Finite Group Databasebeta

← #10592441⋊C3 #13141654

Subgroups — #10592441⋊C3 #13141654

Order 1728.

612 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 90 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#64361⋊C3C211yesmaximal, characteristic
2#64144⋊C3C211yesmaximal, characteristic, complemented
3#64095⋊C3C212yesmaximal, complemented
4H(3,𝔽3)×#99C211yesmaximal, characteristic, complemented
5#63633⋊C3C212yesmaximal, complemented
6#10592441C311yesmaximal, characteristic, complemented
7C3×#4742—31nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3—21no—
10C2×#638⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
12#3830⋊C3—22no—
13D4×(C3⋊D3)⋊C3—22no—
14#3828⋊C3—21no—
15C22⋊C4×H(3,𝔽3)C2212yescomplemented
16#3757⋊C3—21no—
17#3757⋊C3C2211yescharacteristic, complemented
18#3752⋊C3—22no—
19C4⋊((C3⋊Dic3)⋊C3)—22no—
20#3750⋊C3—21no—
21#64361C2×C311yescharacteristic
22C3×#1004—31no—
23#64144C2×C311yescharacteristic, complemented
24#64095C2×C312yescomplemented
25C3×#935—31no—
26C3×#886—32no—
27C32×#99—21no—
28C32×#99D311yescharacteristic, complemented
29C32×#99C2×C311yescharacteristic, complemented
30#63633C2×C312yescomplemented
31C3×#806—32no—
32C23×H(3,𝔽3)D411yescharacteristic
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×(C3⋊D3)⋊C3—21no—
35C22×(C3⋊D3)⋊C3D412yes—
36D4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)—22no—
38C2×C4×H(3,𝔽3)D411yescharacteristic
39#638⋊C3—22no—
40#638⋊C3—22no—
41#638⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3D412yes—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4×(C3⋊D3)⋊C3—22no—
49#634⋊C3—21no—
50C3×#458—91no—
51#4742—31no—
52#4742—91no—
53C2×C32×D4—21no—
54C2×C32×D4C2×D311yescharacteristic
55C2×C32×D4C22×C311yescharacteristic
56C2×#638—21no—
57C2×#638C22×C312yes—
58C22×C3⋊Dic3C22×C311yescharacteristic, complemented
59#3830—22no—
60D4×C3⋊D3—22no—
61#3828—21no—
62C2×C3×#67—32no—
63C2×C3×#67—61no—
64C22×C3×Dic3—31no—
65C3×#237—62no—
66C3×D3×D4—62no—
67C3×#235—61no—
68C32×C22⋊C4—22no—
69C32×C22⋊C4C2×D312yescomplemented
70C32×C22⋊C4C22×C312yescomplemented
71#3757—21no—
72#3757C22×C311yescharacteristic, complemented
73#3752—22no—
74C4⋊(C3⋊Dic3)—22no—
75#3750—21no—
76C3×#217—31no—
77C3×#217—61no—
78C3×(C2×D3)⋊C4—62no—
79C3×C4⋊Dic3—62no—
80C3×Dic3⋊C4—61no—
81C22×H(3,𝔽3)—21no—
82C22×H(3,𝔽3)—41no—
83C22×H(3,𝔽3)—41no—
84C22×H(3,𝔽3)C2×D411yescharacteristic
85C22×H(3,𝔽3)C2×D412yes—
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3—42no—
88C2×(C3⋊D3)⋊C3—81no—
89C4×H(3,𝔽3)—21no—
90C4×H(3,𝔽3)—42no—
91(C3⋊Dic3)⋊C3—22no—
92(C3⋊Dic3)⋊C3—41no—
93(C3⋊Dic3)⋊C3—41no—
94C3×H(5,𝔽2)—91no—
95#1004—31no—
96#1004—91no—
97C3×#123—92no—
98C3×#123—91no—
99#935—31no—
100#935—91no—
101#886—32no—
102#886—92no—
103C3×#99—31no—
104C3×#99—31no—
105C3×#99—61no—
106C3×#99—92no—
107C3×#99C3×D311yescharacteristic
108#806—32no—
109#806—92no—
110C23×C32—21no—
111C23×C32—21no—
112C23×C32D1211yescharacteristic
113C23×C32C3×D411yescharacteristic
114C23×C32C22×D311yescharacteristic
115C23×C32C23×C311yescharacteristic
116C22×C3⋊D3—21no—
117C22×C3⋊D3C3×D412yes—
118C22×C3×D3—32no—
119C22×C3×D3—61no—
120C32×D4—22no—
121C32×D4—22no—
122C32×D4—42no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—21no—
126C2×C32×C4—42no—
127C2×C32×C4D1211yescharacteristic
