Finite Group Databasebeta

← #10592687⋊C3 #13142626

Subgroups — #10592687⋊C3 #13142626

Order 1728.

995 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#3752⋊C3C214yesmaximal, complemented
4C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
5#10592687C311yesmaximal, characteristic, complemented
6C3×#4988—31nomaximal
7C23×(C3⋊D3)⋊C3C2212yes—
8C22×C4×H(3,𝔽3)C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C2×C4⋊((C3⋊D3)⋊C3)—22no—
12C2×C4⋊((C3⋊D3)⋊C3)—22no—
13C2×C4⋊((C3⋊D3)⋊C3)C414yescomplemented
14C4⋊C4×H(3,𝔽3)—22no—
15#3752⋊C3—28no—
16C4×(C3⋊Dic3)⋊C3—22no—
17C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
18C22×C3×D12—31no—
19C2×C32×C4⋊C4—21no—
20C2×C32×C4⋊C4D311yescharacteristic, complemented
21C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
22C2×#3752C2×C314yescomplemented
23C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
24C2×C3×(C2×D3)⋊C4—34no—
25C2×C3×C4×Dic3—31no—
26C23×H(3,𝔽3)C2311yescharacteristic
27C22×(C3⋊D3)⋊C3—22no—
28C22×(C3⋊D3)⋊C3—22no—
29C22×(C3⋊D3)⋊C3—24no—
30C22×(C3⋊D3)⋊C3—24no—
31C22×(C3⋊D3)⋊C3C2×C414yes—
32C2×C4×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)—24no—
34C2×C4×H(3,𝔽3)C2×C412yes—
35C2×C4×H(3,𝔽3)D412yes—
36C2×C4×H(3,𝔽3)D412yes—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3D414yescomplemented
40C4⋊((C3⋊D3)⋊C3)—28no—
41C3×#390—91no—
42#4988—31no—
43#4988—91no—
44C23×C3⋊D3C22×C312yes—
45C23×C3×D3—32no—
46C22×C32×C4—21no—
47C22×C32×C4—22no—
48C22×C32×C4C2×D312yes—
49C22×C32×C4C2×D311yescharacteristic
50C22×C32×C4C22×C312yes—
51C22×C32×C4C22×C311yescharacteristic
52C22×C3⋊Dic3C22×C312yes—
53C2×C4⋊(C3⋊D3)—22no—
54C2×C4⋊(C3⋊D3)—22no—
55C2×C4⋊(C3⋊D3)C3×C414yescomplemented
56C22×C3×Dic3—32no—
57C2×C3×D12—34no—
58C2×C3×D12—62no—
59C2×C3×D12—62no—
60C32×C4⋊C4—24no—
61C32×C4⋊C4—22no—
62C32×C4⋊C4—22no—
63#3752—28no—
64C4×C3⋊Dic3—22no—
65C3×(C2×D3)⋊C4—68no—
66C3×C4×Dic3—62no—
67C22×H(3,𝔽3)C22×C411yescharacteristic
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C22×H(3,𝔽3)C2×D412yes—
71C22×H(3,𝔽3)#4112yes—
72C2×(C3⋊D3)⋊C3—24no—
73C2×(C3⋊D3)⋊C3—28no—
74C2×(C3⋊D3)⋊C3—44no—
75C2×(C3⋊D3)⋊C3—42no—
76C2×(C3⋊D3)⋊C3—42no—
77C4×H(3,𝔽3)—42no—
78C4×H(3,𝔽3)C22⋊C414yescomplemented
79(C3⋊Dic3)⋊C3—24no—
80C22×C3×D4—91no—
81C22×D12—31no—
82C22×D12—91no—
83C2×C3×C4⋊C4—31no—
84C2×C3×C4⋊C4—31no—
85C2×C3×C4⋊C4—61no—
86C2×C3×C4⋊C4C3×D311yescharacteristic
87C2×C3×C22⋊C4—94no—
88C2×C3×C42—91no—
89C2×(C2×D3)⋊C4—34no—
90C2×(C2×D3)⋊C4—94no—
91C2×C4×Dic3—31no—
92C2×C4×Dic3—91no—
93C23×C32—21no—
94C23×C32C22×D311yescharacteristic
95C23×C32C23×C311yescharacteristic
96C22×C3⋊D3—24no—
97C22×C3⋊D3—22no—
98C22×C3⋊D3—22no—
99C22×C3⋊D3—24no—
100C22×C3⋊D3C2×C3×C414yes—
101C22×C3×D3—34no—
102C22×C3×D3—62no—
103C22×C3×D3—64no—
104C22×C3×D3—64no—
105C22×C3×D3—62no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—22no—
109C2×C32×C4—24no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—44no—
115C2×C32×C4C4×D312yes—
116C2×C32×C4D1212yes—
117C2×C32×C4#6712yes—
118C2×C32×C4C2×C3×C412yes—
119C2×C32×C4C3×D412yes—
120C2×C32×C4C3×D412yes—
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3—22no—
123C2×C3⋊Dic3C3×D414yescomplemented
124C4⋊(C3⋊D3)—28no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C3×D12—68no—
129#390—91noSylow
