Finite Group Databasebeta

← #13140217

Subgroups — #13140217

Order 1728.

1139 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊#272C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C2×#88451C214yesmaximal, complemented
5#10591323—31nomaximal
6#10591318—31nomaximal
7C23×#272C2212yes—
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×C4⋊#272—22no—
10C2×C4⋊#272—22no—
11C2×C4⋊#272C414yescomplemented
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C4⋊((C3⋊Dic3)⋊C3)—22no—
15C4×(C3⋊Dic3)⋊C3—22no—
16#88451—28no—
17C22×C3×D12—31no—
18C22×C3×D12—31no—
19C22×C3×D12—61no—
20C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
21C2×C4×C3⋊Dic3D311yescharacteristic, complemented
22C2×C3×C4⋊Dic3—31no—
23C2×C3×C4×Dic3—31no—
24C2×(C2×D3)⋊Dic3—34no—
25C2×(C2×D3)⋊Dic3—34no—
26C23×H(3,𝔽3)C2311yescharacteristic
27C22×#272—22no—
28C22×#272—22no—
29C22×#272—24no—
30C22×#272—24no—
31C22×#272C2×C414yes—
32C2×C4×H(3,𝔽3)C2×C412yes—
33C2×C4×H(3,𝔽3)D412yes—
34C2×C4×H(3,𝔽3)D412yes—
35C4⋊#272—28no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—24no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3D414yescomplemented
41#5221—91no—
42#4988—91no—
43#4939—91no—
44C23×C3×D3—32no—
45C23×C3×D3—32no—
46C23×C3×D3—62no—
47C22×C32×C4—21no—
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C22×C3⋊Dic3C2×D312yes—
51C22×C3⋊Dic3C2×D312yes—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—32no—
54C2×C3×D12—34no—
55C2×C3×D12—34no—
56C2×C3×D12—64no—
57C2×C3×D12—64no—
58C2×C3×D12—64no—
59C2×C3×D12—62no—
60C2×C3×D12—62no—
61C2×C3×D12—62no—
62C2×C3×D12—62no—
63C4⋊(C3⋊Dic3)—22no—
64C4×C3⋊Dic3—22no—
65C3×C4⋊Dic3—62no—
66C3×C4×Dic3—62no—
67(C2×D3)⋊Dic3—68no—
68(C2×D3)⋊Dic3—68no—
69C22×H(3,𝔽3)C22×C411yescharacteristic
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C22×H(3,𝔽3)C2×D411yescharacteristic
72C22×H(3,𝔽3)C2×D412yes—
73C22×H(3,𝔽3)#4112yes—
74C2×#272—24no—
75C2×#272—28no—
76C2×#272—44no—
77C2×#272—42no—
78C2×#272—42no—
79C4×H(3,𝔽3)C22⋊C414yescomplemented
80(C3⋊Dic3)⋊C3—24no—
81(C3⋊Dic3)⋊C3—42no—
82C22×C3×D4—91no—
83C22×D12—31no—
84C22×D12—31no—
85C22×D12—61no—
86C2×C3×C4⋊C4—91no—
87C2×C3×C42—91no—
88C2×#217—94no—
89C2×(C2×D3)⋊C4—94no—
90C2×(C2×D3)⋊C4—94no—
91C2×C4⋊Dic3—31no—
92C2×C4⋊Dic3—91no—
93C2×C4×Dic3—31no—
94C2×C4×Dic3—91no—
95C23×C32—21no—
96C23×C32C22×D311yescharacteristic
97C23×C32C22×D311yescharacteristic
98C22×C3×D3—34no—
99C22×C3×D3—34no—
100C22×C3×D3—64no—
101C22×C3×D3—64no—
102C22×C3×D3—64no—
103C22×C3×D3—64no—
104C22×C3×D3—62no—
105C22×C3×D3—62no—
106C22×C3×D3—64no—
107C22×C3×D3—64no—
108C22×C3×D3—64no—
109C22×C3×D3—62no—
110C22×C3×D3—62no—
111C22×C3×D3—124no—
112C22×C3×D3—124no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4C4×D312yes—
117C2×C32×C4C4×D312yes—
118C2×C32×C4D1212yes—
119C2×C32×C4D1212yes—
120C2×C32×C4#6712yes—
121C2×C32×C4#6712yes—
122C2×C3⋊Dic3—22no—
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—24no—
125C2×C3⋊Dic3—22no—
126C2×C3⋊Dic3D1214yescomplemented
127C2×C3×Dic3—34no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—64no—
131C2×C3×Dic3—62no—
132C3×D12—616no—
133C3×D12—68no—
134C3×D12—68no—
