Finite Group Databasebeta

← #10592652⋊C3 #13142502

Subgroups — #10592652⋊C3 #13142502

Order 1728.

727 conjugacy classes of subgroups.

The subgroup lattice has 425 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
4#63658⋊C3C211yesmaximal, characteristic, complemented
5#63658⋊C3C212yesmaximal, complemented
6#10592652C311yesmaximal, characteristic, complemented
7C3×#4953—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—
16C2×C32×C22⋊C4—21no—
17C2×C32×C22⋊C4D311yescharacteristic
18C2×C32×C22⋊C4C2×C311yescharacteristic
19C2×#3757C2×C312yes—
20C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
21C2×C3×#217—32no—
22C2×C3×C4⋊Dic3—31no—
23#63658C2×C312yescomplemented
24#63658C2×C311yescharacteristic, complemented
25C3×C3⋊#95—32no—
26C3×C3⋊#95—31no—
27C23×H(3,𝔽3)—21no—
28C23×H(3,𝔽3)—21no—
29C23×H(3,𝔽3)—22no—
30C23×H(3,𝔽3)—22no—
31C23×H(3,𝔽3)D412yes—
32C23×H(3,𝔽3)C2311yescharacteristic
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)D412yes—
37C2×(C3⋊Dic3)⋊C3—22no—
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—
43C3×#397—91no—
44#4953—31no—
45#4953—91no—
46C24×C32—21no—
47C24×C32C2×D311yescharacteristic
48C24×C32C22×C311yescharacteristic
49C22×C32×C4—21no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C22×C311yescharacteristic
54C22×C32×C4C22×C311yescharacteristic
55C22×C3⋊Dic3C22×C312yes—
56C22×C3⋊Dic3C22×C312yes—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—32no—
59C32×C22⋊C4—24no—
60C32×C22⋊C4—22no—
61C32×C22⋊C4—22no—
62#3757—24no—
63C4⋊(C3⋊Dic3)—22no—
64C3×#217—64no—
65C3×C4⋊Dic3—62no—
66C22×H(3,𝔽3)—22no—
67C22×H(3,𝔽3)—24no—
68C22×H(3,𝔽3)—41no—
69C22×H(3,𝔽3)—41no—
70C22×H(3,𝔽3)—42no—
71C22×H(3,𝔽3)C2×D411yescharacteristic
72C22×H(3,𝔽3)C2×D411yescharacteristic
73C22×H(3,𝔽3)#4111yescharacteristic
74C22×H(3,𝔽3)#4112yes—
75C22×H(3,𝔽3)#4112yes—
76C4×H(3,𝔽3)—22no—
77C4×H(3,𝔽3)—41no—
78(C3⋊Dic3)⋊C3—42no—
79(C3⋊Dic3)⋊C3—42no—
80C2×C3×C4⋊C4—91no—
81C2×C3×C22⋊C4—31no—
82C2×C3×C22⋊C4—31no—
83C2×C3×C22⋊C4—61no—
84C2×C3×C22⋊C4—92no—
85C2×C3×C22⋊C4C3×D311yescharacteristic
86C2×#217—32no—
87C2×#217—92no—
88C2×C4⋊Dic3—31no—
89C2×C4⋊Dic3—91no—
90C3×#95—91no—
91C3×#95—92no—
92C3⋊#95—31no—
93C3⋊#95—32no—
94C3⋊#95—91no—
95C3⋊#95—92no—
96C23×C32—22no—
97C23×C32—22no—
98C23×C32—21no—
99C23×C32—22no—
100C23×C32—22no—
101C23×C32—21no—
102C23×C32—21no—
103C23×C32—21no—
104C23×C32—22no—
105C23×C32—22no—
106C23×C32—22no—
107C23×C32—21no—
108C23×C32—42no—
109C23×C32—42no—
110C23×C32D1212yes—
111C23×C32C3×D412yes—
112C23×C32C22×D311yescharacteristic
113C23×C32C23×C311yescharacteristic
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
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—22no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—22no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138#397—91noSylow
139C2×H(3,𝔽3)—42no—
140C2×H(3,𝔽3)#12111yescharacteristic
141C2×H(3,𝔽3)#12312yes—
142C2×H(3,𝔽3)#12311yescharacteristic
143C2×H(3,𝔽3)#12411yescharacteristic