128C2×C32×C4C3×D411yescharacteristic
129#638—22no—
130#638—22no—
131#638—22no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—22no—
136C2×C3⋊Dic3C3×D412yes—
137C4⋊(C3⋊D3)—21no—
138C4×C3⋊D3—22no—
139#634—21no—
140C3×#67—62no—
141C3×#67—62no—
142C3×#67—62no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—62no—
148C3×D12—61no—
149C3×C4×D3—62no—
150C3×Dic6—61no—
151#458—91noSylow
152C2×H(3,𝔽3)—21no—
153C2×H(3,𝔽3)—22no—
154C2×H(3,𝔽3)—41no—
155C2×H(3,𝔽3)#12011yescharacteristic
156(C3⋊D3)⋊C3—42no—
157C3×#41—181no—
158C3×#41—182no—
159C2×C3×D4—31no—
160C2×C3×D4—31no—
161C2×C3×D4—61no—
162C2×C3×D4—92no—
163C2×C3×D4—182no—
164C2×C3×D4—181no—
165C2×C3×D4C2×C3×D311yescharacteristic
166C22×C3×C4—91no—
167C2×#67—32no—
168C2×#67—61no—
169C2×#67—92no—
170C2×#67—181no—
171C22×Dic3—31no—
172C22×Dic3—91no—
173#237—62no—
174#237—182no—
175D3×D4—62no—
176D3×D4—182no—
177#235—61no—
178#235—181no—
179C3×C4⋊C4—181no—
180C3×C4⋊C4—182no—
181C3×C22⋊C4—32no—
182C3×C22⋊C4—32no—
183C3×C22⋊C4—62no—
184C3×C22⋊C4—91no—
185C3×C22⋊C4—182no—
186C3×C22⋊C4—181no—
187C3×C22⋊C4C2×C3×D312yes—
188#217—31no—
189#217—61no—
190#217—91no—
191#217—181no—
192(C2×D3)⋊C4—62no—
193(C2×D3)⋊C4—182no—
194C4⋊Dic3—62no—
195C4⋊Dic3—182no—
196Dic3⋊C4—61no—
197Dic3⋊C4—181no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—22no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—81no—
208C22×C32—81no—
209C22×C32C2×D1211yescharacteristic
210C22×C32D3×D412yes—
211C22×C32C2×C3×D411yescharacteristic
212C22×C32C2×C3×D412yes—
213C2×C3⋊D3—22no—
214C2×C3⋊D3—42no—
215C2×C3⋊D3—81no—
216C2×C3×D3—62no—
217C2×C3×D3—122no—
218C2×C3×D3—241no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—41no—
222C32×C4—42no—
223C32×C4—42no—
224C32×C4—82no—
225C3⋊Dic3—22no—
226C3⋊Dic3—41no—
227C3⋊Dic3—41no—
228C3×Dic3—62no—
229C3×Dic3—121no—
230C3×Dic3—121no—
231H(5,𝔽2)—91no—
232#123—91no—
233#123—92no—
234#99—92no—
235#99(C3⋊D3)⋊C311yescharacteristic, complemented
236H(3,𝔽3)#45811yescharacteristic, complemented, Sylow
237C23×C3—31no—
238C23×C3—31no—
239C23×C3—31no—
240C23×C3—31no—
241C23×C3—61no—
242C23×C3—61no—
243C23×C3—92no—
244C23×C3—181no—
245C23×C3C3×D1211yescharacteristic
246C23×C3C22×C3×D311yescharacteristic
247C22×D3—32no—
248C22×D3—61no—
249C22×D3—92no—
250C22×D3—181no—
251C3×Q8—181no—
252C3×D4—22no—
253C3×D4—62no—
254C3×D4—62no—
255C3×D4—122no—
256C3×D4—181no—
257C3×D4—182no—
258C3×D4—182no—
259C3×D4—182no—
260C2×C3×C4—22no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—61no—
266C2×C3×C4—92no—
267C2×C3×C4—122no—
268C2×C3×C4—182no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4—182no—
273C2×C3×C4C3×D1211yescharacteristic
274#67—62no—
275#67—62no—
276#67—62no—
277#67—182no—
278#67—182no—
279#67—182no—
280C2×Dic3—32no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—62no—
285C2×Dic3—92no—
286C2×Dic3—181no—
287C2×Dic3—181no—
288C2×Dic3—181no—
289C2×Dic3—182no—
290D12—61no—
291D12—181no—
292C4×D3—62no—
293C4×D3—182no—
294Dic6—61no—
295Dic6—181no—
296C2×C32—21no—
297C2×C32—22no—
298C2×C32—21no—
299C2×C32—22no—
300C2×C32—21no—
301C2×C32—42no—
302C2×C32—41no—
303C2×C32—41no—
304C2×C32—41no—
305C2×C32—81no—
306C2×C32#88211yescharacteristic
307C2×C32C3×#12011yescharacteristic
308C3⋊D3—42no—
309C3×D3—122no—
310#41—182no—
