Finite Group Databasebeta

← #10592691⋊C3 #13142630

Subgroups — #10592691⋊C3 #13142630

Order 1728.

687 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 687 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3752⋊C3C212yesmaximal
3C2×#3752⋊C3C211yesmaximal, characteristic
4#63658⋊C3C212yesmaximal, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#10592691C311yesmaximal, characteristic, complemented
7C3×#4992—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)—22no—
14#3752⋊C3—24no—
15#3752⋊C3—22no—
16C2×C32×C4⋊C4—21no—
17C2×C32×C4⋊C4D311yescharacteristic, complemented
18C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
19C2×#3752C2×C312yes—
20C2×#3752C2×C311yescharacteristic
21C2×C3×(C2×D3)⋊C4—32no—
22C2×C3×(C2×D3)⋊C4—31no—
23#63658C2×C312yescomplemented
24#63658C2×C311yescharacteristic, complemented
25C3×C3⋊#95—32no—
26C3×C3⋊#95—31no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C22×(C3⋊D3)⋊C3—21no—
29C22×(C3⋊D3)⋊C3—21no—
30C22×(C3⋊D3)⋊C3—22no—
31C22×(C3⋊D3)⋊C3—22no—
32C22×(C3⋊D3)⋊C3D412yes—
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)—22no—
35C2×C4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)D412yes—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—21no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C3×#397—91no—
44#4992—31no—
45#4992—91no—
46C23×C3⋊D3C22×C311yescharacteristic
47C23×C3×D3—31no—
48C22×C32×C4—21no—
49C22×C32×C4—22no—
50C22×C32×C4C2×D312yes—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C22×C312yes—
53C22×C32×C4C22×C311yescharacteristic
54C22×C3⋊Dic3C22×C312yes—
55C22×C3⋊Dic3C22×C311yescharacteristic
56C22×C3×Dic3—32no—
57C22×C3×Dic3—31no—
58C32×C4⋊C4—24no—
59C32×C4⋊C4—22no—
60C32×C4⋊C4—22no—
61#3752—24no—
62#3752—22no—
63C3×(C2×D3)⋊C4—64no—
64C3×(C2×D3)⋊C4—62no—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C22×H(3,𝔽3)C2×D411yescharacteristic
67C22×H(3,𝔽3)#4111yescharacteristic
68C22×H(3,𝔽3)#4112yes—
69C22×H(3,𝔽3)#4112yes—
70C2×(C3⋊D3)⋊C3—22no—
71C2×(C3⋊D3)⋊C3—24no—
72C2×(C3⋊D3)⋊C3—41no—
73C2×(C3⋊D3)⋊C3—41no—
74C2×(C3⋊D3)⋊C3—42no—
75C4×H(3,𝔽3)—22no—
76C4×H(3,𝔽3)—42no—
77(C3⋊Dic3)⋊C3—42no—
78(C3⋊Dic3)⋊C3—41no—
79C2×C3×C4⋊C4—31no—
80C2×C3×C4⋊C4—31no—
81C2×C3×C4⋊C4—61no—
82C2×C3×C4⋊C4C3×D311yescharacteristic
83C2×C3×C22⋊C4—91no—
84C2×C3×C22⋊C4—92no—
85C2×(C2×D3)⋊C4—32no—
86C2×(C2×D3)⋊C4—31no—
87C2×(C2×D3)⋊C4—91no—
88C2×(C2×D3)⋊C4—92no—
89C3×#95—91no—
90C3×#95—92no—
91C3⋊#95—32no—
92C3⋊#95—31no—
93C3⋊#95—91no—
94C3⋊#95—92no—
95C23×C32—21no—
96C23×C32C22×D311yescharacteristic
97C23×C32C23×C311yescharacteristic
98C22×C3⋊D3—21no—
99C22×C3⋊D3—22no—
100C22×C3⋊D3—21no—
101C22×C3⋊D3—22no—
102C22×C3⋊D3C3×D412yes—
103C22×C3×D3—32no—
104C22×C3×D3—62no—
105C22×C3×D3—61no—
106C22×C3×D3—62no—
107C22×C3×D3—61no—
108C2×C32×C4—24no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—24no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4#6712yes—
122C2×C32×C4C3×D412yes—
