Finite Group Databasebeta

← #13140179

Subgroups — #13140179

Order 1728.

727 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 727 conjugacy classes of subgroups — too many to draw. Only its 123 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×#3752⋊C3C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×#88450C212yesmaximal
4C2×#88449C212yesmaximal, complemented
5#11720574C211yesmaximal, characteristic, complemented
6#10591350—31nomaximal
7#10591317—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2212yes—
13#3752⋊C3—22no—
14C4⋊((C3⋊Dic3)⋊C3)—22no—
15#88450—24no—
16#88449—24no—
17C2×#3752D311yescharacteristic
18C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
19C2×C3×(C2×D3)⋊C4—31no—
20C2×C3×C4⋊Dic3—31no—
21C2×Dic3⋊Dic3—32no—
22C2×Dic3⋊Dic3—32no—
23C2×#3722—32no—
24C2×(C2×D3)⋊Dic3—32no—
25C3×C3⋊#95—31no—
26C3×C3⋊#95—31no—
27C3×C3⋊#95—61no—
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)D412yes—
36C2×#1084—22no—
37C2×#1084—22no—
38C2×#1084D414yes—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3—21no—
44#4993—91no—
45#4949—91no—
46#4677—91no—
47C23×C3⋊D3C2×D311yescharacteristic
48C23×C3×D3—31no—
49C22×C32×C4—21no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C2×D311yescharacteristic
52C22×C3⋊Dic3C2×D312yes—
53C22×C3⋊Dic3C2×D311yescharacteristic
54C22×C3×Dic3—32no—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—31no—
58C22×C3×Dic3—62no—
59#3752—22no—
60C4⋊(C3⋊Dic3)—22no—
61C3×(C2×D3)⋊C4—62no—
62C3×C4⋊Dic3—62no—
63Dic3⋊Dic3—64no—
64Dic3⋊Dic3—64no—
65#3722—64no—
66(C2×D3)⋊Dic3—64no—
67C22×H(3,𝔽3)C2×D412yes—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C22×H(3,𝔽3)C2×Q811yescharacteristic
70C22×H(3,𝔽3)#4112yes—
71C22×H(3,𝔽3)#4111yescharacteristic
72C2×(C3⋊D3)⋊C3—24no—
73C2×(C3⋊D3)⋊C3—22no—
74C2×(C3⋊D3)⋊C3—42no—
75C2×(C3⋊D3)⋊C3—41no—
76C2×(C3⋊D3)⋊C3—41no—
77C4×H(3,𝔽3)—22no—
78#1084—24no—
79(C3⋊Dic3)⋊C3—41no—
80(C3⋊Dic3)⋊C3—42no—
81C2×C3×C4⋊C4—91no—
82C2×C3×C22⋊C4—91no—
83C2×#217—92no—
84C2×(C2×D3)⋊C4—31no—
85C2×(C2×D3)⋊C4—91no—
86C2×(C2×D3)⋊C4—92no—
87C2×(C2×D3)⋊C4—92no—
88C2×C4⋊Dic3—31no—
89C2×C4⋊Dic3—91no—
90C2×C4⋊Dic3—92no—
91C2×Dic3⋊C4—92no—
92C2×Dic3⋊C4—92no—
93C3×#95—91no—
94C3⋊#95—31no—
95C3⋊#95—31no—
96C3⋊#95—61no—
97C23×C32—21no—
98C23×C32C22×D311yescharacteristic
99C23×C32C22×D311yescharacteristic
100C22×C3⋊D3—21no—
101C22×C3⋊D3—22no—
102C22×C3⋊D3—22no—
103C22×C3⋊D3—21no—
104C22×C3⋊D3Dic612yes—
105C22×C3×D3—32no—
106C22×C3×D3—62no—
107C22×C3×D3—62no—
108C22×C3×D3—61no—
109C22×C3×D3—61no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—42no—
114C2×C32×C4#6712yes—
115C2×C32×C4#6712yes—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3—22no—
120C2×C3⋊Dic3—21no—
121C2×C3×Dic3—34no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—122no—
134C2×C3×Dic3—122no—
135#399—271noSylow
136C2×H(3,𝔽3)C22⋊Q812yes—
137C2×H(3,𝔽3)C22⋊Q811yescharacteristic
