Finite Group Databasebeta

← #10592692⋊C3 #13142618

Subgroups — #10592692⋊C3 #13142618

Order 1728.

743 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3752⋊C3C211yesmaximal, characteristic
3C2×#3752⋊C3C212yesmaximal
4C2×#3750⋊C3C212yesmaximal, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#10592692C311yesmaximal, characteristic, complemented
7C3×#4993—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C4⋊C4×H(3,𝔽3)—22no—
14#3752⋊C3—22no—
15#3752⋊C3—24no—
16#3750⋊C3—24no—
17C2×C32×C4⋊C4—21no—
18C2×C32×C4⋊C4D311yescharacteristic, complemented
19C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
20C2×#3752C2×C311yescharacteristic
21C2×#3752C2×C312yes—
22C2×#3750C2×C312yescomplemented
23C2×C3×(C2×D3)⋊C4—32no—
24C2×C3×(C2×D3)⋊C4—31no—
25C2×C3×Dic3⋊C4—32no—
26#63658C2×C311yescharacteristic, complemented
27C3×C3⋊#95—31no—
28C23×H(3,𝔽3)C2311yescharacteristic
29C22×(C3⋊D3)⋊C3—22no—
30C22×(C3⋊D3)⋊C3—21no—
31C22×(C3⋊D3)⋊C3—22no—
32C22×(C3⋊D3)⋊C3—21no—
33C22×(C3⋊D3)⋊C3Q812yes—
34C2×C4×H(3,𝔽3)—22no—
35C2×C4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)—22no—
38C2×C4×H(3,𝔽3)D412yes—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D414yes—
44C3×#399—91no—
45#4993—31no—
46#4993—91no—
47C23×C3⋊D3C22×C311yescharacteristic
48C23×C3×D3—31no—
49C22×C32×C4—21no—
50C22×C32×C4—22no—
51C22×C32×C4C2×D312yes—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C22×C312yes—
54C22×C32×C4C22×C311yescharacteristic
55C22×C3⋊Dic3C22×C311yescharacteristic
56C22×C3⋊Dic3C22×C312yes—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—31no—
59C32×C4⋊C4—24no—
60C32×C4⋊C4—22no—
61C32×C4⋊C4—22no—
62#3752—22no—
63#3752—24no—
64#3750—24no—
65C3×(C2×D3)⋊C4—64no—
66C3×(C2×D3)⋊C4—62no—
67C3×Dic3⋊C4—64no—
68C22×H(3,𝔽3)C2×D412yes—
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C22×H(3,𝔽3)C2×Q811yescharacteristic
71C22×H(3,𝔽3)#4112yes—
72C22×H(3,𝔽3)#4111yescharacteristic
73C2×(C3⋊D3)⋊C3—24no—
74C2×(C3⋊D3)⋊C3—22no—
75C2×(C3⋊D3)⋊C3—42no—
76C2×(C3⋊D3)⋊C3—41no—
77C2×(C3⋊D3)⋊C3—41no—
78C4×H(3,𝔽3)—22no—
79C4×H(3,𝔽3)—42no—
80(C3⋊Dic3)⋊C3—24no—
81(C3⋊Dic3)⋊C3—41no—
82C2×C3×C4⋊C4—31no—
83C2×C3×C4⋊C4—31no—
84C2×C3×C4⋊C4—61no—
85C2×C3×C4⋊C4—92no—
86C2×C3×C4⋊C4C3×D311yescharacteristic
87C2×C3×C22⋊C4—91no—
88C2×C3×C22⋊C4—92no—
89C2×(C2×D3)⋊C4—32no—
90C2×(C2×D3)⋊C4—31no—
91C2×(C2×D3)⋊C4—92no—
92C2×(C2×D3)⋊C4—91no—
93C2×Dic3⋊C4—32no—
94C2×Dic3⋊C4—92no—
95C3×#95—91no—
96C3⋊#95—31no—
97C3⋊#95—91no—
98C23×C32—21no—
99C23×C32C22×D311yescharacteristic
100C23×C32C23×C311yescharacteristic
101C22×C3⋊D3—22no—
102C22×C3⋊D3—21no—
103C22×C3⋊D3—22no—
104C22×C3⋊D3—21no—
105C22×C3⋊D3C3×Q812yes—
106C22×C3×D3—32no—
107C22×C3×D3—62no—
108C22×C3×D3—62no—
109C22×C3×D3—61no—
110C22×C3×D3—61no—
111C2×C32×C4—24no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—24no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—42no—
123C2×C32×C4—42no—
124C2×C32×C4#6712yes—
125C2×C32×C4C3×D412yes—
126C2×C3⋊Dic3—22no—
127C2×C3⋊Dic3—22no—
