Finite Group Databasebeta

← #10592459⋊C3 #13141774

Subgroups — #10592459⋊C3 #13141774

Order 1728.

463 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 94 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
1#64149⋊C3C211yesmaximal, characteristic
2#64097⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic, complemented
6C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
7#63636⋊C3C211yesmaximal, characteristic, complemented
8#10592459C311yesmaximal, characteristic, complemented
9C3×#4760—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×#638⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×#634⋊C3—21no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16C4⋊C4×H(3,𝔽3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64149C2×C311yescharacteristic
22#64097C2×C311yescharacteristic, complemented
23C3×#940—31no—
24C3×#888—31no—
25C32×#102—21no—
26C32×#102D311yescharacteristic, complemented
27C32×#102C2×C311yescharacteristic, complemented
28#63659C2×C311yescharacteristic, complemented
29#63648C2×C311yescharacteristic, complemented
30C8×C3⋊Dic3C2×C311yescharacteristic, complemented
31#63636C2×C311yescharacteristic, complemented
32C3×#832—31no—
33C3×#821—31no—
34C3×C8×Dic3—31no—
35C3×#809—31no—
36C23×H(3,𝔽3)—21no—
37C22×(C3⋊D3)⋊C3—21no—
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42#638⋊C3—42no—
43#638⋊C3—42no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49#634⋊C3—42no—
50C8×H(3,𝔽3)—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52C3×#486—91no—
53#4760—31no—
54#4760—91no—
55C2×C32×D4—21no—
56C2×C32×D4C2×D311yescharacteristic
57C2×C32×D4C22×C311yescharacteristic
58C2×#638—21no—
59C2×#638—21no—
60C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
61C2×#634—21no—
62C2×C3×#67—61no—
63C2×C3×#67—61no—
64C2×C3×D12—31no—
65C2×C3×Dic6—61no—
66C2×C32×C8—21no—
67C2×C32×C8C2×D311yescharacteristic
68C2×C32×C8C22×C311yescharacteristic
69C32×C4⋊C4—21no—
70C32×C4⋊C4C2×D311yescharacteristic
71C32×C4⋊C4C22×C311yescharacteristic
72C4⋊(C3⋊Dic3)C22×C311yescharacteristic
73#3750—21no—
74C4×C3⋊Dic3C22×C311yescharacteristic
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76C3×C4⋊Dic3—31no—
77C3×Dic3⋊C4—61no—
78C3×C4×Dic3—31no—
79C2×C3×C3⋊C8—31no—
80C22×H(3,𝔽3)—41no—
81C22×H(3,𝔽3)—41no—
82C22×H(3,𝔽3)—41no—
83C22×H(3,𝔽3)C2×D411yescharacteristic
84C2×(C3⋊D3)⋊C3—41no—
85C2×(C3⋊D3)⋊C3—41no—
86C2×(C3⋊D3)⋊C3—41no—
87C4×H(3,𝔽3)—41no—
88C4×H(3,𝔽3)#4111yescharacteristic
89C4×H(3,𝔽3)#4111yescharacteristic
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3D812yes—
92(C3⋊Dic3)⋊C3SD1612yes—
93C3×C4⋊Q8—91no—
94C3×#127—91no—
95#940—31no—
96#940—91no—
97#888—31no—
98#888—91no—
99C3×#102—31no—
100C3×#102—31no—
101C3×#102—61no—
102C3×#102—91no—
103C3×#102—91no—
104C3×#102—91no—
105C3×#102C3×D311yescharacteristic
106C3×C4×C8—91no—
107#832—31no—
108#832—91no—
109#821—31no—
110#821—91no—
111C8×Dic3—31no—
112C8×Dic3—91no—
113#809—31no—
114#809—91no—
115C23×C32—21no—
116C23×C32—21no—
117C23×C32—41no—
118C22×C3⋊D3—21no—
119C22×C3×D3—61no—
120C32×D4—22no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—21no—
125C32×D4—41no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—41no—
