Finite Group Databasebeta

← #10592752⋊C3 #13142898

Subgroups — #10592752⋊C3 #13142898

Order 1728.

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

normal subgroups only

This group has 638 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
2H(3,𝔽3)×#123C211yesmaximal, characteristic, complemented
3#64138⋊C3C212yesmaximal, complemented
4#63661⋊C3C211yesmaximal, characteristic, complemented
5#63633⋊C3C212yesmaximal, complemented
6#10592752C311yesmaximal, characteristic, complemented
7C3×#5053—31nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
10C2×#638⋊C3—21no—
11C2×#638⋊C3C2212yes—
12#3830⋊C3—22no—
13D4×(C3⋊D3)⋊C3—22no—
14#3828⋊C3—21no—
15C4⋊C4×H(3,𝔽3)—21no—
16C22⋊C4×H(3,𝔽3)—21no—
17C22⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
18#3757⋊C3C2212yescomplemented
19#3752⋊C3—22no—
20#3750⋊C3—22no—
21#64361C2×C311yescharacteristic
22C3×#1004—31no—
23C32×#123—21no—
24C32×#123D311yescharacteristic, complemented
25C32×#123C2×C311yescharacteristic, complemented
26#64138C2×C312yescomplemented
27C3×#929—32no—
28#63661C2×C311yescharacteristic, complemented
29#63633C2×C312yescomplemented
30C3×#834—31no—
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)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—22no—
40C2×C4×H(3,𝔽3)D411yescharacteristic
41#638⋊C3—22no—
42#638⋊C3—22no—
43#638⋊C3—22no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—22no—
46C2×(C3⋊Dic3)⋊C3D412yes—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4×(C3⋊D3)⋊C3—22no—
49#634⋊C3—21no—
50C3×#458—91no—
51#5053—31no—
52#5053—91no—
53C2×C32×D4—21no—
54C2×C32×D4C2×D311yescharacteristic
55C2×C32×D4C22×C311yescharacteristic
56C22×C32×C4—21no—
57C22×C32×C4C2×D311yescharacteristic, complemented
58C22×C32×C4C22×C311yescharacteristic, complemented
59C2×#638—21no—
60C2×#638C22×C312yes—
61#3830—22no—
62D4×C3⋊D3—22no—
63#3828—21no—
64C2×C3×#67—32no—
65C2×C3×#67—61no—
66C3×#237—62no—
67C3×D3×D4—62no—
68C3×#235—61no—
69C32×C4⋊C4—22no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4—21no—
72C32×C22⋊C4—22no—
73C32×C22⋊C4—21no—
74C32×C22⋊C4—21no—
75C32×C22⋊C4—21no—
76C32×C22⋊C4C2×D311yescharacteristic, complemented
77C32×C22⋊C4C22×C311yescharacteristic, complemented
78#3757C22×C312yescomplemented
79#3752—22no—
80#3750—22no—
81C3×#217—32no—
82C3×(C2×D3)⋊C4—62no—
83C3×Dic3⋊C4—62no—
84C22×H(3,𝔽3)—21no—
85C22×H(3,𝔽3)—41no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)C2×D412yes—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×(C3⋊D3)⋊C3—22no—
90C2×(C3⋊D3)⋊C3—42no—
91C2×(C3⋊D3)⋊C3—81no—
92C4×H(3,𝔽3)—21no—
93C4×H(3,𝔽3)—41no—
94C4×H(3,𝔽3)—41no—
95(C3⋊Dic3)⋊C3—22no—
96(C3⋊Dic3)⋊C3—42no—
97C3×H(5,𝔽2)—91no—
98#1004—31no—
99#1004—91no—
100C3×#123—31no—
101C3×#123—31no—
102C3×#123—61no—
103C3×#123—92no—
104C3×#123C3×D311yescharacteristic
105#929—32no—
106#929—92no—
107C3×#99—92no—
108C3×#99—91no—
109#834—31no—
110#834—91no—
111#806—32no—
112#806—92no—
113C23×C32—21no—
114C23×C32—21no—
115C23×C32#6711yescharacteristic
116C23×C32C3×D411yescharacteristic
117C23×C32C22×D311yescharacteristic
118C23×C32C23×C311yescharacteristic
119C22×C3⋊D3—21no—
120C22×C3⋊D3C3×D412yes—
121C22×C3×D3—32no—
122C22×C3×D3—61no—
123C32×D4—22no—
124C32×D4—22no—
125C32×D4—42no—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—22no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—42no—
136C2×C32×C4#6711yescharacteristic
