Finite Group Databasebeta

← #13141156

Subgroups — #13141156

Order 1728.

783 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 125 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×#88552C211yesmaximal, characteristic
2C2×#3757⋊C3C212yesmaximal
3C2×#88449C211yesmaximal, characteristic, complemented
4C2×#88448C212yesmaximal, complemented
5#11720334C211yesmaximal, characteristic, complemented
6#10591715—32nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12#88552—22no—
13#3757⋊C3—24no—
14#88449—22no—
15#88448—24no—
16C2×#3757D312yes—
17C2×C3×#217—32no—
18C2×C3×#217—32no—
19C2×C3×#217—61no—
20C2×#3724—32no—
21C2×Dic3⋊Dic3—32no—
22C2×Dic3⋊Dic3—32no—
23#63579—32no—
24C23×H(3,𝔽3)—21no—
25C23×H(3,𝔽3)—22no—
26C23×H(3,𝔽3)—21no—
27C23×H(3,𝔽3)—22no—
28C23×H(3,𝔽3)Q812yes—
29C23×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—22no—
31C2×#1084—21no—
32C2×#1084—22no—
33C2×#1084D412yes—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3D414yes—
40#4949—91no—
41#4949—92no—
42C24×C32—21no—
43C24×C32C2×D312yes—
44C22×C3⋊Dic3C2×D312yes—
45C22×C3⋊Dic3C2×D312yes—
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—32no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3—61no—
52#3757—24no—
53C3×#217—64no—
54C3×#217—64no—
55C3×#217—64no—
56#3724—64no—
57Dic3⋊Dic3—64no—
58Dic3⋊Dic3—64no—
59C22×H(3,𝔽3)—22no—
60C22×H(3,𝔽3)—24no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)—42no—
63C22×H(3,𝔽3)—41no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C22×H(3,𝔽3)C2×D412yes—
66C22×H(3,𝔽3)C2×Q811yescharacteristic
67C22×H(3,𝔽3)#4112yes—
68C22×H(3,𝔽3)#4111yescharacteristic
69#1084—22no—
70#1084—41no—
71(C3⋊Dic3)⋊C3—24no—
72(C3⋊Dic3)⋊C3—42no—
73C2×C3×C22⋊C4—92no—
74C2×C3×C22⋊C4—91no—
75C2×#217—32no—
76C2×#217—32no—
77C2×#217—61no—
78C2×#217—92no—
79C2×C4⋊Dic3—91no—
80C2×C4⋊Dic3—92no—
81C2×Dic3⋊C4—92no—
82C2×Dic3⋊C4—92no—
83C2×Dic3⋊C4—92no—
84C3⋊#95—91no—
85C3⋊#95—92no—
86C23×C32—22no—
87C23×C32—22no—
88C23×C32—22no—
89C23×C32—22no—
90C23×C32—22no—
91C23×C32—22no—
92C23×C32—22no—
93C23×C32—22no—
94C23×C32—22no—
95C23×C32—21no—
96C23×C32—42no—
97C23×C32—42no—
98C23×C32Dic614yes—
99C23×C32C22×D312yes—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3#6714yes—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—122no—
121C2×C3×Dic3—122no—
122#399—271noSylow
123C2×H(3,𝔽3)—42no—
124C2×H(3,𝔽3)C22⋊Q811yescharacteristic
125C2×H(3,𝔽3)C22⋊Q812yes—
126C2×H(3,𝔽3)#12311yescharacteristic
127C2×H(3,𝔽3)#12312yes—
128C2×H(3,𝔽3)#12711yescharacteristic
129C24×C3—32no—
130C24×C3—61no—
131C24×C3D3211yescharacteristic
132C22×C3×C4—92no—
133C22×C3×C4—91no—
134C22×C3×C4—92no—
135C22×C3×C4—91no—
136C22×Dic3—32no—
137C22×Dic3—32no—
138C22×Dic3—32no—
139C22×Dic3—32no—
140C22×Dic3—61no—
141C22×Dic3—61no—
142C22×Dic3—92no—
143C22×Dic3—92no—
144C3×C22⋊C4—182no—
145C3×C22⋊C4—184no—
146#217—64no—
147#217—64no—
148#217—64no—
149#217—184no—
150C4⋊Dic3—182no—
151C4⋊Dic3—184no—
152Dic3⋊C4—184no—
153Dic3⋊C4—184no—
154Dic3⋊C4—184no—
155C22×C32—24no—
156C22×C32—24no—
157C22×C32—24no—
158C22×C32—24no—
159C22×C32—22no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32—44no—
165C22×C32—44no—
166C22×C32—42no—
167C22×C32—42no—
168C22×C32—42no—
169C22×C32—42no—
170C22×C32—42no—
171C22×C32—42no—
172C22×C32C2×Dic612yes—
173C22×C32#23512yes—
174C22×C32D3×D412yes—
175C22×C32D3×D412yes—
176C22×C32#23712yes—
177C22×C32#23712yes—
178C22×C32C2×#6712yes—
179C3⋊Dic3—24no—
180C3⋊Dic3—42no—
181C3×Dic3—64no—
182C3×Dic3—64no—
183C3×Dic3—122no—
184C3×Dic3—122no—
185C3×Dic3—122no—
186C3×Dic3—122no—
187C2×C4⋊C4—271no—
188C2×C4⋊C4—272no—
189C2×C22⋊C4—91no—
190C2×C22⋊C4—92no—
191#95—271no—
192H(3,𝔽3)#39911yescharacteristic, complemented, Sylow
