Finite Group Databasebeta

← #10592467⋊C3 #13141783

Subgroups — #10592467⋊C3 #13141783

Order 1728.

386 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 86 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#64145⋊C3C211yesmaximal, characteristic
2#64099⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic
4#63659⋊C3C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic, complemented
6C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592467C311yesmaximal, characteristic, complemented
9C3×#4768—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#3757⋊C3—21no—
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64145C2×C311yescharacteristic
22#64099C2×C311yescharacteristic, complemented
23C3×#936—31no—
24C3×#890—31no—
25C32×#102—21no—
26C32×#102D311yescharacteristic
27C32×#102C2×C311yescharacteristic
28#63659C2×C311yescharacteristic
29#63643C2×C311yescharacteristic, complemented
30C8×C3⋊Dic3C2×C311yescharacteristic, complemented
31#63637C2×C311yescharacteristic, complemented
32C3×#832—31no—
33C3×#816—31no—
34C3×C8×Dic3—31no—
35C3×#810—31no—
36C23×H(3,𝔽3)—21no—
37D4×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45#634⋊C3—21no—
46#634⋊C3—21no—
47C8×H(3,𝔽3)—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49C3×#488—91no—
50#4768—31no—
51#4768—91no—
52C2×C32×D4—21no—
53C2×C32×D4C2×D311yescharacteristic
54C2×C32×D4C22×C311yescharacteristic
55C2×#634C22×C311yescharacteristic
56C2×C3×Dic6—31no—
57C2×C32×C8—21no—
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C22×C311yescharacteristic
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C22×C311yescharacteristic
63#3757—21no—
64#3757—21no—
65C4⋊(C3⋊Dic3)—21no—
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67#3750—21no—
68C4×C3⋊Dic3C22×C311yescharacteristic
69C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
70C3×#217—61no—
71C3×#217—61no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—61no—
74C3×Dic3⋊C4—61no—
75C3×C4×Dic3—31no—
76C2×C3×C3⋊C8—31no—
77C22×H(3,𝔽3)—41no—
78C22×H(3,𝔽3)—41no—
79C22×H(3,𝔽3)—41no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)#4111yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88C3×#125—91no—
89C3×#124—91no—
90#936—31no—
91#936—91no—
92#890—31no—
93#890—91no—
94C3×#103—91no—
95C3×#103—91no—
96C3×#102—31no—
97C3×#102—31no—
98C3×#102—61no—
99C3×#102—91no—
100C3×#102C3×D311yescharacteristic
101C3×C4×C8—91no—
102#832—31no—
103#832—91no—
104#816—31no—
105#816—91no—
106C8×Dic3—31no—
107C8×Dic3—91no—
108#810—31no—
109#810—91no—
110C23×C32—21no—
111C23×C32—21no—
112C23×C32—41no—
113C32×D4—22no—
114C32×D4—21no—
115C32×D4—21no—
116C32×D4—21no—
117C32×D4—21no—
118C32×D4—41no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—41no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C23×C311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3C3×D411yescharacteristic
128C2×C3⋊Dic3C3×D411yescharacteristic
129#634—21no—
130#634—21no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C3×Dic6—61no—
136C3×Dic6—61no—
137C32×C8—21no—
138C32×C8—21no—
139C32×C8—41no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—61no—
142#488—91noSylow
143C2×H(3,𝔽3)—81no—
144C2×H(3,𝔽3)#12411yescharacteristic
145C2×H(3,𝔽3)#13511yescharacteristic
146C2×H(3,𝔽3)#13511yescharacteristic
147C2×C3×Q8—91no—
148C2×C3×D4—31no—
149C2×C3×D4—31no—
150C2×C3×D4—61no—
151C2×C3×D4C2×C3×D311yescharacteristic
152C2×Dic6—31no—
153C2×Dic6—91no—
154C2×C3×C8—31no—
155C2×C3×C8—31no—
156C2×C3×C8—61no—
157C2×C3×C8—91no—
158C2×C3×C8C2×C3×D311yescharacteristic
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—181no—
164C3×C4⋊C4—181no—
165C3×C4⋊C4C2×C3×D311yescharacteristic
166C3×C22⋊C4—181no—
167C3×C22⋊C4—181no—
168C3×C42—91no—
169#217—61no—
170#217—61no—
171#217—181no—
172#217—181no—
173C4⋊Dic3—31no—
174C4⋊Dic3—61no—
175C4⋊Dic3—91no—
176C4⋊Dic3—181no—
177Dic3⋊C4—61no—
178Dic3⋊C4—181no—
179C4×Dic3—31no—
180C4×Dic3—91no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—91no—
183C22×C32—21no—
184C22×C32—42no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—41no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32—81no—
192C22×C32—81no—