130C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
131C2×H(3,𝔽3)C4×D412yes—
132C2×H(3,𝔽3)#12112yes—
133C2×H(3,𝔽3)#12411yescharacteristic
134C2×H(3,𝔽3)#12711yescharacteristic
135(C3⋊D3)⋊C3—44no—
136C24×C3—92no—
137C23×D3—32no—
138C23×D3—92no—
139C2×C3×D4—94no—
140C2×C3×D4—182no—
141C2×C3×D4—182no—
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—32no—
145C22×C3×C4—32no—
146C22×C3×C4—61no—
147C22×C3×C4—62no—
148C22×C3×C4—92no—
149C22×C3×C4C2×C3×D312yes—
150C22×C3×C4C2×C3×D311yescharacteristic
151C22×Dic3—32no—
152C22×Dic3—92no—
153C2×D12—34no—
154C2×D12—62no—
155C2×D12—62no—
156C2×D12—94no—
157C2×D12—182no—
158C2×D12—182no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—62no—
161C3×C4⋊C4—62no—
162C3×C4⋊C4—64no—
163C3×C22⋊C4—188no—
164C3×C42—182no—
165(C2×D3)⋊C4—68no—
166(C2×D3)⋊C4—188no—
167C4×Dic3—62no—
168C4×Dic3—182no—
169C22×C32—21no—
170C22×C32—21no—
171C22×C32—22no—
172C22×C32—22no—
173C22×C32—21no—
174C22×C32C2×C4×D311yescharacteristic
175C22×C32C2×D1211yescharacteristic
176C22×C32D3×D412yes—
177C22×C32#23912yes—
178C22×C32C2×#6711yescharacteristic
179C22×C32C22×C3×C411yescharacteristic
180C22×C32C2×C3×D411yescharacteristic
181C22×C32C2×C3×D412yes—
182C22×C32C2×C3×D411yescharacteristic
183C22×C32C3×#4112yes—
184C2×C3⋊D3—24no—
185C2×C3⋊D3—28no—
186C2×C3⋊D3—44no—
187C2×C3⋊D3—42no—
188C2×C3⋊D3—42no—
189C2×C3×D3—68no—
190C2×C3×D3—64no—
191C2×C3×D3—124no—
192C2×C3×D3—122no—
193C2×C3×D3—122no—
194C32×C4—24no—
195C32×C4—44no—
196C32×C4—42no—
197C32×C4—42no—
198C32×C4(C2×D3)⋊C414yescomplemented
199C32×C4C3×C22⋊C414yescomplemented
200C3⋊Dic3—24no—
201C3×Dic3—64no—
202C22×D4—91no—
203C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
204C2×C22⋊C4—94no—
205C2×C42—91no—
206H(3,𝔽3)#39011yescharacteristic, complemented, Sylow
207C23×C3—31no—
208C23×C3—31no—
209C23×C3—61no—
210C23×C3—94no—
211C23×C3—184no—
212C23×C3—182no—
213C23×C3—182no—
214C23×C3—184no—
215C23×C3C22×C3×D311yescharacteristic
216C22×D3—34no—
217C22×D3—62no—
218C22×D3—64no—
219C22×D3—64no—
220C22×D3—62no—
221C22×D3—94no—
222C22×D3—184no—
223C22×D3—182no—
224C22×D3—182no—
225C22×D3—184no—
226C3×D4—188no—
227C2×C3×C4—24no—
228C2×C3×C4—22no—
229C2×C3×C4—32no—
230C2×C3×C4—32no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—32no—
234C2×C3×C4—32no—
235C2×C3×C4—64no—
236C2×C3×C4—64no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—64no—
240C2×C3×C4—62no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—94no—
244C2×C3×C4—124no—
245C2×C3×C4—182no—
246C2×C3×C4—182no—
247C2×C3×C4C3×C4×D312yes—
248C2×C3×C4C3×D1212yes—
249C2×C3×C4C3×#6712yes—
250C2×Dic3—34no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—94no—
254C2×Dic3—182no—
255C2×Dic3—182no—
256D12—68no—
257D12—188no—
258C2×C32—22no—
259C2×C32—22no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32#89312yes—
264C2×C32#89512yes—
265C2×C32C2×(C2×D3)⋊C411yescharacteristic
266C2×C32#94011yescharacteristic
267C2×C32#94711yescharacteristic
268C2×C32C2×C3×C22⋊C411yescharacteristic
269C2×C32C3×C4×D412yes—
270C2×C32C3×#12112yes—
271C2×C32C3×#12411yescharacteristic
272C2×C32C3×#12711yescharacteristic
273C3⋊D3—44no—
274C3×D3—124no—
275C24—92no—
276C2×D4—94no—
277C2×D4—182no—
278C2×D4—182no—
279C22×C4—92no—
280C22×C4C2×(C3⋊D3)⋊C312yes—