135#390—271noSylow
136C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
137C2×H(3,𝔽3)C4×D412yes—
138C2×H(3,𝔽3)#12112yes—
139C2×H(3,𝔽3)#12411yescharacteristic
140C2×H(3,𝔽3)#12711yescharacteristic
141#272—44no—
142C24×C3—92no—
143C23×D3—32no—
144C23×D3—32no—
145C23×D3—62no—
146C2×C3×D4—94no—
147C2×C3×D4—182no—
148C2×C3×D4—182no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—61no—
152C22×C3×C4—92no—
153C22×C3×C4—92no—
154C22×C3×C4D3211yescharacteristic
155C22×Dic3—32no—
156C22×Dic3—32no—
157C22×Dic3—92no—
158C22×Dic3—92no—
159C2×D12—34no—
160C2×D12—34no—
161C2×D12—64no—
162C2×D12—62no—
163C2×D12—62no—
164C2×D12—62no—
165C2×D12—62no—
166C2×D12—64no—
167C2×D12—64no—
168C3×C4⋊C4—182no—
169C3×C42—182no—
170#217—188no—
171(C2×D3)⋊C4—188no—
172(C2×D3)⋊C4—188no—
173C4⋊Dic3—62no—
174C4⋊Dic3—182no—
175C4×Dic3—62no—
176C4×Dic3—182no—
177C22×C32—22no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—21no—
182C22×C32C2×C4×D311yescharacteristic
183C22×C32C2×C4×D311yescharacteristic
184C22×C32C2×D1212yes—
185C22×C32C2×D1211yescharacteristic
186C22×C32C2×D1211yescharacteristic
187C22×C32#23512yes—
188C22×C32D3×D412yes—
189C22×C32#23912yes—
190C22×C32C2×#6711yescharacteristic
191C22×C32C2×#6711yescharacteristic
192C2×C3×D3—68no—
193C2×C3×D3—68no—
194C2×C3×D3—68no—
195C2×C3×D3—64no—
196C2×C3×D3—64no—
197C2×C3×D3—128no—
198C2×C3×D3—124no—
199C2×C3×D3—128no—
200C2×C3×D3—124no—
201C2×C3×D3—122no—
202C2×C3×D3—122no—
203C2×C3×D3—124no—
204C2×C3×D3—124no—
205C2×C3×D3—122no—
206C2×C3×D3—122no—
207C32×C4—24no—
208C32×C4(C2×D3)⋊C414yescomplemented
209C32×C4(C2×D3)⋊C414yescomplemented
210C3⋊Dic3—24no—
211C3⋊Dic3—42no—
212C3×Dic3—64no—
213C3×Dic3—122no—
214C22×D4—91no—
215C2×C4⋊C4—91no—
216C2×C22⋊C4—274no—
217C2×C42—91no—
218H(3,𝔽3)#39011yescharacteristic, complemented, Sylow
219C23×C3—31no—
220C23×C3—31no—
221C23×C3—61no—
222C23×C3—94no—
223C23×C3—184no—
224C23×C3—184no—
225C23×C3—182no—
226C23×C3—182no—
227C23×C3C2×D3211yescharacteristic
228C22×D3—34no—
229C22×D3—34no—
230C22×D3—64no—
231C22×D3—64no—
232C22×D3—64no—
233C22×D3—62no—
234C22×D3—62no—
235C22×D3—62no—
236C22×D3—62no—
237C22×D3—64no—
238C22×D3—64no—
239C22×D3—64no—
240C22×D3—64no—
241C22×D3—124no—
242C22×D3—124no—
243C3×D4—188no—
244C2×C3×C4—32no—
245C2×C3×C4—32no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—32no—
249C2×C3×C4—32no—
250C2×C3×C4—62no—
251C2×C3×C4—62no—
252C2×C3×C4—62no—
253C2×C3×C4—94no—
254C2×C3×C4—182no—
255C2×C3×C4—182no—
256C2×C3×C4—184no—
257C2×C3×C4—182no—
258C2×C3×C4#62412yes—
259C2×C3×C4#62612yes—
260C2×C3×C4#62612yes—
261C2×Dic3—34no—
262C2×Dic3—62no—
263C2×Dic3—62no—
264C2×Dic3—64no—
265C2×Dic3—62no—
266C2×Dic3—94no—
267C2×Dic3—182no—
268C2×Dic3—182no—
269C2×Dic3—182no—
270C2×Dic3—184no—
271D12—68no—
272D12—68no—
273D12—616no—
274C2×C32—21no—
275C2×C32—22no—
276C2×C32—22no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32C4×D1212yes—
280C2×C32#87411yescharacteristic
281C2×C32#87511yescharacteristic
282C2×C32#89312yes—
283C2×C32#89512yes—
284C2×C32C2×(C2×D3)⋊C411yescharacteristic
285C2×C32C2×(C2×D3)⋊C411yescharacteristic
286C2×C32#93012yes—
287C2×C32#94011yescharacteristic