193C22×C32D3×D411yescharacteristic
194C22×C32C2×C3×D411yescharacteristic
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4—81no—
200C32×C4#23511yescharacteristic
201C32×C4#23711yescharacteristic
202C32×C4C3×#4111yescharacteristic
203C32×C4C3×#4111yescharacteristic
204C3⋊Dic3—21no—
205C3⋊Dic3—21no—
206C3⋊Dic3—41no—
207C3⋊Dic3—41no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—121no—
211C3×Dic3—121no—
212#125—91no—
213#124—91no—
214#103—91no—
215#103—91no—
216#102—91no—
217#102(C3⋊D3)⋊C311yescharacteristic
218C4×C8—91no—
219H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
220C23×C3—21no—
221C23×C3—61no—
222C23×C3—61no—
223C23×C3—121no—
224C3×Q8—181no—
225C3×Q8—181no—
226C3×D4—21no—
227C3×D4—21no—
228C3×D4—61no—
229C3×D4—61no—
230C3×D4—61no—
231C3×D4—61no—
232C3×D4—62no—
233C3×D4—121no—
234C2×C3×C4—21no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—121no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4C22×C3×D311yescharacteristic
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—91no—
251C2×Dic3—91no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254Dic6—61no—
255Dic6—61no—
256Dic6—181no—
257Dic6—181no—
258C3×C8—21no—
259C3×C8—61no—
260C3×C8—61no—
261C3×C8—121no—
262C3×C8—181no—
263C3⋊C8—61no—
264C3⋊C8—181no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—82no—
269C2×C32—81no—
270C2×C32—81no—
271C2×C32#88511yescharacteristic
272C2×C32#91211yescharacteristic
273C2×C32#91611yescharacteristic
274C2×C32C3×#12411yescharacteristic
275C2×C32C3×#13511yescharacteristic
276C2×C32C3×#13511yescharacteristic
277C2×Q8—91no—
278C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
279C2×C8—91no—
280C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
281C4⋊C4—91no—
282C4⋊C4—181no—
283C4⋊C4—181no—
284C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
285C22⋊C4—181no—
286C22⋊C4—181no—
287C42—91no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—61no—
294C22×C3—121no—
295C22×C3—121no—
296C22×C3—121no—
297C22×C3—121no—
298C22×C3—121no—
299C22×C3—121no—
300C22×C3—122no—
301C22×C3—241no—
302C22×C3—241no—
303C22×C3C3×D3×D411yescharacteristic
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—41no—
309C3×C4—61no—
310C3×C4—61no—
311C3×C4—121no—
312C3×C4—121no—
313C3×C4—181no—
314C3×C4—181no—
315C3×C4—241no—
316C3×C4—361no—
317C3×C4—361no—
318C3×C4C3×#23511yescharacteristic
319C3×C4C3×#23711yescharacteristic
320Dic3—61no—
321Dic3—61no—
322Dic3—121no—
323Dic3—121no—
324Dic3—181no—
325Dic3—181no—
326Dic3—361no—
327Dic3—361no—
328C32—21no—
329C32#476811yescharacteristic, complemented
330C32C3×#48811yescharacteristic, complemented
331C23—21no—
332Q8—181no—
333Q8—181no—
334D4—21no—
335D4—21no—
336C2×C4—21no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
342C8—21no—
343C8—181no—
344C2×C3—31no—
345C2×C3—31no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—61no—
351C2×C3—61no—
352C2×C3—61no—
353C2×C3—81no—
354C2×C3—241no—
355C2×C3—241no—
356C2×C3—242no—
357C2×C3C3×#88511yescharacteristic
358C2×C3C3×#91211yescharacteristic
359C2×C3C3×#91611yescharacteristic
360C22—41no—
361C22—41no—
362C22—41no—
363C22D4×(C3⋊D3)⋊C311yescharacteristic
364C4—41no—
365C4—181no—
366C4—181no—
367C4—361no—
368C4—361no—
369C4#3828⋊C311yescharacteristic
370C4#3830⋊C311yescharacteristic
371C3—31no—
372C3—31no—
373C3—61no—
374C3C3×#476811yescharacteristic
375C2—81no—
376C2#64094⋊C311yescharacteristic
377C2#64121⋊C311yescharacteristic
378C2#64125⋊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 radicalC2×D4×H(3,𝔽3)C22
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#488
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. #64099⋊C3C2semidirect
5. #63643⋊C3C2semidirect
6. C8×(C3⋊Dic3)⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. #10592467C3semidirect
22. #64099C2×C3semidirect
29. #63643C2×C3semidirect
30. C8×C3⋊Dic3C2×C3semidirect
31. #63637C2×C3semidirect
219. H(3,𝔽3)#488semidirect
329. C32#4768semidirect
330. C32C3×#488semidirect

Maximal subgroups

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

subgroupindex
#64145⋊C32
#64099⋊C32
H(3,𝔽3)×#1022
#63659⋊C32
#63643⋊C32
C8×(C3⋊Dic3)⋊C32
#63637⋊C32
#105924673
C3×#47683