Finite Group Databasebeta

← #10592517⋊C3 #13141900

Subgroups — #10592517⋊C3 #13141900

Order 1728.

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

normal subgroups only

This group has 462 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#64153⋊C3C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic
3C2×#3744⋊C3C211yesmaximal, characteristic
4#64102⋊C3C211yesmaximal, characteristic
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592517C311yesmaximal, characteristic, complemented
9C3×#4818—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C2×#634⋊C3C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#3755⋊C3—22no—
17#3752⋊C3—21no—
18#3750⋊C3—21no—
19#3750⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3744⋊C3—22no—
23#64153C2×C311yescharacteristic, complemented
24C2×#3755C2×C311yescharacteristic
25C2×#3744C2×C311yescharacteristic
26#64102C2×C311yescharacteristic
27C3×#944—31no—
28C2×C3×#215—31no—
29C2×C3×#204—31no—
30C3×#893—31no—
31C32×#103—21no—
32C32×#103D311yescharacteristic, complemented
33C32×#103C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35#63637C2×C311yescharacteristic, complemented
36C3×Dic3⋊C8—31no—
37C3×#810—31no—
38C22×(C3⋊D3)⋊C3—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)D412yes—
49C4×(C3⋊D3)⋊C3—42no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52C8×H(3,𝔽3)—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54C3×#461—91no—
55#4818—31no—
56#4818—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C22×C311yescharacteristic
60C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
61C2×C4×C3⋊D3—21no—
62C2×#634C22×C311yescharacteristic
63C2×C3×D12—31no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—31no—
66C2×C32×C8—21no—
67C2×C32×C8C2×D311yescharacteristic
68C2×C32×C8C22×C311yescharacteristic
69C32×C4⋊C4—21no—
70C32×C4⋊C4C2×D311yescharacteristic
71C32×C4⋊C4C22×C311yescharacteristic
72#3755—22no—
73#3752—21no—
74#3750—21no—
75#3750—21no—
76C4×C3⋊Dic3C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78#3744—22no—
79C3×#215—62no—
80C3×(C2×D3)⋊C4—61no—
81C3×Dic3⋊C4—61no—
82C3×Dic3⋊C4—61no—
83C3×C4×Dic3—31no—
84C2×C3×C3⋊C8—31no—
85C3×#204—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×(C3⋊D3)⋊C3—22no—
88C2×(C3⋊D3)⋊C3—41no—
89C2×(C3⋊D3)⋊C3—41no—
90C4×H(3,𝔽3)—41no—
91C4×H(3,𝔽3)—41no—
92C4×H(3,𝔽3)C2×D411yescharacteristic
93C4×H(3,𝔽3)#4111yescharacteristic
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—41no—
96(C3⋊Dic3)⋊C3SD1612yescomplemented
97C2×C3×SD16—91no—
98C2×C3×SD16—91no—
99C3×C4⋊Q8—91no—
100C3×C4×D4—91no—
101#944—31no—
102#944—91no—
103C2×#215—31no—
104C2×#215—91no—
105C2×#204—31no—
106C2×#204—91no—
107#893—31no—
108#893—91no—
109C3×C4⋊C8—91no—
110C3×#103—31no—
111C3×#103—31no—
112C3×#103—61no—
113C3×#103—91no—
114C3×#103C3×D311yescharacteristic
115Dic3⋊C8—31no—
116Dic3⋊C8—91no—
117#810—31no—
118#810—91no—
119C22×C3⋊D3—21no—
120C22×C3×D3—61no—
121C32×Q8—22no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—41no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—41no—
133C2×C32×C4—41no—