144C2×H(3,𝔽3)#12612yes—
145C24×C3—31no—
146C24×C3—31no—
147C24×C3—61no—
148C24×C3C2×C3×D311yescharacteristic
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—31no—
152C22×C3×C4—31no—
153C22×C3×C4—61no—
154C22×C3×C4—61no—
155C22×C3×C4—92no—
156C22×C3×C4—92no—
157C22×C3×C4C2×C3×D311yescharacteristic
158C22×C3×C4C2×C3×D311yescharacteristic
159C22×Dic3—32no—
160C22×Dic3—32no—
161C22×Dic3—92no—
162C22×Dic3—92no—
163C3×C4⋊C4—182no—
164C3×C22⋊C4—22no—
165C3×C22⋊C4—62no—
166C3×C22⋊C4—62no—
167C3×C22⋊C4—64no—
168C3×C22⋊C4—184no—
169#217—64no—
170#217—184no—
171C4⋊Dic3—62no—
172C4⋊Dic3—182no—
173C22×C32—24no—
174C22×C32—24no—
175C22×C32—22no—
176C22×C32—22no—
177C22×C32—24no—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—22no—
181C22×C32—22no—
182C22×C32—21no—
183C22×C32—44no—
184C22×C32—44no—
185C22×C32—42no—
186C22×C32—42no—
187C22×C32—42no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32—42no—
193C22×C32C2×D1211yescharacteristic
194C22×C32#23511yescharacteristic
195C22×C32#23712yes—
196C22×C32#23712yes—
197C22×C32C2×#6711yescharacteristic
198C22×C32C2×C3×D411yescharacteristic
199C22×C32C2×C3×D411yescharacteristic
200C22×C32C3×#4111yescharacteristic
201C22×C32C3×#4112yes—
202C22×C32C3×#4112yes—
203C32×C4—22no—
204C32×C4—22no—
205C32×C4—42no—
206C32×C4—42no—
207C32×C4—41no—
208C32×C4—41no—
209C3⋊Dic3—42no—
210C3⋊Dic3—42no—
211C3×Dic3—122no—
212C3×Dic3—122no—
213C2×C4⋊C4—91no—
214C2×C22⋊C4—92no—
215C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic
216#95—91no—
217#95—92no—
218H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
219C23×C3—21no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—21no—
223C23×C3—32no—
224C23×C3—32no—
225C23×C3—31no—
226C23×C3—31no—
227C23×C3—61no—
228C23×C3—61no—
229C23×C3—61no—
230C23×C3—61no—
231C23×C3—62no—
232C23×C3—62no—
233C23×C3—62no—
234C23×C3—62no—
235C23×C3—62no—
236C23×C3—62no—
237C23×C3—62no—
238C23×C3—61no—
239C23×C3—122no—
240C23×C3—122no—
241C23×C3C3×D1212yes—
242C23×C3C22×C3×D311yescharacteristic
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—21no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—62no—
249C2×C3×C4—62no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—122no—
257C2×C3×C4—122no—
258C2×C3×C4—182no—
259C2×C3×C4—182no—
260C2×C3×C4—182no—
261C2×C3×C4—182no—
262C2×C3×C4—182no—
263C2×C3×C4—182no—
264C2×C3×C4C3×#6712yes—
265C2×Dic3—62no—
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—62no—
269C2×Dic3—62no—
270C2×Dic3—62no—
271C2×Dic3—182no—
272C2×Dic3—182no—
273C2×Dic3—182no—
274C2×Dic3—182no—
275C2×Dic3—182no—
276C2×Dic3—182no—
277C2×C32—22no—
278C2×C32—22no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—44no—
283C2×C32—42no—
284C2×C32—42no—
285C2×C32#87912yes—
286C2×C32#88612yes—
287C2×C32#93011yescharacteristic
288C2×C32#93511yescharacteristic
289C2×C32#93611yescharacteristic
290C2×C32C3×#12111yescharacteristic
291C2×C32C3×#12312yes—
292C2×C32C3×#12311yescharacteristic
293C2×C32C3×#12411yescharacteristic
