3×C4—24no—
352C3×C4—24no—
353C3×C4—38no—
354C3×C4—41no—
355C3×C4—44no—
356C3×C4—84no—
357C3×C4—124no—
358C3×C4C4×D3214yes—
359C3×C4D3×D1214yes—
360C3×C4C2×C4×C3⋊D311yescharacteristic
361C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
362Dic3—41no—
363Dic3#375212yes—
364C32—24no—
365C32#477011yescharacteristic, complemented
366C32#500114yescomplemented
367C23—31no—
368C23—181no—
369C23—544no—
370C23—541no—
371C23—1082no—
372C23—1082no—
373D4—92no—
374D4—181no—
375D4—272no—
376D4—544no—
377D4—541no—
378C2×C4—21no—
379C2×C4—31no—
380C2×C4—32no—
381C2×C4—32no—
382C2×C4—61no—
383C2×C4C2×D3×C3⋊D311yescharacteristic
384C8—181no—
385C8—541no—
386C2×C3—24no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—68no—
390C2×C3—362no—
391C2×C3D3×(C2×D3)⋊C414yes—
392C2×C3#6392514yes—
393C2×C3#6393914yes—
394C2×C3C2×#375211yescharacteristic
395C2×C3#6415811yescharacteristic
396C2×C3#6415811yescharacteristic
397D3—22no—
398D3—128no—
399D3—368no—
400D3—362no—
401D3—728no—
402C22—32no—
403C22—61no—
404C22—61no—
405C22—182no—
406C22—361no—
407C22—361no—
408C22—542no—
409C22—1088no—
410C22—1081no—
411C22—1081no—
412C22D3×#63811yescharacteristic
413C4—32no—
414C4—41no—
415C4—121no—
416C4C4×D3×C3⋊D311yescharacteristic
417C4D3×C4⋊(C3⋊D3)11yescharacteristic
418C3—24no—
419C3#1059059214yescomplemented
420C3#1059270011yescharacteristic, complemented
421C2—62no—
422C2—362no—
423C2—1082no—
424C2D3×#375211yescharacteristic
425C2#1172354411yescharacteristic
426C2#1172356811yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×D3×C4⋊(C3⋊D3)C2
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#418
LCSG ▷ 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
2. C32×#892C2semidirect
3. C2×#88786C2semidirect
4. #11722179C2semidirect
5. C3×#63636C2semidirect
6. #11722035C2semidirect
7. #11722033C2semidirect
16. D3×C4⋊(C3⋊D3)C4semidirect
17. C33×C4⋊C4C22semidirect
18. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
19. C2×C3×C3⋊(C3⋊C8)C22semidirect
21. C32×C4⋊Dic3C22semidirect
28. C3×#892D3semidirect
35. #63636D3semidirect
77. C32×C4⋊C4C2×D3semidirect
78. C32×C4⋊C4C2×D3semidirect
85. C2×C3⋊(C3⋊C8)C2×D3semidirect
87. C3×C4⋊Dic3C2×D3semidirect
112. #892C3⋊D3semidirect
191. C3×C4⋊C4D32semidirect
192. C3×C4⋊C4C2×C3⋊D3semidirect
196. C4⋊Dic3C2×C3⋊D3semidirect
250. C33#418semidirect
322. C4⋊C4D3×C3⋊D3semidirect
365. C32#4770semidirect
366. C32#5001semidirect
419. C3#10590592semidirect
420. C3#10592700semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊(C3⋊D3)2
C32×#8922
C2×#887862
#117221792
C3×#636362
#117220352
#117220332
#105927003
#105905923