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C23×C311yescharacteristic
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3C3×D411yescharacteristic
138C2×C3⋊Dic3C3×D411yescharacteristic
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)C3×D412yes—
141C4×C3⋊D3—42no—
142#634—21no—
143#634—21no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—31no—
146C2×C3×Dic3—61no—
147C3×D12—32no—
148C3×D12—61no—
149C3×C4×D3—122no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C32×C8—21no—
153C32×C8—21no—
154C32×C8—41no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—61no—
157#461—91noSylow
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)C2×SD1611yescharacteristic
160C2×H(3,𝔽3)#13711yescharacteristic
161(C3⋊D3)⋊C3—42no—
162C2×C3×Q8—31no—
163C2×C3×Q8—31no—
164C2×C3×Q8—61no—
165C2×C3×Q8—91no—
166C2×C3×Q8C2×C3×D311yescharacteristic
167C2×C3×D4—91no—
168C22×C3×C4—181no—
169C2×D12—31no—
170C2×D12—91no—
171C2×C4×D3—61no—
172C2×C4×D3—181no—
173C2×Dic6—31no—
174C2×Dic6—91no—
175C3×SD16—182no—
176C3×SD16—182no—
177C2×C3×C8—31no—
178C2×C3×C8—31no—
179C2×C3×C8—61no—
180C2×C3×C8—91no—
181C2×C3×C8C2×C3×D311yescharacteristic
182C3×C4⋊C4—31no—
183C3×C4⋊C4—31no—
184C3×C4⋊C4—61no—
185C3×C4⋊C4—181no—
186C3×C4⋊C4—181no—
187C3×C4⋊C4C2×C3×D311yescharacteristic
188C3×C22⋊C4—181no—
189C3×C42—91no—
190#215—62no—
191#215—182no—
192(C2×D3)⋊C4—61no—
193(C2×D3)⋊C4—181no—
194Dic3⋊C4—61no—
195Dic3⋊C4—61no—
196Dic3⋊C4—181no—
197Dic3⋊C4—181no—
198C4×Dic3—31no—
199C4×Dic3—91no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—91no—
202#204—62no—
203#204—182no—
204C22×C32—21no—
205C22×C32D3×D411yescharacteristic
206C22×C32C2×C3×D411yescharacteristic
207C2×C3⋊D3—22no—
208C2×C3⋊D3—41no—
209C2×C3⋊D3—41no—
210C2×C3×D3—62no—
211C2×C3×D3—121no—
212C2×C3×D3—121no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—42no—
216C32×C4—41no—
217C32×C4—41no—
218C32×C4—41no—
219C32×C4—41no—
220C32×C4—81no—
221C32×C4#23511yescharacteristic
222C32×C4D3×D411yescharacteristic
223C32×C4C2×C3×D411yescharacteristic
224C32×C4C3×#4111yescharacteristic
225C3⋊Dic3—21no—
226C3⋊Dic3—41no—
227C3⋊Dic3C3×SD1612yescomplemented
228C3×Dic3—32no—
229C3×Dic3—61no—
230C3×Dic3—121no—
231C2×SD16—91no—
232C2×SD16—91no—
233C4⋊Q8—91no—
234C4×D4—91no—
235C4⋊C8—91no—
236#103—91no—
237#103(C3⋊D3)⋊C311yescharacteristic, complemented
238H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
239C23×C3—181no—
240C22×D3—61no—
241C22×D3—181no—
242C3×Q8—21no—
243C3×Q8—21no—
244C3×Q8—61no—
245C3×Q8—61no—
246C3×Q8—61no—
247C3×Q8—61no—
248C3×Q8—62no—
249C3×Q8—121no—
250C3×Q8—181no—
251C3×Q8—181no—
252C3×D4—92no—
253C3×D4—181no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—91no—
264C2×C3×C4—91no—
265C2×C3×C4—121no—
266C2×C3×C4—121no—
267C2×C3×C4—181no—
268C2×C3×C4—362no—
269C2×C3×C4C22×C3×D311yescharacteristic
270C2×Dic3—31no—
271C2×Dic3—31no—
272C2×Dic3—61no—
273C2×Dic3—91no—
274C2×Dic3—91no—
275C2×Dic3—181no—
276D12—32no—
277D12—61no—
278D12—92no—
279D12—181no—
280C4×D3—122no—
281C4×D3—362no—
282Dic6—61no—
283Dic6—61no—
284Dic6—181no—
285Dic6—181no—
286C3×C8—21no—