294C2×C32C3×#12612yes—
295C24C2×(C3⋊D3)⋊C311yescharacteristic
296C22×C4—92no—
297C22×C4—92no—
298C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
299C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
300C4⋊C4—182no—
301C22⋊C4—22no—
302C22⋊C4—184no—
303C22×C3—24no—
304C22×C3—22no—
305C22×C3—31no—
306C22×C3—31no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—32no—
310C22×C3—32no—
311C22×C3—32no—
312C22×C3—32no—
313C22×C3—31no—
314C22×C3—31no—
315C22×C3—42no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—62no—
319C22×C3—62no—
320C22×C3—64no—
321C22×C3—64no—
322C22×C3—64no—
323C22×C3—61no—
324C22×C3—61no—
325C22×C3—62no—
326C22×C3—62no—
327C22×C3—61no—
328C22×C3—124no—
329C22×C3—121no—
330C22×C3—121no—
331C22×C3—121no—
332C22×C3—121no—
333C22×C3—122no—
334C22×C3—122no—
335C22×C3—122no—
336C22×C3—122no—
337C22×C3—124no—
338C22×C3C2×C3×D1211yescharacteristic
339C22×C3C3×#23511yescharacteristic
340C22×C3C3×#23712yes—
341C22×C3C3×#23712yes—
342C22×C3C2×C3×#6711yescharacteristic
343C3×C4—22no—
344C3×C4—41no—
345C3×C4—62no—
346C3×C4—62no—
347C3×C4—122no—
348C3×C4—121no—
349C3×C4—121no—
350C3×C4—122no—
351C3×C4—362no—
352C3×C4—362no—
353Dic3—122no—
354Dic3—122no—
355Dic3—362no—
356Dic3—362no—
357C32—21no—
358C32#495311yescharacteristic, complemented
359C32C3×#39711yescharacteristic, complemented
360C23—21no—
361C23—22no—
362C23—22no—
363C23—21no—
364C23C4⋊((C3⋊D3)⋊C3)12yes—
365C23C22×(C3⋊D3)⋊C311yescharacteristic
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—182no—
370C2×C4—182no—
371C2×C4—182no—
372C2×C4—182no—
373C2×C4—182no—
374C2×C4—182no—
375C2×C4#638⋊C312yes—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—32no—
383C2×C3—32no—
384C2×C3—32no—
385C2×C3—32no—
386C2×C3—42no—
387C2×C3—61no—
388C2×C3—61no—
389C2×C3—61no—
390C2×C3—62no—
391C2×C3—62no—
392C2×C3—122no—
393C2×C3—122no—
394C2×C3—124no—
395C2×C3C3×#87912yes—
396C2×C3C3×#88612yes—
397C2×C3C3×#93011yescharacteristic
398C2×C3C3×#93511yescharacteristic
399C2×C3C3×#93611yescharacteristic
400C22—22no—
401C22—24no—
402C22—41no—
403C22—41no—
404C22—42no—
405C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
406C22#3828⋊C311yescharacteristic
407C22#3830⋊C312yes—
408C22#3830⋊C312yes—
409C22C2×#638⋊C311yescharacteristic
410C4—22no—
411C4—41no—
412C4—362no—
413C4—362no—
414C3—31no—
415C3—31no—
416C3—61no—
417C3C3×#495311yescharacteristic
418C2—42no—
419C2#64088⋊C312yes—
420C2#64095⋊C312yes—
421C2#64139⋊C311yescharacteristic
422C2#64144⋊C311yescharacteristic
423C2#64145⋊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×#397
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. #63658⋊C3C2semidirect
5. #63658⋊C3C2semidirect
6. #10592652C3semidirect
20. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
23. #63658C2×C3semidirect
24. #63658C2×C3semidirect
218. H(3,𝔽3)#397semidirect
358. C32#4953semidirect
359. C32C3×#397semidirect

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
#63658⋊C32
#105926523
C3×#49533