137C2×C32×C4C3×D411yescharacteristic
138#638—22no—
139#638—22no—
140#638—22no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3C3×D412yes—
144C4⋊(C3⋊D3)—21no—
145C4×C3⋊D3—22no—
146#634—21no—
147C3×#67—62no—
148C3×#67—62no—
149C3×#67—62no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—62no—
153C3×D12—61no—
154C3×C4×D3—62no—
155C3×Dic6—61no—
156#458—91noSylow
157C2×H(3,𝔽3)—21no—
158C2×H(3,𝔽3)—22no—
159C2×H(3,𝔽3)—41no—
160C2×H(3,𝔽3)#12011yescharacteristic
161(C3⋊D3)⋊C3—42no—
162C3×#41—181no—
163C3×#41—182no—
164C2×C3×D4—31no—
165C2×C3×D4—31no—
166C2×C3×D4—61no—
167C2×C3×D4—92no—
168C2×C3×D4—182no—
169C2×C3×D4—181no—
170C2×C3×D4C2×C3×D311yescharacteristic
171C22×C3×C4—31no—
172C22×C3×C4—31no—
173C22×C3×C4—61no—
174C22×C3×C4C2×C3×D311yescharacteristic
175C2×#67—32no—
176C2×#67—61no—
177C2×#67—92no—
178C2×#67—181no—
179#237—62no—
180#237—182no—
181D3×D4—62no—
182D3×D4—182no—
183#235—61no—
184#235—181no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—61no—
187C3×C4⋊C4—61no—
188C3×C4⋊C4—62no—
189C3×C4⋊C4—182no—
190C3×C22⋊C4—21no—
191C3×C22⋊C4—31no—
192C3×C22⋊C4—31no—
193C3×C22⋊C4—61no—
194C3×C22⋊C4—61no—
195C3×C22⋊C4—62no—
196C3×C22⋊C4—61no—
197C3×C22⋊C4—92no—
198C3×C22⋊C4—182no—
199C3×C22⋊C4C2×C3×D311yescharacteristic
200#217—32no—
201#217—92no—
202(C2×D3)⋊C4—62no—
203(C2×D3)⋊C4—182no—
204Dic3⋊C4—62no—
205Dic3⋊C4—182no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—22no—
209C22×C32—21no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—42no—
216C22×C32—42no—
217C22×C32D3×D412yes—
218C22×C32C2×#6711yescharacteristic
219C22×C32C2×C3×D412yes—
220C22×C32C2×C3×D411yescharacteristic
221C2×C3⋊D3—22no—
222C2×C3⋊D3—42no—
223C2×C3⋊D3—81no—
224C2×C3×D3—62no—
225C2×C3×D3—122no—
226C2×C3×D3—241no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—41no—
230C32×C4—42no—
231C32×C4—41no—
232C32×C4—41no—
233C32×C4—42no—
234C32×C4—41no—
235C32×C4—41no—
236C3⋊Dic3—22no—
237C3⋊Dic3—42no—
238C3×Dic3—62no—
239C3×Dic3—122no—
240H(5,𝔽2)—91no—
241#123—92no—
242#123(C3⋊D3)⋊C311yescharacteristic, complemented
243#99—91no—
244#99—92no—
245H(3,𝔽3)#45811yescharacteristic, complemented, Sylow
246C23×C3—31no—
247C23×C3—31no—
248C23×C3—31no—
249C23×C3—31no—
250C23×C3—61no—
251C23×C3—61no—
252C23×C3—92no—
253C23×C3—181no—
254C23×C3C3×#6711yescharacteristic
255C23×C3C22×C3×D311yescharacteristic
256C22×D3—32no—
257C22×D3—61no—
258C22×D3—92no—
259C22×D3—181no—
260C3×Q8—181no—
261C3×D4—22no—
262C3×D4—62no—
263C3×D4—62no—
264C3×D4—122no—
265C3×D4—181no—
266C3×D4—182no—
267C3×D4—182no—
268C3×D4—182no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—62no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—62no—
280C2×C3×C4—62no—
281C2×C3×C4—62no—
282C2×C3×C4—61no—
283C2×C3×C4—92no—
284C2×C3×C4—122no—
285C2×C3×C4—182no—
286C2×C3×C4—181no—
287C2×C3×C4—182no—
288C2×C3×C4C3×#6711yescharacteristic
289#67—62no—
290#67—62no—
291#67—62no—
292#67—182no—
293#67—182no—
294#67—182no—
295C2×Dic3—32no—
296C2×Dic3—61no—
297C2×Dic3—62no—
298C2×Dic3—92no—
299C2×Dic3—181no—
300C2×Dic3—182no—
301D12—61no—
302D12—181no—
303C4×D3—62no—
304C4×D3—182no—
305Dic6—61no—
306Dic6—181no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—22no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—42no—
314C2×C32—41no—
315C2×C32—41no—
316C2×C32—81no—
317C2×C32#93711yescharacteristic
318C2×C32C3×#12011yescharacteristic