287C3×C8—61no—
288C3×C8—61no—
289C3×C8—121no—
290C3×C8—181no—
291C3⋊C8—61no—
292C3⋊C8—181no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32#88411yescharacteristic
297C2×C32D3×SD1611yescharacteristic
298C2×C32#91811yescharacteristic
299C2×C32C3×#12111yescharacteristic
300C2×C32C2×C3×SD1611yescharacteristic
301C2×C32C3×#13711yescharacteristic
302C3⋊D3—42no—
303C3×D3—122no—
304C2×Q8—91no—
305C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
306C2×D4—91no—
307C22×C4—181no—
308SD16—182no—
309SD16—182no—
310C2×C8—91no—
311C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
312C4⋊C4—181no—
313C4⋊C4—181no—
314C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
315C22⋊C4—181no—
316C42—91no—
317C22×C3—31no—
318C22×C3—31no—
319C22×C3—61no—
320C22×C3—182no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3C3×D3×D411yescharacteristic
324C2×D3—62no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—182no—
328C2×D3—361no—
329C2×D3—361no—
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—61no—
337C3×C4—61no—
338C3×C4—92no—
339C3×C4—121no—
340C3×C4—121no—
341C3×C4—121no—
342C3×C4—121no—
343C3×C4—122no—
344C3×C4—181no—
345C3×C4—241no—
346C3×C4—361no—
347C3×C4C3×#23511yescharacteristic
348C3×C4C3×D3×D411yescharacteristic
349Dic3—32no—
350Dic3—61no—
351Dic3—92no—
352Dic3—121no—
353Dic3—181no—
354Dic3—361no—
355C32—21no—
356C32#481811yescharacteristic, complemented
357C32C3×#46111yescharacteristic, complemented
358C23—181no—
359Q8—21no—
360Q8—21no—
361Q8—181no—
362Q8—181no—
363D4—92no—
364D4—181no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—91no—
368C2×C4—91no—
369C2×C4—181no—
370C2×C4—362no—
371C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
372C8—21no—
373C8—181no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—61no—
383C2×C3—362no—
384C2×C3C3×#88411yescharacteristic
385C2×C3C3×D3×SD1611yescharacteristic
386C2×C3C3×#91811yescharacteristic
387D3—122no—
388D3—362no—
389C22—182no—
390C22—361no—
391C22—361no—
392C22D4×(C3⋊D3)⋊C311yescharacteristic
393C4—41no—
394C4—41no—
395C4—92no—
396C4—181no—
397C4—361no—
398C4#3828⋊C311yescharacteristic
399C4D4×(C3⋊D3)⋊C311yescharacteristic
400C3—31no—
401C3—31no—
402C3—61no—
403C3C3×#481811yescharacteristic
404C2—362no—
405C2#64093⋊C311yescharacteristic
406C2SD16×(C3⋊D3)⋊C311yescharacteristic
407C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#461
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. #64153⋊C3C2semidirect
5. H(3,𝔽3)×#103C2semidirect
6. #63641⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. #10592517C3semidirect
23. #64153C2×C3semidirect
32. C32×#103D3semidirect
33. C32×#103C2×C3semidirect
34. #63641C2×C3semidirect
35. #63637C2×C3semidirect
96. (C3⋊Dic3)⋊C3SD16semidirect
227. C3⋊Dic3C3×SD16semidirect
237. #103(C3⋊D3)⋊C3semidirect
238. H(3,𝔽3)#461semidirect
356. C32#4818semidirect
357. C32C3×#461semidirect

Maximal subgroups

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

subgroupindex
#64153⋊C32
C2×#3755⋊C32
C2×#3744⋊C32
#64102⋊C32
H(3,𝔽3)×#1032
#63641⋊C32
#63637⋊C32
#105925173
C3×#48183