Finite Group Databasebeta

← #10592648⋊C3 #13142500

Subgroups — #10592648⋊C3 #13142500

Order 1728.

783 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#3757⋊C3C212yesmaximal
3C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4C2×#3750⋊C3C212yesmaximal, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#10592648C311yesmaximal, characteristic, complemented
7C3×#4949—31nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C22⋊C4×H(3,𝔽3)—22no—
14#3757⋊C3—24no—
15C4⋊((C3⋊Dic3)⋊C3)—22no—
16#3750⋊C3—24no—
17C2×C32×C22⋊C4—21no—
18C2×C32×C22⋊C4D311yescharacteristic
19C2×C32×C22⋊C4C2×C311yescharacteristic
20C2×#3757C2×C312yes—
21C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
22C2×#3750C2×C312yescomplemented
23C2×C3×#217—32no—
24C2×C3×C4⋊Dic3—31no—
25C2×C3×Dic3⋊C4—32no—
26#63658C2×C311yescharacteristic, complemented
27C3×C3⋊#95—31no—
28C23×H(3,𝔽3)—21no—
29C23×H(3,𝔽3)—22no—
30C23×H(3,𝔽3)—21no—
31C23×H(3,𝔽3)—22no—
32C23×H(3,𝔽3)Q812yes—
33C23×H(3,𝔽3)C2311yescharacteristic
34C2×C4×H(3,𝔽3)—22no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)D412yes—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3D414yes—
44C3×#399—91no—
45#4949—31no—
46#4949—91no—
47C24×C32—21no—
48C24×C32C2×D311yescharacteristic
49C24×C32C22×C311yescharacteristic
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C22×C311yescharacteristic
55C22×C32×C4C22×C311yescharacteristic
56C22×C3⋊Dic3C22×C312yes—
57C22×C3⋊Dic3C22×C312yes—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—32no—
60C32×C22⋊C4—24no—
61C32×C22⋊C4—22no—
62C32×C22⋊C4—22no—
63#3757—24no—
64C4⋊(C3⋊Dic3)—22no—
65#3750—24no—
66C3×#217—64no—
67C3×C4⋊Dic3—62no—
68C3×Dic3⋊C4—64no—
69C22×H(3,𝔽3)—22no—
70C22×H(3,𝔽3)—24no—
71C22×H(3,𝔽3)—42no—
72C22×H(3,𝔽3)—41no—
73C22×H(3,𝔽3)—41no—
74C22×H(3,𝔽3)C2×D412yes—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C22×H(3,𝔽3)C2×Q811yescharacteristic
77C22×H(3,𝔽3)#4111yescharacteristic
78C22×H(3,𝔽3)#4112yes—
79C4×H(3,𝔽3)—22no—
80C4×H(3,𝔽3)—41no—
81(C3⋊Dic3)⋊C3—24no—
82(C3⋊Dic3)⋊C3—42no—
83C2×C3×C4⋊C4—92no—
84C2×C3×C4⋊C4—91no—
85C2×C3×C22⋊C4—31no—
86C2×C3×C22⋊C4—31no—
87C2×C3×C22⋊C4—61no—
88C2×C3×C22⋊C4—92no—
89C2×C3×C22⋊C4C3×D311yescharacteristic
90C2×#217—32no—
91C2×#217—92no—
92C2×C4⋊Dic3—31no—
93C2×C4⋊Dic3—91no—
94C2×Dic3⋊C4—32no—
95C2×Dic3⋊C4—92no—
96C3×#95—91no—
97C3⋊#95—31no—
98C3⋊#95—91no—
99C23×C32—22no—
100C23×C32—22no—
101C23×C32—21no—
102C23×C32—22no—
103C23×C32—22no—
104C23×C32—21no—
105C23×C32—22no—
106C23×C32—21no—
107C23×C32—21no—
108C23×C32—22no—
109C23×C32—22no—
110C23×C32—21no—
111C23×C32—42no—
112C23×C32—42no—
113C23×C32Dic612yes—
114C23×C32C3×Q812yes—
115C23×C32C22×D311yescharacteristic
116C23×C32C23×C311yescharacteristic
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—22no—
123C2×C32×C4—21no—
124C2×C32×C4—22no—
125C2×C32×C4—42no—
126C2×C32×C4—42no—
127C2×C32×C4#6712yes—
128C2×C32×C4C3×D412yes—
129C2×C3⋊Dic3—22no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3—22no—
133C2×C3⋊Dic3—22no—
134C2×C3⋊Dic3C3×D414yes—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141#399—91noSylow
142C2×H(3,𝔽3)—42no—
143C2×H(3,𝔽3)C22⋊Q812yes—
