Finite Group Databasebeta

← #13141161

Subgroups — #13141161

Order 1728.

727 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 727 conjugacy classes of subgroups — too many to draw. Only its 117 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
4#11720334C212yesmaximal, complemented
5#11720334C211yesmaximal, characteristic, complemented
6#10591717—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—
15C2×#3757D312yes—
16C2×C3×#217—32no—
17C2×C3×#217—32no—
18C2×C3×#217—61no—
19C2×Dic3⋊Dic3—32no—
20#63579—32no—
21#63579—32no—
22#63579—32no—
23C23×H(3,𝔽3)—21no—
24C23×H(3,𝔽3)—21no—
25C23×H(3,𝔽3)—22no—
26C23×H(3,𝔽3)—22no—
27C23×H(3,𝔽3)D412yes—
28C23×H(3,𝔽3)C2311yescharacteristic
29C2×#1084—22no—
30C2×#1084—21no—
31C2×#1084—22no—
32C2×#1084D412yes—
33C2×(C3⋊Dic3)⋊C3—22no—
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—
39#4953—91no—
40#4952—92no—
41C24×C32—21no—
42C24×C32C2×D312yes—
43C22×C3⋊Dic3C2×D312yes—
44C22×C3⋊Dic3C2×D312yes—
45C22×C3×Dic3—32no—
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3—61no—
51#3757—24no—
52C3×#217—64no—
53C3×#217—64no—
54C3×#217—64no—
55Dic3⋊Dic3—64no—
56C22×H(3,𝔽3)—22no—
57C22×H(3,𝔽3)—24no—
58C22×H(3,𝔽3)—41no—
59C22×H(3,𝔽3)—41no—
60C22×H(3,𝔽3)—42no—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C22×H(3,𝔽3)#4111yescharacteristic
64C22×H(3,𝔽3)#4112yes—
65C22×H(3,𝔽3)#4112yes—
66#1084—22no—
67#1084—41no—
68(C3⋊Dic3)⋊C3—42no—
69(C3⋊Dic3)⋊C3—42no—
70C2×C3×C22⋊C4—92no—
71C2×C3×C22⋊C4—91no—
72C2×#217—32no—
73C2×#217—32no—
74C2×#217—61no—
75C2×#217—92no—
76C2×C4⋊Dic3—91no—
77C2×Dic3⋊C4—92no—
78C3⋊#95—91no—
79C3⋊#95—92no—
80C3⋊#95—92no—
81C3⋊#95—92no—
82C3⋊#95—92no—
83C23×C32—22no—
84C23×C32—22no—
85C23×C32—22no—
86C23×C32—22no—
87C23×C32—22no—
88C23×C32—22no—
89C23×C32—22no—
90C23×C32—22no—
91C23×C32—22no—
92C23×C32—21no—
93C23×C32—42no—
94C23×C32—42no—
95C23×C32#6714yes—
96C23×C32C22×D312yes—
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3—22no—
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—22no—
103C2×C3×Dic3—34no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
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—122no—
118C2×C3×Dic3—122no—
119#397—271noSylow
120C2×H(3,𝔽3)—42no—
121C2×H(3,𝔽3)#12111yescharacteristic
122C2×H(3,𝔽3)#12311yescharacteristic
123C2×H(3,𝔽3)#12312yes—
124C2×H(3,𝔽3)#12411yescharacteristic
125C2×H(3,𝔽3)#12612yes—
126C24×C3—32no—
127C24×C3—61no—
128C24×C3D3211yescharacteristic
129C22×C3×C4—91no—
130C22×C3×C4—92no—
131C22×C3×C4—92no—
132C22×C3×C4—91no—
133C22×Dic3—32no—
134C22×Dic3—32no—
135C22×Dic3—32no—
136C22×Dic3—32no—
137C22×Dic3—61no—
138C22×Dic3—61no—
139C22×Dic3—92no—
140C22×Dic3—92no—
141C3×C22⋊C4—182no—
142C3×C22⋊C4—184no—
143#217—64no—
144#217—64no—
145#217—64no—
146#217—184no—
147C4⋊Dic3—182no—
148Dic3⋊C4—184no—
149C22×C32—24no—
150C22×C32—24no—
151C22×C32—24no—
152C22×C32—24no—
153C22×C32—22no—
154C22×C32—22no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—21no—
158C22×C32—44no—
159C22×C32—42no—
160C22×C32—44no—
161C22×C32—42no—
162C22×C32—42no—
163C22×C32—42no—
164C22×C32—42no—
165C22×C32—42no—
166C22×C32#23512yes—
167C22×C32#23512yes—
168C22×C32D3×D412yes—
169C22×C32#23712yes—
170C22×C32#23712yes—
171C22×C32#23712yes—
172C22×C32C2×#6712yes—
173C3⋊Dic3—42no—
174C3⋊Dic3—42no—
175C3×Dic3—64no—
176C3×Dic3—122no—
177C3×Dic3—122no—
178C3×Dic3—122no—
179C3×Dic3—122no—
180C3×Dic3—122no—
181C2×C4⋊C4—271no—
182C2×C22⋊C4—91no—
183C2×C22⋊C4—92no—
184#95—271no—
185#95—272no—
186H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—34no—
