Finite Group Databasebeta

← #10592464⋊C3 #13141780

Subgroups — #10592464⋊C3 #13141780

Order 1728.

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

normal subgroups only

This group has 400 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#64097⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic
4#63659⋊C3C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#63642⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592464C311yesmaximal, characteristic, complemented
9C3×#4765—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3—21no—
12C2×#634⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#3757⋊C3—21no—
16#3757⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64145C2×C311yescharacteristic
22#64097C2×C311yescharacteristic, complemented
23C3×#936—31no—
24C3×#888—31no—
25C32×#102—21no—
26C32×#102D311yescharacteristic
27C32×#102C2×C311yescharacteristic
28#63659C2×C311yescharacteristic
29#63643C2×C311yescharacteristic, complemented
30#63642C2×C311yescharacteristic, complemented
31#63637C2×C311yescharacteristic, complemented
32C3×#832—31no—
33C3×#816—31no—
34C3×#815—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—
47#634⋊C3—42no—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50C3×#489—91no—
51#4765—31no—
52#4765—91no—
53C2×C32×D4—21no—
54C2×C32×D4C2×D311yescharacteristic
55C2×C32×D4C22×C311yescharacteristic
56C2×#634—21no—
57C2×#634C22×C311yescharacteristic
58C2×C3×Dic6—31no—
59C2×C3×Dic6—61no—
60C2×C32×C8—21no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C22×C311yescharacteristic
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C22×C311yescharacteristic
66#3757—21no—
67#3757—21no—
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic
69#3750—21no—
70C4×C3⋊Dic3C22×C311yescharacteristic
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C3×#217—61no—
73C3×#217—61no—
74C3×C4⋊Dic3—31no—
75C3×Dic3⋊C4—61no—
76C3×C4×Dic3—31no—
77C2×C3×C3⋊C8—31no—
78C22×H(3,𝔽3)—41no—
79C22×H(3,𝔽3)—41no—
80C22×H(3,𝔽3)—41no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C4×H(3,𝔽3)—41no—
83C4×H(3,𝔽3)#4111yescharacteristic
84C4×H(3,𝔽3)#4111yescharacteristic
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89C3×C4⋊Q8—91no—
90C3×#124—91no—
91#936—31no—
92#936—91no—
93#888—31no—
94#888—91no—
95C3×#103—91no—
96C3×#103—91no—
97C3×#102—31no—
98C3×#102—31no—
99C3×#102—61no—
100C3×#102—91no—
101C3×#102C3×D311yescharacteristic
102C3×#97—91no—
103#832—31no—
104#832—91no—
105#816—31no—
106#816—91no—
107#815—31no—
108#815—91no—
109#810—31no—
110#810—91no—
111C23×C32—21no—
112C23×C32—21no—
113C23×C32—41no—
114C32×D4—22no—
115C32×D4—21no—
116C32×D4—21no—
117C32×D4—21no—
118C32×D4—21no—
119C32×D4—41no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—41no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C23×C311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3C3×D411yescharacteristic
129C2×C3⋊Dic3C3×D411yescharacteristic
130#634—21no—
131#634—21no—
132#634—42no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C3×Dic6—61no—
138C3×Dic6—61no—
139C3×Dic6—122no—
140C32×C8—21no—
141C32×C8—21no—
142C32×C8—41no—
143C3⋊(C3⋊C8)—21no—
144C3×C3⋊C8—61no—
145#489—91noSylow
146C2×H(3,𝔽3)—81no—
147C2×H(3,𝔽3)#12411yescharacteristic
148C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
149C2×H(3,𝔽3)#13711yescharacteristic
150C2×C3×Q8—91no—
151C2×C3×Q8—181no—
152C2×C3×D4—31no—
153C2×C3×D4—31no—
154C2×C3×D4—61no—
155C2×C3×D4C2×C3×D311yescharacteristic
156C2×Dic6—31no—
157C2×Dic6—61no—
158C2×Dic6—91no—
159C2×Dic6—181no—
160C2×C3×C8—31no—
161C2×C3×C8—31no—
162C2×C3×C8—61no—
163C2×C3×C8—91no—
164C2×C3×C8C2×C3×D311yescharacteristic
165C3×C4⋊C4—31no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—181no—
170C3×C4⋊C4C2×C3×D311yescharacteristic
171C3×C22⋊C4—181no—
172C3×C22⋊C4—181no—
173C3×C42—91no—
174#217—61no—
175#217—61no—
176#217—181no—
177#217—181no—
178C4⋊Dic3—31no—
179C4⋊Dic3—91no—
180Dic3⋊C4—61no—
181Dic3⋊C4—181no—
182C4×Dic3—31no—
183C4×Dic3—91no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—91no—
186C22×C32—21no—
187C22×C32—42no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—81no—
195C22×C32—81no—
196C22×C32D3×D411yescharacteristic