144C2×H(3,𝔽3)C22⋊Q811yescharacteristic
145C2×H(3,𝔽3)#12312yes—
146C2×H(3,𝔽3)#12311yescharacteristic
147C2×H(3,𝔽3)#12711yescharacteristic
148C24×C3—31no—
149C24×C3—31no—
150C24×C3—61no—
151C24×C3C2×C3×D311yescharacteristic
152C22×C3×C4—31no—
153C22×C3×C4—31no—
154C22×C3×C4—31no—
155C22×C3×C4—31no—
156C22×C3×C4—61no—
157C22×C3×C4—61no—
158C22×C3×C4—92no—
159C22×C3×C4—92no—
160C22×C3×C4C2×C3×D311yescharacteristic
161C22×C3×C4C2×C3×D311yescharacteristic
162C22×Dic3—32no—
163C22×Dic3—32no—
164C22×Dic3—92no—
165C22×Dic3—92no—
166C3×C4⋊C4—182no—
167C3×C4⋊C4—184no—
168C3×C22⋊C4—22no—
169C3×C22⋊C4—62no—
170C3×C22⋊C4—62no—
171C3×C22⋊C4—64no—
172C3×C22⋊C4—184no—
173#217—64no—
174#217—184no—
175C4⋊Dic3—62no—
176C4⋊Dic3—182no—
177Dic3⋊C4—64no—
178Dic3⋊C4—184no—
179C22×C32—24no—
180C22×C32—24no—
181C22×C32—22no—
182C22×C32—24no—
183C22×C32—22no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—22no—
187C22×C32—22no—
188C22×C32—21no—
189C22×C32—44no—
190C22×C32—44no—
191C22×C32—42no—
192C22×C32—42no—
193C22×C32—42no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—42no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32C2×Dic611yescharacteristic
200C22×C32#23511yescharacteristic
201C22×C32D3×D412yes—
202C22×C32#23712yes—
203C22×C32C2×#6711yescharacteristic
204C22×C32C2×C3×D412yes—
205C22×C32C2×C3×D411yescharacteristic
206C22×C32C2×C3×Q811yescharacteristic
207C22×C32C3×#4111yescharacteristic
208C22×C32C3×#4112yes—
209C32×C4—22no—
210C32×C4—22no—
211C32×C4—42no—
212C32×C4—42no—
213C32×C4—41no—
214C32×C4—41no—
215C3⋊Dic3—24no—
216C3⋊Dic3—42no—
217C3×Dic3—64no—
218C3×Dic3—122no—
219C2×C4⋊C4—92no—
220C2×C4⋊C4—91no—
221C2×C22⋊C4—92no—
222C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic
223#95—91no—
224H(3,𝔽3)#39911yescharacteristic, complemented, Sylow
225C23×C3—21no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—21no—
229C23×C3—32no—
230C23×C3—32no—
231C23×C3—31no—
232C23×C3—31no—
233C23×C3—62no—
234C23×C3—61no—
235C23×C3—61no—
236C23×C3—62no—
237C23×C3—61no—
238C23×C3—61no—
239C23×C3—62no—
240C23×C3—62no—
241C23×C3—62no—
242C23×C3—62no—
243C23×C3—62no—
244C23×C3—61no—
245C23×C3—122no—
246C23×C3—122no—
247C23×C3C3×Dic612yes—
248C23×C3C22×C3×D311yescharacteristic
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—32no—
253C2×C3×C4—32no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—62no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—94no—
263C2×C3×C4—122no—
264C2×C3×C4—122no—
265C2×C3×C4—182no—
266C2×C3×C4—182no—
267C2×C3×C4—182no—
268C2×C3×C4—182no—
269C2×C3×C4—182no—
270C2×C3×C4C3×#6712yes—
271C2×Dic3—34no—
272C2×Dic3—62no—
273C2×Dic3—62no—
274C2×Dic3—62no—
275C2×Dic3—62no—
276C2×Dic3—62no—
277C2×Dic3—94no—
278C2×Dic3—182no—
279C2×Dic3—182no—
280C2×Dic3—182no—
281C2×Dic3—182no—
282C2×Dic3—182no—
283C2×C32—22no—
284C2×C32—22no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—44no—
289C2×C32—42no—
290C2×C32—42no—
291C2×C32#87812yes—
292C2×C32#88312yes—
293C2×C32#92411yescharacteristic
294C2×C32#93511yescharacteristic
295C2×C32#94011yescharacteristic
296C2×C32C3×C22⋊Q812yes—
297C2×C32C3×C22⋊Q811yescharacteristic
298C2×C32C3×#12312yes—