128C2×C3⋊Dic3—22no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3C3×D414yes—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—61no—
136#399—91noSylow
137C2×H(3,𝔽3)C22⋊Q812yes—
138C2×H(3,𝔽3)C22⋊Q811yescharacteristic
139C2×H(3,𝔽3)#12312yes—
140C2×H(3,𝔽3)#12311yescharacteristic
141C2×H(3,𝔽3)#12711yescharacteristic
142(C3⋊D3)⋊C3—42no—
143C24×C3—91no—
144C23×D3—31no—
145C23×D3—91no—
146C22×C3×C4—31no—
147C22×C3×C4—31no—
148C22×C3×C4—32no—
149C22×C3×C4—32no—
150C22×C3×C4—61no—
151C22×C3×C4—62no—
152C22×C3×C4—92no—
153C22×C3×C4—91no—
154C22×C3×C4C2×C3×D312yes—
155C22×C3×C4C2×C3×D311yescharacteristic
156C22×Dic3—32no—
157C22×Dic3—31no—
158C22×Dic3—92no—
159C22×Dic3—91no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—62no—
162C3×C4⋊C4—62no—
163C3×C4⋊C4—64no—
164C3×C4⋊C4—184no—
165C3×C22⋊C4—182no—
166C3×C22⋊C4—184no—
167(C2×D3)⋊C4—64no—
168(C2×D3)⋊C4—62no—
169(C2×D3)⋊C4—184no—
170(C2×D3)⋊C4—182no—
171Dic3⋊C4—64no—
172Dic3⋊C4—184no—
173C22×C32—22no—
174C22×C32—22no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32#23512yes—
179C22×C32D3×D412yes—
180C22×C32D3×Q811yescharacteristic
181C22×C32#23911yescharacteristic
182C22×C32C2×#6711yescharacteristic
183C22×C32C2×C3×D412yes—
184C22×C32C2×C3×D411yescharacteristic
185C22×C32C2×C3×Q811yescharacteristic
186C22×C32C3×#4112yes—
187C22×C32C3×#4111yescharacteristic
188C2×C3⋊D3—24no—
189C2×C3⋊D3—22no—
190C2×C3⋊D3—42no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C2×C3×D3—64no—
194C2×C3×D3—62no—
195C2×C3×D3—122no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C32×C4—24no—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—44no—
202C32×C4—42no—
203C32×C4—42no—
204C3⋊Dic3—24no—
205C3⋊Dic3—41no—
206C3×Dic3—64no—
207C3×Dic3—121no—
208C2×C4⋊C4—92no—
209C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
210C2×C22⋊C4—92no—
211C2×C22⋊C4—91no—
212#95—91no—
213H(3,𝔽3)#39911yescharacteristic, complemented, Sylow
214C23×C3—31no—
215C23×C3—31no—
216C23×C3—61no—
217C23×C3—92no—
218C23×C3—182no—
219C23×C3—181no—
220C23×C3—182no—
221C23×C3—181no—
222C23×C3C22×C3×D311yescharacteristic
223C22×D3—32no—
224C22×D3—62no—
225C22×D3—62no—
226C22×D3—61no—
227C22×D3—61no—
228C22×D3—92no—
229C22×D3—182no—
230C22×D3—181no—
231C22×D3—182no—
232C22×D3—181no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—32no—
238C2×C3×C4—32no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—64no—
242C2×C3×C4—62no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—62no—
246C2×C3×C4—62no—
247C2×C3×C4—62no—
248C2×C3×C4—64no—
249C2×C3×C4—62no—
250C2×C3×C4—94no—
251C2×C3×C4—122no—
252C2×C3×C4—122no—
253C2×C3×C4—182no—
254C2×C3×C4—182no—
255C2×C3×C4—181no—
256C2×C3×C4—182no—
257C2×C3×C4C3×#6712yes—
258C2×Dic3—34no—
259C2×Dic3—62no—
260C2×Dic3—62no—
261C2×Dic3—62no—
262C2×Dic3—61no—
263C2×Dic3—94no—
264C2×Dic3—182no—
265C2×Dic3—182no—
266C2×Dic3—182no—
267C2×Dic3—181no—
268C2×C32—22no—
269C2×C32—22no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32#89412yes—
274C2×C32#89612yes—
275C2×C32#92911yescharacteristic
276C2×C32#94011yescharacteristic
277C2×C32#94611yescharacteristic
278C2×C32C3×C22⋊Q812yes—
