Finite Group Databasebeta

← #10592744⋊C3 #13142882

Subgroups — #10592744⋊C3 #13142882

Order 1728.

410 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 92 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
1H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic
2#64135⋊C3C211yesmaximal, characteristic
3#63662⋊C3C211yesmaximal, characteristic, complemented
4#63662⋊C3C211yesmaximal, characteristic, complemented
5#63657⋊C3C211yesmaximal, characteristic
6#63635⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592744C311yesmaximal, characteristic, complemented
9C3×#5045—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C22⋊C4×H(3,𝔽3)—21no—
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C32×C22⋊Q8—21no—
22C32×C22⋊Q8D311yescharacteristic
23C32×C22⋊Q8C2×C311yescharacteristic
24#64135C2×C311yescharacteristic
25C3×#926—31no—
26#63662C2×C311yescharacteristic, complemented
27#63662C2×C311yescharacteristic, complemented
28#63657C2×C311yescharacteristic
29#63635C2×C311yescharacteristic, complemented
30#63634C2×C311yescharacteristic, complemented
31C3×#835—31no—
32C3×#835—31no—
33C3×#830—31no—
34C3×#808—31no—
35C3×#807—31no—
36C23×H(3,𝔽3)D411yescharacteristic
37Q8×H(3,𝔽3)—21no—
38Q8×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49C3×#485—91no—
50#5045—31no—
51#5045—91no—
52C2×C32×Q8—21no—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C32×Q8C22×C311yescharacteristic
55C22×C32×C4—21no—
56C22×C32×C4C2×D311yescharacteristic
57C22×C32×C4C22×C311yescharacteristic
58C32×C4⋊C4—22no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4C2×D311yescharacteristic
63C32×C4⋊C4C22×C311yescharacteristic
64C32×C22⋊C4—22no—
65C32×C22⋊C4—21no—
66C32×C22⋊C4—21no—
67#3757—21no—
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic
69C4⋊(C3⋊Dic3)C22×C311yescharacteristic
70C4×C3⋊Dic3—21no—
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
73C3×#217—61no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×C4×Dic3—61no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C22×H(3,𝔽3)—21no—
80C22×H(3,𝔽3)—21no—
81C22×H(3,𝔽3)—21no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)—41no—
85C4×H(3,𝔽3)—41no—
86C4×H(3,𝔽3)#4111yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88(C3⋊Dic3)⋊C3—41no—
89(C3⋊Dic3)⋊C3—41no—
90C3×C22⋊Q8—31no—
91C3×C22⋊Q8—31no—
92C3×C22⋊Q8—61no—
93C3×C22⋊Q8C3×D311yescharacteristic
94C3×#117—91no—
95#926—31no—
96#926—91no—
97C3×#107—91no—
98C3×#106—91no—
99C3×#103—91no—
100C3×#103—91no—
101C3×C22⋊C8—91no—
102#835—31no—
103#835—31no—
104#835—91no—
105#835—91no—
106#830—31no—
107#830—91no—
108#808—31no—
109#808—91no—
110#807—31no—
111#807—91no—
112C23×C32—21no—
113C23×C32#6711yescharacteristic
114C23×C32C3×D411yescharacteristic
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8—21no—
119C32×Q8—41no—
120C32×Q8—41no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—41no—
134C2×C32×C4—41no—
135C2×C32×C4#6711yescharacteristic
136C2×C32×C4C3×D411yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C23×C311yescharacteristic
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—21no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C3⋊(C3⋊C8)—21no—
144C3⋊(C3⋊C8)—21no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147#485—91noSylow
148C2×H(3,𝔽3)—41no—
149C2×H(3,𝔽3)#12311yescharacteristic
150C2×H(3,𝔽3)#13511yescharacteristic
151C2×H(3,𝔽3)#13711yescharacteristic
152C2×C3×Q8—31no—
153C2×C3×Q8—31no—
154C2×C3×Q8—61no—
155C2×C3×Q8C2×C3×D311yescharacteristic
156C22×C3×C4—31no—
157C22×C3×C4—31no—
158C22×C3×C4—61no—
159C22×C3×C4C2×C3×D311yescharacteristic
160C2×C3×C8—91no—
161C2×C3×C8—91no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4—61no—
167C3×C4⋊C4—62no—
168C3×C4⋊C4—61no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4—91no—
171C3×C4⋊C4C2×C3×D311yescharacteristic
172C3×C22⋊C4—21no—
173C3×C22⋊C4—61no—
174C3×C22⋊C4—61no—
175C3×C22⋊C4—62no—
176C3×C22⋊C4—181no—
177C3×C42—181no—
178#217—61no—
179#217—181no—
180C4⋊Dic3—31no—
181C4⋊Dic3—31no—
182C4⋊Dic3—91no—
183C4⋊Dic3—91no—
184C4×Dic3—61no—
185C4×Dic3—181no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—91no—