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C23×C311yescharacteristic
132#638—42no—
133#638—42no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3C3×D411yescharacteristic
136C2×C3⋊Dic3C3×D411yescharacteristic
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139#634—42no—
140C3×#67—122no—
141C3×#67—122no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×Dic6—122no—
148C32×C8—21no—
149C32×C8—21no—
150C32×C8—41no—
151C3⋊(C3⋊C8)—21no—
152C3×C3⋊C8—61no—
153#486—91noSylow
154C2×H(3,𝔽3)—81no—
155C2×H(3,𝔽3)#12411yescharacteristic
156C2×H(3,𝔽3)C2×D811yescharacteristic
157C2×H(3,𝔽3)C2×SD1611yescharacteristic
158(C3⋊D3)⋊C3—81no—
159C2×C3×Q8—181no—
160C2×C3×D4—31no—
161C2×C3×D4—31no—
162C2×C3×D4—61no—
163C2×C3×D4—91no—
164C2×C3×D4—181no—
165C2×C3×D4—181no—
166C2×C3×D4C2×C3×D311yescharacteristic
167C2×#67—61no—
168C2×#67—61no—
169C2×#67—181no—
170C2×#67—181no—
171C2×D12—31no—
172C2×D12—91no—
173C2×Dic6—61no—
174C2×Dic6—181no—
175C2×C3×C8—31no—
176C2×C3×C8—31no—
177C2×C3×C8—61no—
178C2×C3×C8—91no—
179C2×C3×C8C2×C3×D311yescharacteristic
180C3×C4⋊C4—31no—
181C3×C4⋊C4—31no—
182C3×C4⋊C4—61no—
183C3×C4⋊C4—91no—
184C3×C4⋊C4—181no—
185C3×C4⋊C4C2×C3×D311yescharacteristic
186C3×C42—91no—
187C4⋊Dic3—31no—
188C4⋊Dic3—91no—
189Dic3⋊C4—61no—
190Dic3⋊C4—181no—
191C4×Dic3—31no—
192C4×Dic3—91no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—91no—
195C22×C32—21no—
196C22×C32—42no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32—81no—
204C22×C32—81no—
205C22×C32D3×D411yescharacteristic
206C22×C32C2×C3×D411yescharacteristic
207C2×C3⋊D3—41no—
208C2×C3⋊D3—41no—
209C2×C3⋊D3—41no—
210C2×C3×D3—121no—
211C2×C3×D3—121no—
212C2×C3×D3—121no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4—81no—
218C32×C4#23511yescharacteristic
219C32×C4#23711yescharacteristic
220C32×C4C3×#4111yescharacteristic
221C32×C4C3×#4111yescharacteristic
222C3⋊Dic3—41no—
223C3⋊Dic3C3×D812yes—
224C3⋊Dic3C3×SD1612yes—
225C3×Dic3—32no—
226C3×Dic3—32no—
227C3×Dic3—121no—
228C4⋊Q8—91no—
229#127—91no—
230#102—91no—
231#102—91no—
232#102—91no—
233#102(C3⋊D3)⋊C311yescharacteristic, complemented
234C4×C8—91no—
235H(3,𝔽3)#48611yescharacteristic, complemented, Sylow
236C23×C3—21no—
237C23×C3—61no—
238C23×C3—61no—
239C23×C3—121no—
240C23×C3—181no—
241C22×D3—61no—
242C22×D3—181no—
243C3×Q8—362no—
244C3×D4—21no—
245C3×D4—21no—
246C3×D4—61no—
247C3×D4—61no—
248C3×D4—61no—
249C3×D4—61no—
250C3×D4—62no—
251C3×D4—121no—
252C3×D4—181no—
253C3×D4—181no—
254C3×D4—362no—
255C3×D4—362no—
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×C4C22×C3×D311yescharacteristic
267#67—122no—
268#67—122no—
269#67—362no—
270#67—362no—
271C2×Dic3—31no—
272C2×Dic3—31no—
273C2×Dic3—61no—
274C2×Dic3—91no—
275C2×Dic3—91no—
276C2×Dic3—181no—
277D12—61no—
278D12—61no—
279D12—181no—
280D12—181no—
281Dic6—122no—
282Dic6—362no—
283C3×C8—21no—
284C3×C8—61no—
285C3×C8—61no—
286C3×C8—121no—
287C3×C8—181no—
288C3⋊C8—61no—
289C3⋊C8—181no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—82no—
294C2×C32—81no—
295C2×C32—81no—
296C2×C32#88511yescharacteristic
297C2×C32D3×D811yescharacteristic
298C2×C32D3×SD1611yescharacteristic
299C2×C32C3×#12411yescharacteristic