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—22no—
126C2×C3⋊Dic3—22no—
127C2×C3⋊Dic3—22no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—61no—
133#397—91noSylow
134C2×H(3,𝔽3)#12111yescharacteristic
135C2×H(3,𝔽3)#12312yes—
136C2×H(3,𝔽3)#12311yescharacteristic
137C2×H(3,𝔽3)#12411yescharacteristic
138C2×H(3,𝔽3)#12612yes—
139(C3⋊D3)⋊C3—42no—
140C24×C3—91no—
141C23×D3—31no—
142C23×D3—91no—
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—32no—
146C22×C3×C4—32no—
147C22×C3×C4—61no—
148C22×C3×C4—62no—
149C22×C3×C4—92no—
150C22×C3×C4—91no—
151C22×C3×C4C2×C3×D312yes—
152C22×C3×C4C2×C3×D311yescharacteristic
153C22×Dic3—32no—
154C22×Dic3—31no—
155C22×Dic3—92no—
156C22×Dic3—91no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—62no—
159C3×C4⋊C4—62no—
160C3×C4⋊C4—64no—
161C3×C22⋊C4—182no—
162C3×C22⋊C4—184no—
163(C2×D3)⋊C4—64no—
164(C2×D3)⋊C4—62no—
165(C2×D3)⋊C4—182no—
166(C2×D3)⋊C4—184no—
167C22×C32—22no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32—22no—
171C22×C32—21no—
172C22×C32#23512yes—
173C22×C32D3×D411yescharacteristic
174C22×C32#23711yescharacteristic
175C22×C32#23912yes—
176C22×C32C2×#6711yescharacteristic
177C22×C32C2×C3×D411yescharacteristic
178C22×C32C2×C3×D411yescharacteristic
179C22×C32C3×#4112yes—
180C22×C32C3×#4111yescharacteristic
181C22×C32C3×#4112yes—
182C2×C3⋊D3—22no—
183C2×C3⋊D3—24no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C2×C3⋊D3—42no—
187C2×C3×D3—64no—
188C2×C3×D3—62no—
189C2×C3×D3—122no—
190C2×C3×D3—121no—
191C2×C3×D3—121no—
192C32×C4—24no—
193C32×C4—22no—
194C32×C4—22no—
195C32×C4—44no—
196C32×C4—42no—
197C32×C4—42no—
198C3⋊Dic3—42no—
199C3⋊Dic3—41no—
200C3×Dic3—122no—
201C3×Dic3—121no—
202C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
203C2×C22⋊C4—92no—
204C2×C22⋊C4—91no—
205#95—92no—
206#95—91no—
207H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
208C23×C3—31no—
209C23×C3—31no—
210C23×C3—61no—
211C23×C3—92no—
212C23×C3—181no—
213C23×C3—182no—
214C23×C3—181no—
215C23×C3—182no—
216C23×C3C22×C3×D311yescharacteristic
217C22×D3—32no—
218C22×D3—62no—
219C22×D3—61no—
220C22×D3—62no—
221C22×D3—61no—
222C22×D3—92no—
223C22×D3—181no—
224C22×D3—182no—
225C22×D3—181no—
226C22×D3—182no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—64no—
236C2×C3×C4—62no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—62no—
242C2×C3×C4—64no—
243C2×C3×C4—62no—
244C2×C3×C4—122no—
245C2×C3×C4—122no—
246C2×C3×C4—182no—
247C2×C3×C4—181no—
248C2×C3×C4—182no—
249C2×C3×C4—182no—
250C2×C3×C4—182no—
251C2×C3×C4C3×#6712yes—
252C2×Dic3—62no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—61no—
257C2×Dic3—182no—
258C2×Dic3—181no—
259C2×Dic3—182no—
260C2×Dic3—182no—
261C2×Dic3—182no—
262C2×C32—22no—
263C2×C32—22no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32#89412yes—
268C2×C32#89812yes—
269C2×C32#92911yescharacteristic
270C2×C32#93811yescharacteristic
271C2×C32#94711yescharacteristic
272C2×C32C3×#12111yescharacteristic