288C2×C32#94711yescharacteristic
289C3×D3—128no—
290C3×D3—124no—
291C3×D3—124no—
292C24—92no—
293C2×D4—94no—
294C2×D4—182no—
295C2×D4—182no—
296C22×C4—92no—
297C22×C4—92no—
298C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
299C4⋊C4—182no—
300C22⋊C4—548no—
301C42—182no—
302C22×C3—32no—
303C22×C3—32no—
304C22×C3—32no—
305C22×C3—32no—
306C22×C3—31no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—31no—
312C22×C3—62no—
313C22×C3—62no—
314C22×C3—61no—
315C22×C3—61no—
316C22×C3—61no—
317C22×C3—188no—
318C22×C3—184no—
319C22×C3—362no—
320C22×C3—364no—
321C22×C3—362no—
322C22×C3#379912yes—
323C22×C3D3×D1212yes—
324C22×C3C2×#62411yescharacteristic
325C22×C3C2×#62611yescharacteristic
326C22×C3C2×#62611yescharacteristic
327C2×D3—64no—
328C2×D3—64no—
329C2×D3—68no—
330C2×D3—68no—
331C2×D3—68no—
332C2×D3—128no—
333C2×D3—122no—
334C2×D3—122no—
335C2×D3—124no—
336C2×D3—124no—
337C2×D3—122no—
338C2×D3—122no—
339C2×D3—124no—
340C2×D3—128no—
341C2×D3—124no—
342C3×C4—34no—
343C3×C4—34no—
344C3×C4—64no—
345C3×C4—184no—
346C3×C4—362no—
347C3×C4#372214yes—
348Dic3—64no—
349Dic3—122no—
350Dic3—184no—
351Dic3—362no—
352C32—21no—
353C32#493911yescharacteristic, complemented
354C32#498811yescharacteristic, complemented
355C23—94no—
356C23—184no—
357C23—184no—
358C23—182no—
359C23—182no—
360C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
361D4—188no—
362C2×C4—94no—
363C2×C4—182no—
364C2×C4—182no—
365C2×C4—184no—
366C2×C4—182no—
367C2×C4#619912yes—
368C2×C4#620012yes—
369C2×C4#620012yes—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—32no—
375C2×C3—32no—
376C2×C3—32no—
377C2×C3—32no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—62no—
383C2×C3—62no—
384C2×C3—61no—
385C2×C3—364no—
386C2×C3#6386411yescharacteristic
387C2×C3#6389412yes—
388C2×C3#6390912yes—
389C2×C3#6391111yescharacteristic
390C2×C3C2×#372211yescharacteristic
391D3—124no—
392D3—124no—
393D3—128no—
394C22—184no—
395C22—188no—
396C22—362no—
397C22—364no—
398C22—362no—
399C22#8865212yes—
400C22#8865412yes—
401C22C2×#619911yescharacteristic
402C22C2×#620011yescharacteristic
403C22C2×#620011yescharacteristic
404C4—184no—
405C4—362no—
406C4#8845114yescomplemented
407C3—31no—
408C3—31no—
409C3—61no—
410C3#1059132011yescharacteristic
411C2—364no—
412C2#1172111411yescharacteristic
413C2#1172114912yes—
414C2#1172115612yes—
415C2#1172115811yescharacteristic
416C2C2×#8845111yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC22×C4⋊#272C2
cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#390
LCSG ▷ C22×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
3. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
4. C2×#88451C2semidirect
11. C2×C4⋊#272C4semidirect
20. C2×C4⋊(C3⋊Dic3)D3semidirect
21. C2×C4×C3⋊Dic3D3semidirect
40. C2×(C3⋊Dic3)⋊C3D4semidirect
79. C4×H(3,𝔽3)C22⋊C4semidirect
126. C2×C3⋊Dic3D12semidirect
208. C32×C4(C2×D3)⋊C4semidirect
209. C32×C4(C2×D3)⋊C4semidirect
218. H(3,𝔽3)#390semidirect
353. C32#4939semidirect
354. C32#4988semidirect
406. C4#88451semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×C4×(C3⋊Dic3)⋊C32
C2×#884512
#105913233
#105913183