300C2×C32C2×C3×D811yescharacteristic
301C2×C32C2×C3×SD1611yescharacteristic
302C3⋊D3—81no—
303C3×D3—241no—
304C2×Q8—181no—
305C2×D4—91no—
306C2×D4—181no—
307C2×D4—181no—
308C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
309C2×C8—91no—
310C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
311C4⋊C4—91no—
312C4⋊C4—181no—
313C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
314C42—91no—
315C22×C3—31no—
316C22×C3—31no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—61no—
321C22×C3—121no—
322C22×C3—121no—
323C22×C3—121no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—121no—
327C22×C3—122no—
328C22×C3—241no—
329C22×C3—241no—
330C22×C3—361no—
331C22×C3—361no—
332C22×C3—361no—
333C22×C3C3×D3×D411yescharacteristic
334C2×D3—121no—
335C2×D3—121no—
336C2×D3—121no—
337C2×D3—361no—
338C2×D3—361no—
339C2×D3—361no—
340C3×C4—31no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—41no—
345C3×C4—61no—
346C3×C4—61no—
347C3×C4—92no—
348C3×C4—92no—
349C3×C4—121no—
350C3×C4—121no—
351C3×C4—241no—
352C3×C4—361no—
353C3×C4C3×#23511yescharacteristic
354C3×C4C3×#23711yescharacteristic
355Dic3—32no—
356Dic3—32no—
357Dic3—92no—
358Dic3—92no—
359Dic3—121no—
360Dic3—361no—
361C32—21no—
362C32#476011yescharacteristic, complemented
363C32C3×#48611yescharacteristic, complemented
364C23—21no—
365C23—181no—
366Q8—362no—
367D4—21no—
368D4—21no—
369D4—181no—
370D4—181no—
371D4—362no—
372D4—362no—
373C2×C4—21no—
374C2×C4—91no—
375C2×C4—91no—
376C2×C4—181no—
377C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
378C8—21no—
379C8—181no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—61no—
387C2×C3—61no—
388C2×C3—61no—
389C2×C3—81no—
390C2×C3—241no—
391C2×C3—241no—
392C2×C3—242no—
393C2×C3—721no—
394C2×C3C3×#88511yescharacteristic
395C2×C3C3×D3×D811yescharacteristic
396C2×C3C3×D3×SD1611yescharacteristic
397D3—241no—
398D3—721no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—361no—
403C22—361no—
404C22—361no—
405C22D4×(C3⋊D3)⋊C311yescharacteristic
406C4—41no—
407C4—92no—
408C4—92no—
409C4—361no—
410C4#3828⋊C311yescharacteristic
411C4#3830⋊C311yescharacteristic
412C3—31no—
413C3—31no—
414C3—61no—
415C3C3×#476011yescharacteristic
416C2—81no—
417C2—721no—
418C2#64094⋊C311yescharacteristic
419C2D8×(C3⋊D3)⋊C311yescharacteristic
420C2SD16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
prime radical#64149⋊C3C2
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#486
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. #64097⋊C3C2semidirect
3. H(3,𝔽3)×#102C2semidirect
4. #63659⋊C3C2semidirect
5. #63648⋊C3C2semidirect
6. C8×(C3⋊Dic3)⋊C3C2semidirect
7. #63636⋊C3C2semidirect
8. #10592459C3semidirect
22. #64097C2×C3semidirect
26. C32×#102D3semidirect
27. C32×#102C2×C3semidirect
28. #63659C2×C3semidirect
29. #63648C2×C3semidirect
30. C8×C3⋊Dic3C2×C3semidirect
31. #63636C2×C3semidirect
233. #102(C3⋊D3)⋊C3semidirect
235. H(3,𝔽3)#486semidirect
362. C32#4760semidirect
363. C32C3×#486semidirect

Maximal subgroups

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

subgroupindex
#64149⋊C32
#64097⋊C32
H(3,𝔽3)×#1022
#63659⋊C32
#63648⋊C32
C8×(C3⋊Dic3)⋊C32
#63636⋊C32
#105924593
C3×#47603