299C2×C32C3×#12311yescharacteristic
300C2×C32C3×#12711yescharacteristic
301C24C2×(C3⋊D3)⋊C311yescharacteristic
302C22×C4—92no—
303C22×C4—92no—
304C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
305C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
306C4⋊C4—184no—
307C4⋊C4—182no—
308C22⋊C4—22no—
309C22⋊C4—184no—
310C22×C3—24no—
311C22×C3—22no—
312C22×C3—31no—
313C22×C3—31no—
314C22×C3—31no—
315C22×C3—31no—
316C22×C3—32no—
317C22×C3—32no—
318C22×C3—32no—
319C22×C3—32no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—42no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—62no—
326C22×C3—62no—
327C22×C3—64no—
328C22×C3—64no—
329C22×C3—64no—
330C22×C3—61no—
331C22×C3—61no—
332C22×C3—62no—
333C22×C3—62no—
334C22×C3—61no—
335C22×C3—124no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—121no—
340C22×C3—122no—
341C22×C3—122no—
342C22×C3—122no—
343C22×C3—122no—
344C22×C3—124no—
345C22×C3C2×C3×Dic611yescharacteristic
346C22×C3C3×#23511yescharacteristic
347C22×C3C3×D3×D412yes—
348C22×C3C3×#23712yes—
349C22×C3C2×C3×#6711yescharacteristic
350C3×C4—22no—
351C3×C4—41no—
352C3×C4—62no—
353C3×C4—62no—
354C3×C4—122no—
355C3×C4—121no—
356C3×C4—121no—
357C3×C4—122no—
358C3×C4—184no—
359C3×C4—362no—
360Dic3—64no—
361Dic3—122no—
362Dic3—184no—
363Dic3—362no—
364C32—21no—
365C32#494911yescharacteristic, complemented
366C32C3×#39911yescharacteristic, complemented
367C23—21no—
368C23—22no—
369C23—22no—
370C23—21no—
371C23#634⋊C312yes—
372C23C22×(C3⋊D3)⋊C311yescharacteristic
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—94no—
377C2×C4—182no—
378C2×C4—182no—
379C2×C4—182no—
380C2×C4—182no—
381C2×C4—182no—
382C2×C4#638⋊C312yes—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—32no—
390C2×C3—32no—
391C2×C3—32no—
392C2×C3—32no—
393C2×C3—42no—
394C2×C3—61no—
395C2×C3—61no—
396C2×C3—61no—
397C2×C3—62no—
398C2×C3—62no—
399C2×C3—122no—
400C2×C3—122no—
401C2×C3—124no—
402C2×C3C3×#87812yes—
403C2×C3C3×#88312yes—
404C2×C3C3×#92411yescharacteristic
405C2×C3C3×#93511yescharacteristic
406C2×C3C3×#94011yescharacteristic
407C22—22no—
408C22—24no—
409C22—41no—
410C22—41no—
411C22—42no—
412C22C2×#634⋊C311yescharacteristic
413C22#3828⋊C311yescharacteristic
414C22D4×(C3⋊D3)⋊C312yes—
415C22#3830⋊C312yes—
416C22C2×#638⋊C311yescharacteristic
417C4—22no—
418C4—41no—
419C4—184no—
420C4—362no—
421C3—31no—
422C3—31no—
423C3—61no—
424C3C3×#494911yescharacteristic
425C2—42no—
426C2#64087⋊C312yes—
427C2#64092⋊C312yes—
428C2#64133⋊C311yescharacteristic
429C2#64144⋊C311yescharacteristic
430C2#64149⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC24×H(3,𝔽3)C22
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#399
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
4. C2×#3750⋊C3C2semidirect
5. #63658⋊C3C2semidirect
6. #10592648C3semidirect
21. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
22. C2×#3750C2×C3semidirect
26. #63658C2×C3semidirect
224. H(3,𝔽3)#399semidirect
365. C32#4949semidirect
366. C32C3×#399semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×H(3,𝔽3)2
C2×#3757⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#3750⋊C32
#63658⋊C32
#105926483
C3×#49493