138C2×H(3,𝔽3)#12312yes—
139C2×H(3,𝔽3)#12311yescharacteristic
140C2×H(3,𝔽3)#12711yescharacteristic
141(C3⋊D3)⋊C3—42no—
142C24×C3—91no—
143C23×D3—31no—
144C23×D3—91no—
145C22×C3×C4—31no—
146C22×C3×C4—31no—
147C22×C3×C4—61no—
148C22×C3×C4—92no—
149C22×C3×C4—91no—
150C22×C3×C4—92no—
151C22×C3×C4D3211yescharacteristic
152C22×Dic3—32no—
153C22×Dic3—32no—
154C22×Dic3—32no—
155C22×Dic3—31no—
156C22×Dic3—62no—
157C22×Dic3—91no—
158C22×Dic3—92no—
159C3×C4⋊C4—182no—
160C3×C22⋊C4—182no—
161#217—184no—
162(C2×D3)⋊C4—62no—
163(C2×D3)⋊C4—182no—
164(C2×D3)⋊C4—184no—
165(C2×D3)⋊C4—184no—
166C4⋊Dic3—62no—
167C4⋊Dic3—182no—
168C4⋊Dic3—184no—
169Dic3⋊C4—184no—
170Dic3⋊C4—184no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—22no—
174C22×C32—22no—
175C22×C32—21no—
176C22×C32C2×Dic611yescharacteristic
177C22×C32#23511yescharacteristic
178C22×C32#23512yes—
179C22×C32D3×D412yes—
180C22×C32D3×D412yes—
181C22×C32#23712yes—
182C22×C32D3×Q811yescharacteristic
183C22×C32#23911yescharacteristic
184C22×C32C2×#6711yescharacteristic
185C22×C32C2×#6711yescharacteristic
186C2×C3⋊D3—24no—
187C2×C3⋊D3—22no—
188C2×C3⋊D3—42no—
189C2×C3⋊D3—41no—
190C2×C3⋊D3—41no—
191C2×C3×D3—64no—
192C2×C3×D3—62no—
193C2×C3×D3—122no—
194C2×C3×D3—121no—
195C2×C3×D3—121no—
196C32×C4—22no—
197C32×C4—22no—
198C32×C4—42no—
199C3⋊Dic3—41no—
200C3⋊Dic3—42no—
201C3×Dic3—64no—
202C3×Dic3—64no—
203C3×Dic3—124no—
204C3×Dic3—122no—
205C3×Dic3—121no—
206C2×C4⋊C4—91no—
207C2×C4⋊C4—272no—
208C2×C22⋊C4—91no—
209C2×C22⋊C4—272no—
210#95—91no—
211H(3,𝔽3)#39911yescharacteristic, complemented, Sylow
212C23×C3—31no—
213C23×C3—31no—
214C23×C3—61no—
215C23×C3—92no—
216C23×C3—182no—
217C23×C3—181no—
218C23×C3—181no—
219C23×C3—182no—
220C23×C3C2×D3211yescharacteristic
221C22×D3—32no—
222C22×D3—61no—
223C22×D3—62no—
224C22×D3—62no—
225C22×D3—61no—
226C22×D3—92no—
227C22×D3—182no—
228C22×D3—181no—
229C22×D3—181no—
230C22×D3—182no—
231C2×C3×C4—22no—
232C2×C3×C4—32no—
233C2×C3×C4—32no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4—62no—
237C2×C3×C4—94no—
238C2×C3×C4—122no—
239C2×C3×C4—182no—
240C2×C3×C4—182no—
241C2×C3×C4—182no—
242C2×C3×C4—182no—
243C2×C3×C4—182no—
244C2×C3×C4—181no—
245C2×C3×C4—182no—
246C2×C3×C4#62512yes—
247C2×Dic3—34no—
248C2×Dic3—34no—
249C2×Dic3—64no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—62no—
257C2×Dic3—62no—
258C2×Dic3—61no—
259C2×Dic3—122no—
260C2×Dic3—122no—
261C2×Dic3—182no—
262C2×Dic3—182no—
263C2×Dic3—182no—
264C2×Dic3—181no—
265C2×Dic3—182no—
266C2×C32—22no—
267C2×C32—21no—
268C2×C32—22no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32#87812yes—
272C2×C32#88312yes—
273C2×C32#89412yes—
274C2×C32#89612yes—
275C2×C32#92411yescharacteristic
276C2×C32#92911yescharacteristic
277C2×C32#93511yescharacteristic
278C2×C32#94011yescharacteristic
279C2×C32#94011yescharacteristic
280C2×C32#94611yescharacteristic
281C3⋊D3—42no—