190C22×C32—22no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32C2×#6711yescharacteristic
201C22×C32C2×C3×D411yescharacteristic
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—41no—
207C32×C4—42no—
208C32×C4—42no—
209C32×C4—41no—
210C32×C4—41no—
211C32×C4—41no—
212C32×C4—41no—
213C32×C4#23711yescharacteristic
214C32×C4#23711yescharacteristic
215C32×C4C3×#4111yescharacteristic
216C32×C4C3×#4111yescharacteristic
217C3⋊Dic3—41no—
218C3⋊Dic3—41no—
219C3×Dic3—121no—
220C3×Dic3—121no—
221C22⋊Q8(C3⋊D3)⋊C311yescharacteristic
222#117—91no—
223#107—91no—
224#106—91no—
225#103—91no—
226#103—91no—
227C22⋊C8—91no—
228H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
229C23×C3—31no—
230C23×C3—31no—
231C23×C3—61no—
232C23×C3C3×#6711yescharacteristic
233C3×Q8—21no—
234C3×Q8—21no—
235C3×Q8—61no—
236C3×Q8—61no—
237C3×Q8—61no—
238C3×Q8—61no—
239C3×Q8—121no—
240C3×Q8—121no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—62no—
258C2×C3×C4—62no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—121no—
262C2×C3×C4—121no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4C3×#6711yescharacteristic
266C2×C3×C4C22×C3×D311yescharacteristic
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—181no—
270C2×Dic3—181no—
271C3×C8—181no—
272C3×C8—181no—
273C3⋊C8—61no—
274C3⋊C8—61no—
275C3⋊C8—181no—
276C3⋊C8—181no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—42no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32#93511yescharacteristic
284C2×C32#94211yescharacteristic
285C2×C32#95011yescharacteristic
286C2×C32C3×#12311yescharacteristic
287C2×C32C3×#13511yescharacteristic
288C2×C32C3×#13711yescharacteristic
289C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
290C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
291C2×C8—91no—
292C2×C8—91no—
293C4⋊C4—21no—
294C4⋊C4—91no—
295C4⋊C4—91no—
296C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—181no—
299C42—181no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—61no—
306C22×C3—61no—
307C22×C3—61no—
308C22×C3—61no—
309C22×C3—61no—
310C22×C3—61no—
311C22×C3—62no—
312C22×C3—61no—
313C22×C3—121no—
314C22×C3—121no—
315C22×C3C2×C3×#6711yescharacteristic
316C3×C4—21no—
317C3×C4—31no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—41no—
322C3×C4—41no—
323C3×C4—61no—
324C3×C4—61no—
325C3×C4—61no—
326C3×C4—61no—
327C3×C4—121no—
328C3×C4—121no—
329C3×C4—121no—
330C3×C4—121no—
331C3×C4—121no—
332C3×C4—122no—
333C3×C4—122no—
334C3×C4—361no—
335C3×C4—361no—
336C3×C4C3×#23711yescharacteristic
337C3×C4C3×#23711yescharacteristic
338Dic3—121no—
339Dic3—121no—
340Dic3—361no—
341Dic3—361no—
342C32—21no—
343C32#504511yescharacteristic, complemented
344C32C3×#48511yescharacteristic, complemented
345C23#638⋊C311yescharacteristic
346Q8—21no—
347Q8—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4#638⋊C311yescharacteristic
355C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
356C8—181no—
357C8—181no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—41no—
365C2×C3—61no—
366C2×C3—61no—
367C2×C3—61no—
368C2×C3—121no—
369C2×C3—121no—
370C2×C3—122no—
371C2×C3C3×#93511yescharacteristic
372C2×C3C3×#94211yescharacteristic
373C2×C3C3×#95011yescharacteristic
374C22—21no—
375C22—21no—
376C22—21no—
377C22C2×#638⋊C311yescharacteristic
378C4—21no—
379C4—41no—
380C4—41no—
381C4—361no—
382C4—361no—
383C4#3830⋊C311yescharacteristic
384C4#3830⋊C311yescharacteristic
385C3—31no—
386C3—31no—
387C3—61no—
388C3C3×#504511yescharacteristic
389C2—41no—
390C2#64144⋊C311yescharacteristic
391C2#64151⋊C311yescharacteristic
392C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC23×H(3,𝔽3)D4
hypercenterC22⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#485
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #63662⋊C3C2semidirect
4. #63662⋊C3C2semidirect
6. #63635⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592744C3semidirect
26. #63662C2×C3semidirect
27. #63662C2×C3semidirect
29. #63635C2×C3semidirect
30. #63634C2×C3semidirect
228. H(3,𝔽3)#485semidirect
343. C32#5045semidirect
344. C32C3×#485semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
#64135⋊C32
#63662⋊C32
#63657⋊C32
#63635⋊C32
#63634⋊C32
#105927443
C3×#50453