319C3⋊D3—42no—
320C3×D3—122no—
321#41—182no—
322#41—181no—
323C2×D4—92no—
324C2×D4—182no—
325C2×D4—181no—
326C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
327C22×C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
328C4⋊C4—21no—
329C4⋊C4—182no—
330C22⋊C4—21no—
331C22⋊C4—92no—
332C22⋊C4—182no—
333C22⋊C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
334C22×C3—21no—
335C22×C3—32no—
336C22×C3—32no—
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3—62no—
344C22×C3—61no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—121no—
348C22×C3—121no—
349C22×C3—121no—
350C22×C3—122no—
351C22×C3—122no—
352C22×C3—182no—
353C22×C3—362no—
354C22×C3—721no—
355C22×C3C3×D3×D412yes—
356C22×C3C2×C3×#6711yescharacteristic
357C2×D3—62no—
358C2×D3—122no—
359C2×D3—182no—
360C2×D3—241no—
361C2×D3—362no—
362C2×D3—721no—
363C3×C4—21no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—61no—
367C3×C4—61no—
368C3×C4—121no—
369C3×C4—121no—
370C3×C4—121no—
371C3×C4—122no—
372C3×C4—121no—
373C3×C4—121no—
374C3×C4—122no—
375C3×C4—182no—
376C3×C4—362no—
377Dic3—62no—
378Dic3—122no—
379Dic3—182no—
380Dic3—362no—
381C32—21no—
382C32#505311yescharacteristic, complemented
383C32C3×#45811yescharacteristic, complemented
384C23—92no—
385C23—181no—
386C23#638⋊C311yescharacteristic
387C23C22×(C3⋊D3)⋊C311yescharacteristic
388Q8—181no—
389D4—22no—
390D4—181no—
391D4—182no—
392D4—182no—
393D4—182no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—92no—
398C2×C4—182no—
399C2×C4—181no—
400C2×C4—182no—
401C2×C4#638⋊C311yescharacteristic
402C2×C3—22no—
403C2×C3—21no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—41no—
407C2×C3—62no—
408C2×C3—61no—
409C2×C3—61no—
410C2×C3—62no—
411C2×C3—62no—
412C2×C3—61no—
413C2×C3—122no—
414C2×C3—121no—
415C2×C3—121no—
416C2×C3—241no—
417C2×C3—362no—
418C2×C3C3×#93711yescharacteristic
419D3—122no—
420D3—362no—
421C22—21no—
422C22—41no—
423C22—41no—
424C22—182no—
425C22—362no—
426C22—721no—
427C22D4×(C3⋊D3)⋊C312yes—
428C22C2×#638⋊C311yescharacteristic
429C4—21no—
430C4—41no—
431C4—41no—
432C4—182no—
433C4—362no—
434C3—31no—
435C3—31no—
436C3—61no—
437C3C3×#505311yescharacteristic
438C2—21no—
439C2—22no—
440C2—41no—
441C2—362no—
442C2#64146⋊C311yescharacteristic
subgroupG/H
centerC2#64146⋊C3
commutator, cosocleC23×C32C23×C3
FrattiniC23×C3C22×C3×D3
FittingH(3,𝔽3)×#123C2
socleC2×C3C3×#937
prime radical#64361⋊C3C2
hypercenter#123(C3⋊D3)⋊C3
hypocenterC32C3×#458
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#123C2semidirect
3. #64138⋊C3C2semidirect
4. #63661⋊C3C2semidirect
5. #63633⋊C3C2semidirect
6. #10592752C3semidirect
9. C22×C4×H(3,𝔽3)C22semidirect
17. C22⋊C4×H(3,𝔽3)C22semidirect
18. #3757⋊C3C22semidirect
24. C32×#123D3semidirect
25. C32×#123C2×C3semidirect
26. #64138C2×C3semidirect
28. #63661C2×C3semidirect
29. #63633C2×C3semidirect
57. C22×C32×C4C2×D3semidirect
58. C22×C32×C4C22×C3semidirect
76. C32×C22⋊C4C2×D3semidirect
77. C32×C22⋊C4C22×C3semidirect
78. #3757C22×C3semidirect
242. #123(C3⋊D3)⋊C3semidirect
245. H(3,𝔽3)#458semidirect
327. C22×C4C2×(C3⋊D3)⋊C3semidirect
333. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
382. C32#5053semidirect
383. C32C3×#458semidirect

Maximal subgroups

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

subgroupindex
#64361⋊C32
H(3,𝔽3)×#1232
#64138⋊C32
#63661⋊C32
#63633⋊C32
#105927523
C3×#50533