Finite Group Databasebeta

← #10592455⋊C3 #13141803

Subgroups — #10592455⋊C3 #13141803

Order 1728.

519 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 116 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⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic
3#64102⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic, complemented
6C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592455C311yesmaximal, characteristic, complemented
9C3×#4756—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3—21no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#3757⋊C3—21no—
17#3753⋊C3C414yescomplemented
18#3752⋊C3—21no—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22D4×C3⋊Dic3C2×C311yescharacteristic, complemented
23C2×#3753C2×C311yescharacteristic
24#64102C2×C311yescharacteristic, complemented
25C3×D4×Dic3—31no—
26C2×C3×#213—31no—
27C3×#893—31no—
28C32×#102—21no—
29C32×#102D311yescharacteristic, complemented
30C32×#102C2×C311yescharacteristic, complemented
31#63648C2×C311yescharacteristic, complemented
32C8×C3⋊Dic3C2×C311yescharacteristic, complemented
33#63634C2×C311yescharacteristic, complemented
34C3×#821—31no—
35C3×C8×Dic3—31no—
36C3×#807—31no—
37C23×H(3,𝔽3)—21no—
38C22×(C3⋊D3)⋊C3—21no—
39D4×H(3,𝔽3)—21no—
40D4×H(3,𝔽3)C2×C412yescomplemented
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—42no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
49C4×(C3⋊D3)⋊C3—42no—
50C8×H(3,𝔽3)—21no—
51(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
52C3×C4×D8—91no—
53#4756—31no—
54#4756—91no—
55C2×C32×D4—21no—
56C2×C32×D4C2×D311yescharacteristic
57C2×C32×D4C22×C311yescharacteristic
58C22×C3⋊Dic3—21no—
59C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
60C2×C4×C3⋊D3—21no—
61C22×C3×Dic3—61no—
62C2×C3×D12—31no—
63C2×C3×C4×D3—61no—
64C2×C32×C8—21no—
65C2×C32×C8C2×D311yescharacteristic
66C2×C32×C8C22×C311yescharacteristic
67C32×C4⋊C4—21no—
68C32×C4⋊C4C2×D311yescharacteristic
69C32×C4⋊C4C22×C311yescharacteristic
70#3757—21no—
71#3753C3×C414yescomplemented
72#3752—21no—
73C4⋊(C3⋊Dic3)C22×C311yescharacteristic
74C4×C3⋊Dic3C22×C311yescharacteristic
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76C3×#217—61no—
77C3×#213—34no—
78C3×(C2×D3)⋊C4—61no—
79C3×C4⋊Dic3—31no—
80C3×C4×Dic3—31no—
81C2×C3×C3⋊C8—31no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)—41no—
84C22×H(3,𝔽3)—41no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3—41no—
88C2×(C3⋊D3)⋊C3—41no—
89C4×H(3,𝔽3)—41no—
90C4×H(3,𝔽3)C22×C411yescharacteristic
91C4×H(3,𝔽3)#4111yescharacteristic
92(C3⋊Dic3)⋊C3—21no—
93(C3⋊Dic3)⋊C3—41no—
94(C3⋊Dic3)⋊C3D812yescomplemented
95C2×C3×D8—91no—
96C3×C4×D4—91no—
97C3×C4×D4—91no—
98D4×Dic3—31no—
99D4×Dic3—91no—
100C2×#213—31no—
101C2×#213—91no—
102#893—31no—
103#893—91no—
104C3×#107—91no—
105C3×#102—31no—
106C3×#102—31no—
107C3×#102—61no—
108C3×#102—91no—
109C3×#102C3×D311yescharacteristic
110C3×C4×C8—91no—
111#821—31no—
112#821—91no—
113C8×Dic3—31no—
114C8×Dic3—91no—
115#807—31no—
116#807—91no—
117C23×C32—21no—
118C23×C32—21no—
119C23×C32—41no—
120C22×C3⋊D3—21no—
121C22×C3×D3—61no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—22no—
125C32×D4—41no—
126C32×D4C4×D312yescomplemented
127C32×D4C2×C3×C412yescomplemented
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—41no—
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C23×C311yescharacteristic
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—42no—