273C2×C32C3×#12312yes—
274C2×C32C3×#12311yescharacteristic
275C2×C32C3×#12411yescharacteristic
276C2×C32C3×#12612yes—
277C3⋊D3—42no—
278C3×D3—122no—
279C24—91no—
280C22×C4—92no—
281C22×C4—91no—
282C22×C4C2×(C3⋊D3)⋊C312yes—
283C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
284C4⋊C4—22no—
285C22⋊C4—184no—
286C22⋊C4—182no—
287C22×C3—32no—
288C22×C3—32no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—32no—
294C22×C3—32no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—62no—
298C22×C3—61no—
299C22×C3—61no—
300C22×C3—62no—
301C22×C3—61no—
302C22×C3—182no—
303C22×C3—184no—
304C22×C3—362no—
305C22×C3—361no—
306C22×C3—361no—
307C22×C3C3×#23512yes—
308C22×C3C3×D3×D411yescharacteristic
309C22×C3C3×#23711yescharacteristic
310C22×C3C3×#23912yes—
311C22×C3C2×C3×#6711yescharacteristic
312C2×D3—64no—
313C2×D3—62no—
314C2×D3—122no—
315C2×D3—121no—
316C2×D3—121no—
317C2×D3—182no—
318C2×D3—184no—
319C2×D3—362no—
320C2×D3—361no—
321C2×D3—361no—
322C3×C4—22no—
323C3×C4—42no—
324C3×C4—62no—
325C3×C4—62no—
326C3×C4—64no—
327C3×C4—122no—
328C3×C4—122no—
329C3×C4—124no—
330C3×C4—361no—
331C3×C4—362no—
332Dic3—122no—
333Dic3—121no—
334Dic3—361no—
335Dic3—362no—
336C32—21no—
337C32#499211yescharacteristic, complemented
338C32C3×#39711yescharacteristic, complemented
339C23—92no—
340C23—182no—
341C23—181no—
342C23—182no—
343C23—181no—
344C23C22×(C3⋊D3)⋊C311yescharacteristic
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—182no—
350C2×C4—182no—
351C2×C4—182no—
352C2×C4—182no—
353C2×C4—181no—
354C2×C4#638⋊C312yes—
355C2×C3—32no—
356C2×C3—32no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—32no—
364C2×C3—32no—
365C2×C3—62no—
366C2×C3—61no—
367C2×C3—61no—
368C2×C3—61no—
369C2×C3—62no—
370C2×C3—362no—
371C2×C3C3×#89412yes—
372C2×C3C3×#89812yes—
373C2×C3C3×#92911yescharacteristic
374C2×C3C3×#93811yescharacteristic
375C2×C3C3×#94711yescharacteristic
376D3—122no—
377D3—362no—
378C22—182no—
379C22—184no—
380C22—362no—
381C22—361no—
382C22—361no—
383C22#3828⋊C312yes—
384C22D4×(C3⋊D3)⋊C311yescharacteristic
385C22#3830⋊C311yescharacteristic
386C22#8872212yes—
387C22C2×#638⋊C311yescharacteristic
388C4—22no—
389C4—42no—
390C4—361no—
391C4—362no—
392C3—31no—
393C3—31no—
394C3—61no—
395C3C3×#499211yescharacteristic
396C2—362no—
397C2#64103⋊C312yes—
398C2#64107⋊C312yes—
399C2#64138⋊C311yescharacteristic
400C2#64147⋊C311yescharacteristic
401C2#64156⋊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×#397
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. #63658⋊C3C2semidirect
5. #63658⋊C3C2semidirect
6. #10592691C3semidirect
17. C2×C32×C4⋊C4D3semidirect
18. C2×C32×C4⋊C4C2×C3semidirect
23. #63658C2×C3semidirect
24. #63658C2×C3semidirect
202. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
207. H(3,𝔽3)#397semidirect
337. C32#4992semidirect
338. C32C3×#397semidirect

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
#63658⋊C32
#105926913
C3×#49923