197C22×C32C2×C3×D411yescharacteristic
198C32×C4—21no—
199C32×C4—21no—
200C32×C4—41no—
201C32×C4—41no—
202C32×C4—81no—
203C32×C4#23511yescharacteristic
204C32×C4#23711yescharacteristic
205C32×C4C3×#4111yescharacteristic
206C32×C4C3×#4111yescharacteristic
207C3⋊Dic3—21no—
208C3⋊Dic3—21no—
209C3⋊Dic3—41no—
210C3⋊Dic3—41no—
211C3×Dic3—61no—
212C3×Dic3—61no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215C4⋊Q8—91no—
216#124—91no—
217#103—91no—
218#103—91no—
219#102—91no—
220#102(C3⋊D3)⋊C311yescharacteristic
221#97—91no—
222H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
223C23×C3—21no—
224C23×C3—61no—
225C23×C3—61no—
226C23×C3—121no—
227C3×Q8—181no—
228C3×Q8—181no—
229C3×Q8—362no—
230C3×D4—21no—
231C3×D4—21no—
232C3×D4—61no—
233C3×D4—61no—
234C3×D4—61no—
235C3×D4—61no—
236C3×D4—62no—
237C3×D4—121no—
238C2×C3×C4—21no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—91no—
245C2×C3×C4—91no—
246C2×C3×C4—121no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4C22×C3×D311yescharacteristic
250C2×Dic3—31no—
251C2×Dic3—31no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—91no—
255C2×Dic3—91no—
256C2×Dic3—181no—
257C2×Dic3—181no—
258Dic6—61no—
259Dic6—61no—
260Dic6—122no—
261Dic6—181no—
262Dic6—181no—
263Dic6—362no—
264C3×C8—21no—
265C3×C8—61no—
266C3×C8—61no—
267C3×C8—121no—
268C3×C8—181no—
269C3⋊C8—61no—
270C3⋊C8—181no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—82no—
275C2×C32—81no—
276C2×C32—81no—
277C2×C32#88511yescharacteristic
278C2×C32#91111yescharacteristic
279C2×C32#91511yescharacteristic
280C2×C32C3×#12411yescharacteristic
281C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
282C2×C32C3×#13711yescharacteristic
283C2×Q8—91no—
284C2×Q8—181no—
285C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
286C2×C8—91no—
287C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
288C4⋊C4—91no—
289C4⋊C4—181no—
290C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
291C22⋊C4—181no—
292C22⋊C4—181no—
293C42—91no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—61no—
300C22×C3—121no—
301C22×C3—121no—
302C22×C3—121no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3—121no—
306C22×C3—122no—
307C22×C3—241no—
308C22×C3—241no—
309C22×C3C3×D3×D411yescharacteristic
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—41no—
315C3×C4—61no—
316C3×C4—61no—
317C3×C4—121no—
318C3×C4—121no—
319C3×C4—181no—
320C3×C4—181no—
321C3×C4—241no—
322C3×C4—361no—
323C3×C4—361no—
324C3×C4C3×#23511yescharacteristic
325C3×C4C3×#23711yescharacteristic
326Dic3—61no—
327Dic3—61no—
328Dic3—121no—
329Dic3—121no—
330Dic3—181no—
331Dic3—181no—
332Dic3—361no—
333Dic3—361no—
334C32—21no—
335C32#476511yescharacteristic, complemented
336C32C3×#48911yescharacteristic, complemented
337C23—21no—
338Q8—181no—
339Q8—181no—
340Q8—362no—
341D4—21no—
342D4—21no—
343C2×C4—21no—
344C2×C4—91no—
345C2×C4—91no—
346C2×C4—181no—
347C2×C4—181no—
348C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
349C8—21no—
350C8—181no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—81no—
361C2×C3—241no—
362C2×C3—241no—
363C2×C3—242no—
364C2×C3C3×#88511yescharacteristic
365C2×C3C3×#91111yescharacteristic
366C2×C3C3×#91511yescharacteristic
367C22—41no—
368C22—41no—
369C22—41no—
370C22D4×(C3⋊D3)⋊C311yescharacteristic
371C4—41no—
372C4—181no—
373C4—181no—
374C4—361no—
375C4—361no—
376C4#3828⋊C311yescharacteristic
377C4#3830⋊C311yescharacteristic
378C3—31no—
379C3—31no—
380C3—61no—
381C3C3×#476511yescharacteristic
382C2—81no—
383C2#64094⋊C311yescharacteristic
384C2#64120⋊C311yescharacteristic
385C2#64124⋊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×#489
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
5. #63643⋊C3C2semidirect
6. #63642⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. #10592464C3semidirect
22. #64097C2×C3semidirect
29. #63643C2×C3semidirect
30. #63642C2×C3semidirect
31. #63637C2×C3semidirect
222. H(3,𝔽3)#489semidirect
335. C32#4765semidirect
336. C32C3×#489semidirect

Maximal subgroups

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

subgroupindex
#64145⋊C32
#64097⋊C32
H(3,𝔽3)×#1022
#63659⋊C32
#63643⋊C32
#63642⋊C32
#63637⋊C32
#105924643
C3×#47653