136C2×C3⋊Dic3C3×D411yescharacteristic
137C2×C3⋊Dic3C3×D411yescharacteristic
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)C2×C3×C412yescomplemented
140C4×C3⋊D3—42no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—122no—
145C3×D12—32no—
146C3×D12—61no—
147C3×C4×D3—122no—
148C32×C8—21no—
149C32×C8—21no—
150C32×C8—41no—
151C3⋊(C3⋊C8)C2×C3×C412yescomplemented
152C3×C3⋊C8—32no—
153C4×D8—91noSylow
154C2×H(3,𝔽3)—42no—
155C2×H(3,𝔽3)C4×D411yescharacteristic
156C2×H(3,𝔽3)C2×D811yescharacteristic
157C2×H(3,𝔽3)#13511yescharacteristic
158(C3⋊D3)⋊C3—42no—
159C2×C3×D4—31no—
160C2×C3×D4—31no—
161C2×C3×D4—61no—
162C2×C3×D4—91no—
163C2×C3×D4C2×C3×D311yescharacteristic
164C22×C3×C4—181no—
165C22×C3×C4—181no—
166C22×Dic3—61no—
167C22×Dic3—181no—
168C2×D12—31no—
169C2×D12—91no—
170C2×C4×D3—61no—
171C2×C4×D3—181no—
172C3×D8—94no—
173C2×C3×C8—31no—
174C2×C3×C8—31no—
175C2×C3×C8—61no—
176C2×C3×C8—91no—
177C2×C3×C8C2×C3×D311yescharacteristic
178C3×C4⋊C4—31no—
179C3×C4⋊C4—31no—
180C3×C4⋊C4—61no—
181C3×C4⋊C4—91no—
182C3×C4⋊C4C2×C3×D311yescharacteristic
183C3×C22⋊C4—181no—
184C3×C22⋊C4—181no—
185C3×C42—91no—
186#217—61no—
187#217—181no—
188#213—34no—
189#213—94no—
190(C2×D3)⋊C4—61no—
191(C2×D3)⋊C4—181no—
192C4⋊Dic3—31no—
193C4⋊Dic3—91no—
194C4×Dic3—31no—
195C4×Dic3—91no—
196C2×C3⋊C8—31no—
197C2×C3⋊C8—91no—
198C22×C32—22no—
199C22×C32—22no—
200C22×C32—21no—
201C22×C32—42no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—41no—
206C22×C32—81no—
207C22×C32—81no—
208C22×C32D3×D411yescharacteristic
209C22×C32C2×C3×D411yescharacteristic
210C2×C3⋊D3—22no—
211C2×C3⋊D3—41no—
212C2×C3⋊D3—41no—
213C2×C3×D3—62no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4—41no—
219C32×C4—41no—
220C32×C4—81no—
221C32×C4C2×C4×D311yescharacteristic
222C32×C4#23711yescharacteristic
223C32×C4C22×C3×C411yescharacteristic
224C32×C4C3×#4111yescharacteristic
225C3⋊Dic3—21no—
226C3⋊Dic3—41no—
227C3⋊Dic3C3×D812yescomplemented
228C3×Dic3—32no—
229C3×Dic3—61no—
230C3×Dic3—121no—
231C2×D8—91no—
232C4×D4—91no—
233C4×D4—91no—
234#107—91no—
235#102—91no—
236#102(C3⋊D3)⋊C311yescharacteristic, complemented
237C4×C8—91no—
238H(3,𝔽3)C4×D811yescharacteristic, complemented, Sylow
239C23×C3—21no—
240C23×C3—61no—
241C23×C3—61no—
242C23×C3—121no—
243C23×C3—181no—
244C22×D3—61no—
245C22×D3—181no—
246C3×D4—21no—
247C3×D4—32no—
248C3×D4—32no—
249C3×D4—61no—
250C3×D4—61no—
251C3×D4—62no—
252C3×D4—92no—
253C3×D4—121no—
254C3×D4—181no—
255C3×D4C3×C4×D312yes—
256C2×C3×C4—21no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—91no—
263C2×C3×C4—91no—
264C2×C3×C4—121no—
265C2×C3×C4—181no—
266C2×C3×C4—362no—
267C2×C3×C4—362no—
268C2×C3×C4C22×C3×D311yescharacteristic
269C2×Dic3—31no—
270C2×Dic3—31no—
271C2×Dic3—61no—
272C2×Dic3—91no—
273C2×Dic3—91no—
274C2×Dic3—122no—
275C2×Dic3—181no—
276C2×Dic3—362no—
277D12—32no—
278D12—61no—
279D12—92no—
280D12—181no—
281C4×D3—122no—
282C4×D3—362no—
283C3×C8—21no—
284C3×C8—61no—
285C3×C8—61no—
286C3×C8—92no—
287C3×C8—121no—
288C3⋊C8—32no—
289C3⋊C8—92no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—42no—
294C2×C32—42no—
295C2×C32—82no—
296C2×C32#67⋊C411yescharacteristic
297C2×C32D3×D811yescharacteristic
298C2×C32#91611yescharacteristic
299C2×C32C3×C4×D411yescharacteristic
300C2×C32C2×C3×D811yescharacteristic
301C2×C32C3×#13511yescharacteristic
302C3⋊D3—42no—
303C3×D3—122no—
304C2×D4—91no—
305C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
306C22×C4—181no—