311#41—181no—
312C2×D4—92no—
313C2×D4—182no—
314C2×D4—181no—
315C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
316C22×C4—91no—
317C4⋊C4—182no—
318C4⋊C4—181no—
319C22⋊C4—91no—
320C22⋊C4—182no—
321C22⋊C4—181no—
322C22⋊C4C2×(C3⋊D3)⋊C312yescomplemented
323C22×C3—21no—
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—32no—
327C22×C3—32no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—61no—
331C22×C3—61no—
332C22×C3—61no—
333C22×C3—62no—
334C22×C3—121no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—182no—
340C22×C3—241no—
341C22×C3—241no—
342C22×C3—362no—
343C22×C3—721no—
344C22×C3C2×C3×D1211yescharacteristic
345C22×C3C3×D3×D412yes—
346C2×D3—62no—
347C2×D3—122no—
348C2×D3—182no—
349C2×D3—241no—
350C2×D3—362no—
351C2×D3—721no—
352C3×C4—21no—
353C3×C4—42no—
354C3×C4—61no—
355C3×C4—61no—
356C3×C4—121no—
357C3×C4—122no—
358C3×C4—122no—
359C3×C4—182no—
360C3×C4—242no—
361C3×C4—361no—
362C3×C4—361no—
363Dic3—62no—
364Dic3—121no—
365Dic3—121no—
366Dic3—182no—
367Dic3—361no—
368Dic3—361no—
369C32—21no—
370C32#474211yescharacteristic, complemented
371C32C3×#45811yescharacteristic, complemented
372C23—92no—
373C23—181no—
374C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
375C23C22×(C3⋊D3)⋊C311yescharacteristic
376Q8—181no—
377D4—22no—
378D4—181no—
379D4—182no—
380D4—182no—
381D4—182no—
382C2×C4—22no—
383C2×C4—92no—
384C2×C4—182no—
385C2×C4—181no—
386C2×C4—181no—
387C2×C4—181no—
388C2×C4—182no—
389C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
390C2×C3—21no—
391C2×C3—22no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—41no—
395C2×C3—61no—
396C2×C3—62no—
397C2×C3—62no—
398C2×C3—61no—
399C2×C3—61no—
400C2×C3—122no—
401C2×C3—121no—
402C2×C3—121no—
403C2×C3—121no—
404C2×C3—241no—
405C2×C3—362no—
406C2×C3C3×#88211yescharacteristic
407D3—122no—
408D3—362no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—182no—
413C22—362no—
414C22—721no—
415C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
416C22D4×(C3⋊D3)⋊C312yes—
417C4—21no—
418C4—42no—
419C4—182no—
420C4—361no—
421C4—361no—
422C3—31no—
423C3—31no—
424C3—61no—
425C3C3×#474211yescharacteristic
426C2—22no—
427C2—21no—
428C2—41no—
429C2—362no—
430C2#64091⋊C311yescharacteristic
subgroupG/H
centerC2#64091⋊C3
commutator, cosocleC23×C32C23×C3
FrattiniC23×C3C22×C3×D3
FittingH(3,𝔽3)×#99C2
socleC2×C3C3×#882
prime radical#64361⋊C3C2
hypercenter#99(C3⋊D3)⋊C3
hypocenterC32C3×#458
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. #64144⋊C3C2semidirect
3. #64095⋊C3C2semidirect
4. H(3,𝔽3)×#99C2semidirect
5. #63633⋊C3C2semidirect
6. #10592441C3semidirect
11. C22×(C3⋊Dic3)⋊C3C22semidirect
15. C22⋊C4×H(3,𝔽3)C22semidirect
17. #3757⋊C3C22semidirect
23. #64144C2×C3semidirect
24. #64095C2×C3semidirect
28. C32×#99D3semidirect
29. C32×#99C2×C3semidirect
30. #63633C2×C3semidirect
58. C22×C3⋊Dic3C22×C3semidirect
69. C32×C22⋊C4C2×D3semidirect
70. C32×C22⋊C4C22×C3semidirect
72. #3757C22×C3semidirect
235. #99(C3⋊D3)⋊C3semidirect
236. H(3,𝔽3)#458semidirect
322. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
370. C32#4742semidirect
371. C32C3×#458semidirect

Maximal subgroups

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

subgroupindex
#64361⋊C32
#64144⋊C32
#64095⋊C32
H(3,𝔽3)×#992
#63633⋊C32
#105924413
C3×#47423