192C23×C3—32no—
193C23×C3—62no—
194C23×C3—62no—
195C23×C3—62no—
196C23×C3—62no—
197C23×C3—62no—
198C23×C3—62no—
199C23×C3—62no—
200C23×C3—62no—
201C23×C3—62no—
202C23×C3—61no—
203C23×C3—122no—
204C23×C3—122no—
205C23×C3#62512yes—
206C23×C3C2×D3211yescharacteristic
207C2×C3×C4—92no—
208C2×C3×C4—182no—
209C2×C3×C4—182no—
210C2×C3×C4—182no—
211C2×C3×C4—182no—
212C2×C3×C4—182no—
213C2×C3×C4—182no—
214C2×C3×C4—182no—
215C2×C3×C4—182no—
216C2×C3×C4—181no—
217C2×Dic3—34no—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—62no—
221C2×Dic3—62no—
222C2×Dic3—62no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—62no—
226C2×Dic3—62no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—62no—
231C2×Dic3—122no—
232C2×Dic3—122no—
233C2×Dic3—182no—
234C2×Dic3—182no—
235C2×Dic3—182no—
236C2×Dic3—182no—
237C2×Dic3—182no—
238C2×Dic3—182no—
239C2×C32—21no—
240C2×C32—22no—
241C2×C32—22no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—44no—
245C2×C32—44no—
246C2×C32#87912yes—
247C2×C32#87912yes—
248C2×C32#88312yes—
249C2×C32#88512yes—
250C2×C32#92912yes—
251C2×C32#93512yes—
252C2×C32#93912yes—
253C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
254C22×C4—91no—
255C22×C4—91no—
256C22×C4—92no—
257C22×C4—92no—
258C4⋊C4—542no—
259C22⋊C4—182no—
260C22⋊C4—184no—
261C22×C3—24no—
262C22×C3—22no—
263C22×C3—32no—
264C22×C3—32no—
265C22×C3—32no—
266C22×C3—32no—
267C22×C3—32no—
268C22×C3—32no—
269C22×C3—32no—
270C22×C3—41no—
271C22×C3—42no—
272C22×C3—41no—
273C22×C3—64no—
274C22×C3—64no—
275C22×C3—64no—
276C22×C3—64no—
277C22×C3—62no—
278C22×C3—62no—
279C22×C3—61no—
280C22×C3—61no—
281C22×C3—61no—
282C22×C3—124no—
283C22×C3—122no—
284C22×C3—122no—
285C22×C3—122no—
286C22×C3—122no—
287C22×C3—122no—
288C22×C3—124no—
289C22×C3—122no—
290C22×C3#380412yes—
291C22×C3#380412yes—
292C22×C3#380511yescharacteristic
293C22×C3C2×#62511yescharacteristic
294C22×C3#381111yescharacteristic
295C3×C4—182no—
296C3×C4—362no—
297C3×C4—362no—
298C3×C4—361no—
299Dic3—64no—
300Dic3—122no—
301Dic3—122no—
302Dic3—122no—
303Dic3—122no—
304Dic3—122no—
305Dic3—362no—
306Dic3—362no—
307C32—21no—
308C32#495212yescomplemented
309C23—22no—
310C23—22no—
311C23—21no—
312C23—21no—
313C23#619812yes—
314C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C2×C4—92no—
316C2×C4—182no—
317C2×C4—182no—
318C2×C4—182no—
319C2×C4—182no—
320C2×C4—182no—
321C2×C4—182no—
322C2×C4—182no—
323C2×C4—181no—
324C2×C4—182no—
325C2×C3—32no—
326C2×C3—32no—
327C2×C3—32no—
328C2×C3—32no—
329C2×C3—32no—
330C2×C3—32no—
331C2×C3—32no—
332C2×C3—42no—
333C2×C3—61no—
334C2×C3—62no—
335C2×C3—62no—
336C2×C3—61no—
337C2×C3—61no—
338C2×C3—124no—
339C2×C3—124no—
340C2×C3#6395311yescharacteristic
341C2×C3#6395612yes—
342C2×C3#6396112yes—
343C2×C3#6397511yescharacteristic
344C2×C3#6398011yescharacteristic
345C22—24no—
346C22—22no—
347C22—41no—
348C22—42no—
349C22—41no—
350C22#8867112yes—
351C22#8867112yes—
352C22#8867211yescharacteristic
353C22C2×#619811yescharacteristic
354C22#8867711yescharacteristic
355C4—182no—
356C4—361no—
357C4—362no—
358C4—362no—
359C3—32no—
360C3—61no—
361C3#1059169411yescharacteristic
362C2—42no—
363C2#1172129411yescharacteristic
364C2#1172129712yes—
365C2#1172130612yes—
366C2#1172131811yescharacteristic
367C2#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)#397
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. #11720334C2semidirect
5. #11720334C2semidirect
186. H(3,𝔽3)#397semidirect
308. C32#4952semidirect

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
#117203342
#105917173