307C22×C4—181no—
308D8—94no—
309C2×C8—91no—
310C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
311C4⋊C4—91no—
312C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
313C22⋊C4—181no—
314C22⋊C4—181no—
315C42—91no—
316C22×C3—22no—
317C22×C3—31no—
318C22×C3—31no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—62no—
322C22×C3—62no—
323C22×C3—61no—
324C22×C3—122no—
325C22×C3—121no—
326C22×C3—121no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3—182no—
330C22×C3—241no—
331C22×C3—241no—
332C22×C3—361no—
333C22×C3—361no—
334C22×C3C3×D3×D411yescharacteristic
335C2×D3—62no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—182no—
339C2×D3—361no—
340C2×D3—361no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—41no—
346C3×C4—61no—
347C3×C4—61no—
348C3×C4—92no—
349C3×C4—121no—
350C3×C4—121no—
351C3×C4—181no—
352C3×C4—241no—
353C3×C4—361no—
354C3×C4C2×C3×C4×D311yescharacteristic
355C3×C4C3×#23711yescharacteristic
356Dic3—32no—
357Dic3—61no—
358Dic3—92no—
359Dic3—121no—
360Dic3—181no—
361Dic3—361no—
362C32—21no—
363C32#475611yescharacteristic, complemented
364C32C3×C4×D811yescharacteristic, complemented
365C23—21no—
366C23—181no—
367D4—21no—
368D4—92no—
369D4—181no—
370D4C4×(C3⋊D3)⋊C312yescomplemented
371C2×C4—21no—
372C2×C4—91no—
373C2×C4—91no—
374C2×C4—181no—
375C2×C4—362no—
376C2×C4—362no—
377C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
378C8—21no—
379C8—92no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—42no—
387C2×C3—61no—
388C2×C3—61no—
389C2×C3—61no—
390C2×C3—122no—
391C2×C3—122no—
392C2×C3—242no—
393C2×C3—362no—
394C2×C3C3×#67⋊C411yescharacteristic
395C2×C3C3×D3×D811yescharacteristic
396C2×C3C3×#91611yescharacteristic
397D3—122no—
398D3—362no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—182no—
403C22—361no—
404C22—361no—
405C22D4×(C3⋊D3)⋊C311yescharacteristic
406C4—41no—
407C4—92no—
408C4—181no—
409C4—361no—
410C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
411C4#3830⋊C311yescharacteristic
412C3—31no—
413C3—31no—
414C3—61no—
415C3C3×#475611yescharacteristic
416C2—42no—
417C2—362no—
418C2#64090⋊C311yescharacteristic
419C2D8×(C3⋊D3)⋊C311yescharacteristic
420C2#64125⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
prime radicalC2×#3753⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×C4×D8
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
1. D4×(C3⋊Dic3)⋊C3C2semidirect
3. #64102⋊C3C2semidirect
4. H(3,𝔽3)×#102C2semidirect
5. #63648⋊C3C2semidirect
6. C8×(C3⋊Dic3)⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592455C3semidirect
17. #3753⋊C3C4semidirect
22. D4×C3⋊Dic3C2×C3semidirect
24. #64102C2×C3semidirect
29. C32×#102D3semidirect
30. C32×#102C2×C3semidirect
31. #63648C2×C3semidirect
32. C8×C3⋊Dic3C2×C3semidirect
33. #63634C2×C3semidirect
40. D4×H(3,𝔽3)C2×C4semidirect
48. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
51. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
71. #3753C3×C4semidirect
94. (C3⋊Dic3)⋊C3D8semidirect
126. C32×D4C4×D3semidirect
127. C32×D4C2×C3×C4semidirect
139. C4⋊(C3⋊D3)C2×C3×C4semidirect
151. C3⋊(C3⋊C8)C2×C3×C4semidirect
227. C3⋊Dic3C3×D8semidirect
236. #102(C3⋊D3)⋊C3semidirect
238. H(3,𝔽3)C4×D8semidirect
363. C32#4756semidirect
364. C32C3×C4×D8semidirect
370. D4C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
D4×(C3⋊Dic3)⋊C32
C2×#3753⋊C32
#64102⋊C32
H(3,𝔽3)×#1022
#63648⋊C32
C8×(C3⋊Dic3)⋊C32
#63634⋊C32
#105924553
C3×#47563