193C23×C3—22no—
194C23×C3—21no—
195C23×C3—22no—
196C23×C3—21no—
197C23×C3—34no—
198C23×C3—32no—
199C23×C3—62no—
200C23×C3—62no—
201C23×C3—62no—
202C23×C3—62no—
203C23×C3—62no—
204C23×C3—62no—
205C23×C3—62no—
206C23×C3—62no—
207C23×C3—62no—
208C23×C3—61no—
209C23×C3—122no—
210C23×C3—122no—
211C23×C3#62712yes—
212C23×C3C2×D3211yescharacteristic
213C2×C3×C4—94no—
214C2×C3×C4—92no—
215C2×C3×C4—182no—
216C2×C3×C4—182no—
217C2×C3×C4—182no—
218C2×C3×C4—182no—
219C2×C3×C4—182no—
220C2×C3×C4—182no—
221C2×C3×C4—182no—
222C2×C3×C4—181no—
223C2×Dic3—34no—
224C2×Dic3—34no—
225C2×Dic3—62no—
226C2×Dic3—62no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—94no—
238C2×Dic3—122no—
239C2×Dic3—122no—
240C2×Dic3—182no—
241C2×Dic3—182no—
242C2×Dic3—182no—
243C2×Dic3—182no—
244C2×Dic3—182no—
245C2×C32—22no—
246C2×C32—22no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—44no—
251C2×C32—44no—
252C2×C32#87812yes—
253C2×C32#87812yes—
254C2×C32#88312yes—
255C2×C32#88312yes—
256C2×C32#92412yes—
257C2×C32#93512yes—
258C2×C32#94012yes—
259C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
260C22×C4—91no—
261C22×C4—92no—
262C22×C4—91no—
263C22×C4—92no—
264C4⋊C4—544no—
265C4⋊C4—542no—
266C22⋊C4—182no—
267C22⋊C4—184no—
268C22×C3—24no—
269C22×C3—22no—
270C22×C3—32no—
271C22×C3—32no—
272C22×C3—32no—
273C22×C3—32no—
274C22×C3—32no—
275C22×C3—32no—
276C22×C3—32no—
277C22×C3—42no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—64no—
281C22×C3—64no—
282C22×C3—64no—
283C22×C3—64no—
284C22×C3—61no—
285C22×C3—62no—
286C22×C3—61no—
287C22×C3—62no—
288C22×C3—61no—
289C22×C3—124no—
290C22×C3—122no—
291C22×C3—122no—
292C22×C3—122no—
293C22×C3—122no—
294C22×C3—124no—
295C22×C3—122no—
296C22×C3—122no—
297C22×C3#380412yes—
298C22×C3#380511yescharacteristic
299C22×C3C2×#62711yescharacteristic
300C22×C3D3×#6712yes—
301C22×C3#381111yescharacteristic
302C3×C4—184no—
303C3×C4—182no—
304C3×C4—362no—
305C3×C4—361no—
306Dic3—64no—
307Dic3—64no—
308Dic3—122no—
309Dic3—122no—
310Dic3—122no—
311Dic3—122no—
312Dic3—184no—
313Dic3—362no—
314C32—21no—
315C32#494912yescomplemented
316C23—22no—
317C23—21no—
318C23—22no—
319C23—21no—
320C23#619612yes—
321C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
322C2×C4—94no—
323C2×C4—92no—
324C2×C4—182no—
325C2×C4—182no—
326C2×C4—182no—
327C2×C4—182no—
328C2×C4—182no—
329C2×C4—182no—
330C2×C4—181no—
331C2×C4—182no—
332C2×C3—32no—
333C2×C3—32no—
334C2×C3—32no—
335C2×C3—32no—
336C2×C3—32no—
337C2×C3—32no—
338C2×C3—32no—
339C2×C3—42no—
340C2×C3—62no—
341C2×C3—61no—
342C2×C3—62no—
343C2×C3—61no—
344C2×C3—61no—
345C2×C3—124no—
346C2×C3—124no—
347C2×C3#6396412yes—
348C2×C3#6396912yes—
349C2×C3#6397511yescharacteristic
350C2×C3#6397911yescharacteristic
351C2×C3#6398211yescharacteristic
352C22—24no—
353C22—22no—
354C22—42no—
355C22—41no—
356C22—41no—
357C22C2×#619611yescharacteristic
358C22#8867112yes—
359C22#8867211yescharacteristic
360C22#8867612yes—
361C22#8867711yescharacteristic
362C4—184no—
363C4—182no—
364C4—361no—
365C4—362no—
366C3—32no—
367C3—61no—
368C3#1059171511yescharacteristic
369C2—42no—
370C2#1172129912yes—
371C2#1172130911yescharacteristic
372C2#1172131112yes—
373C2#1172131711yescharacteristic
374C2#1172131911yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
Fitting, prime radicalC24×H(3,𝔽3)C22
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#399
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
3. C2×#88449C2semidirect
4. C2×#88448C2semidirect
5. #11720334C2semidirect
192. H(3,𝔽3)#399semidirect
315. C32#4949semidirect

Maximal subgroups

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

subgroupindex
C2×#885522
C2×#3757⋊C32
C2×#884492
C2×#884482
#117203342
#105917153