279C2×C32C3×C22⋊Q811yescharacteristic
280C2×C32C3×#12312yes—
281C2×C32C3×#12311yescharacteristic
282C2×C32C3×#12711yescharacteristic
283C3⋊D3—42no—
284C3×D3—122no—
285C24—91no—
286C22×C4—92no—
287C22×C4—91no—
288C22×C4C2×(C3⋊D3)⋊C312yes—
289C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—184no—
292C22⋊C4—184no—
293C22⋊C4—182no—
294C22×C3—32no—
295C22×C3—32no—
296C22×C3—32no—
297C22×C3—32no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—31no—
301C22×C3—31no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—62no—
305C22×C3—62no—
306C22×C3—61no—
307C22×C3—61no—
308C22×C3—61no—
309C22×C3—182no—
310C22×C3—184no—
311C22×C3—361no—
312C22×C3—361no—
313C22×C3—362no—
314C22×C3C3×#23512yes—
315C22×C3C3×D3×D412yes—
316C22×C3C3×D3×Q811yescharacteristic
317C22×C3C3×#23911yescharacteristic
318C22×C3C2×C3×#6711yescharacteristic
319C2×D3—64no—
320C2×D3—62no—
321C2×D3—122no—
322C2×D3—121no—
323C2×D3—121no—
324C2×D3—182no—
325C2×D3—184no—
326C2×D3—361no—
327C2×D3—361no—
328C2×D3—362no—
329C3×C4—22no—
330C3×C4—42no—
331C3×C4—62no—
332C3×C4—62no—
333C3×C4—64no—
334C3×C4—122no—
335C3×C4—122no—
336C3×C4—124no—
337C3×C4—184no—
338C3×C4—361no—
339Dic3—64no—
340Dic3—121no—
341Dic3—184no—
342Dic3—361no—
343C32—21no—
344C32#499311yescharacteristic, complemented
345C32C3×#39911yescharacteristic, complemented
346C23—92no—
347C23—182no—
348C23—182no—
349C23—181no—
350C23—181no—
351C23C22×(C3⋊D3)⋊C311yescharacteristic
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—94no—
357C2×C4—182no—
358C2×C4—182no—
359C2×C4—182no—
360C2×C4—181no—
361C2×C4#638⋊C312yes—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—32no—
369C2×C3—32no—
370C2×C3—32no—
371C2×C3—32no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—61no—
375C2×C3—62no—
376C2×C3—62no—
377C2×C3—362no—
378C2×C3C3×#89412yes—
379C2×C3C3×#89612yes—
380C2×C3C3×#92911yescharacteristic
381C2×C3C3×#94011yescharacteristic
382C2×C3C3×#94611yescharacteristic
383D3—122no—
384D3—362no—
385C22—182no—
386C22—184no—
387C22—361no—
388C22—361no—
389C22—362no—
390C22#3828⋊C312yes—
391C22D4×(C3⋊D3)⋊C312yes—
392C22Q8×(C3⋊D3)⋊C311yescharacteristic
393C22#8872211yescharacteristic
394C22C2×#638⋊C311yescharacteristic
395C4—22no—
396C4—42no—
397C4—184no—
398C4—361no—
399C3—31no—
400C3—31no—
401C3—61no—
402C3C3×#499311yescharacteristic
403C2—362no—
404C2#64103⋊C312yes—
405C2#64105⋊C312yes—
406C2#64138⋊C311yescharacteristic
407C2#64149⋊C311yescharacteristic
408C2#64155⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#399
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×H(3,𝔽3)C2semidirect
4. C2×#3750⋊C3C2semidirect
5. #63658⋊C3C2semidirect
6. #10592692C3semidirect
18. C2×C32×C4⋊C4D3semidirect
19. C2×C32×C4⋊C4C2×C3semidirect
22. C2×#3750C2×C3semidirect
26. #63658C2×C3semidirect
209. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
213. H(3,𝔽3)#399semidirect
344. C32#4993semidirect
345. C32C3×#399semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
C2×#3752⋊C32
C2×#3750⋊C32
#63658⋊C32
#105926923
C3×#49933