282C3×D3—122no—
283C24—91no—
284C22×C4—92no—
285C22×C4—91no—
286C22×C4—92no—
287C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
288C4⋊C4—182no—
289C4⋊C4—544no—
290C22⋊C4—182no—
291C22⋊C4—544no—
292C22×C3—31no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—32no—
297C22×C3—32no—
298C22×C3—32no—
299C22×C3—32no—
300C22×C3—31no—
301C22×C3—31no—
302C22×C3—61no—
303C22×C3—61no—
304C22×C3—62no—
305C22×C3—62no—
306C22×C3—61no—
307C22×C3—182no—
308C22×C3—184no—
309C22×C3—361no—
310C22×C3—362no—
311C22×C3—361no—
312C22×C3D3×Dic611yescharacteristic
313C22×C3#379911yescharacteristic
314C22×C3#380212yes—
315C22×C3#380412yes—
316C22×C3C2×#62511yescharacteristic
317C2×D3—62no—
318C2×D3—64no—
319C2×D3—122no—
320C2×D3—121no—
321C2×D3—121no—
322C2×D3—182no—
323C2×D3—184no—
324C2×D3—361no—
325C2×D3—362no—
326C2×D3—361no—
327C3×C4—22no—
328C3×C4—62no—
329C3×C4—62no—
330C3×C4—122no—
331C3×C4—184no—
332C3×C4—361no—
333C3×C4—362no—
334Dic3—64no—
335Dic3—64no—
336Dic3—124no—
337Dic3—121no—
338Dic3—122no—
339Dic3—362no—
340Dic3—361no—
341C32—21no—
342C32#494911yescharacteristic, complemented
343C32#499311yescharacteristic, complemented
344C23—92no—
345C23—181no—
346C23—182no—
347C23—182no—
348C23—181no—
349C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
350C2×C4—22no—
351C2×C4—94no—
352C2×C4—182no—
353C2×C4—182no—
354C2×C4—182no—
355C2×C4—182no—
356C2×C4—182no—
357C2×C4—181no—
358C2×C4—182no—
359C2×C4#619812yes—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—32no—
363C2×C3—32no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—32no—
367C2×C3—32no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—61no—
371C2×C3—62no—
372C2×C3—61no—
373C2×C3—62no—
374C2×C3—61no—
375C2×C3—362no—
376C2×C3#6385412yes—
377C2×C3#6385611yescharacteristic
378C2×C3#6389912yes—
379C2×C3#6391411yescharacteristic
380C2×C3#6396111yescharacteristic
381D3—122no—
382D3—362no—
383C22—182no—
384C22—184no—
385C22—361no—
386C22—362no—
387C22—361no—
388C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
389C22#8865211yescharacteristic
390C22#8865512yes—
391C22#8867112yes—
392C22C2×#619811yescharacteristic
393C4—22no—
394C4—184no—
395C4—362no—
396C4—361no—
397C3—31no—
398C3—31no—
399C3—61no—
400C3#1059132411yescharacteristic
401C2—362no—
402C2#1172110212yes—
403C2#1172110611yescharacteristic
404C2#1172113412yes—
405C2#1172114411yescharacteristic
406C2#1172130611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#399
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
4. C2×#88449C2semidirect
5. #11720574C2semidirect
18. C2×C4⋊(C3⋊Dic3)D3semidirect
211. H(3,𝔽3)#399semidirect
342. C32#4949semidirect
343. C32#4993semidirect

Maximal subgroups

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

subgroupindex
C2×#3752⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884502
C2×#884492
#117205742
#105913503
#105913173