281C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
282C4⋊C4—22no—
283C22⋊C4—188no—
284C42—182no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—31no—
288C22×C3—31no—
289C22×C3—32no—
290C22×C3—32no—
291C22×C3—32no—
292C22×C3—32no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—61no—
296C22×C3—61no—
297C22×C3—62no—
298C22×C3—62no—
299C22×C3—61no—
300C22×C3—184no—
301C22×C3—188no—
302C22×C3—362no—
303C22×C3—362no—
304C22×C3—364no—
305C22×C3C2×C3×C4×D311yescharacteristic
306C22×C3C2×C3×D1211yescharacteristic
307C22×C3C3×D3×D412yes—
308C22×C3C3×#23912yes—
309C22×C3C2×C3×#6711yescharacteristic
310C2×D3—68no—
311C2×D3—64no—
312C2×D3—124no—
313C2×D3—122no—
314C2×D3—122no—
315C2×D3—184no—
316C2×D3—188no—
317C2×D3—362no—
318C2×D3—362no—
319C2×D3—364no—
320C3×C4—34no—
321C3×C4—34no—
322C3×C4—42no—
323C3×C4—64no—
324C3×C4—122no—
325C3×C4—122no—
326C3×C4—124no—
327C3×C4—184no—
328C3×C4C3×(C2×D3)⋊C414yes—
329Dic3—64no—
330Dic3—184no—
331C32—21no—
332C32#498811yescharacteristic, complemented
333C32C3×#39011yescharacteristic, complemented
334C23—94no—
335C23—182no—
336C23—184no—
337C23—184no—
338C23—182no—
339C23C22×(C3⋊D3)⋊C311yescharacteristic
340D4—188no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—94no—
344C2×C4—182no—
345C2×C4—182no—
346C2×C4C4×(C3⋊D3)⋊C312yes—
347C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
348C2×C4#638⋊C312yes—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—32no—
356C2×C3—32no—
357C2×C3—32no—
358C2×C3—32no—
359C2×C3—61no—
360C2×C3—61no—
361C2×C3—61no—
362C2×C3—62no—
363C2×C3—62no—
364C2×C3—364no—
365C2×C3C3×#89312yes—
366C2×C3C3×#89512yes—
367C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
368C2×C3C3×#94011yescharacteristic
369C2×C3C3×#94711yescharacteristic
370D3—124no—
371D3—364no—
372C22—184no—
373C22—188no—
374C22—362no—
375C22—362no—
376C22—364no—
377C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
378C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
379C22D4×(C3⋊D3)⋊C312yes—
380C22#8872212yes—
381C22C2×#638⋊C311yescharacteristic
382C4—42no—
383C4—184no—
384C4#3752⋊C314yescomplemented
385C3—31no—
386C3—31no—
387C3—61no—
388C3C3×#498811yescharacteristic
389C2—364no—
390C2#64102⋊C312yes—
391C2#64104⋊C312yes—
392C2C2×#3752⋊C311yescharacteristic
393C2#64149⋊C311yescharacteristic
394C2#64156⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
cosocleC23×C32C23×C3
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#390
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×H(3,𝔽3)C2semidirect
3. C2×#3752⋊C3C2semidirect
4. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
5. #10592687C3semidirect
13. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
20. C2×C32×C4⋊C4D3semidirect
21. C2×C32×C4⋊C4C2×C3semidirect
22. C2×#3752C2×C3semidirect
23. C2×C4×C3⋊Dic3C2×C3semidirect
39. C2×(C3⋊Dic3)⋊C3D4semidirect
55. C2×C4⋊(C3⋊D3)C3×C4semidirect
78. C4×H(3,𝔽3)C22⋊C4semidirect
123. C2×C3⋊Dic3C3×D4semidirect
198. C32×C4(C2×D3)⋊C4semidirect
199. C32×C4C3×C22⋊C4semidirect
203. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
206. H(3,𝔽3)#390semidirect
332. C32#4988semidirect
333. C32C3×#390semidirect
384. C4#3752⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
C2×C4⋊C4×H(3,𝔽3)2
C2×#3752⋊C32
C2×C4×(C3⋊Dic3)⋊C32
#105926873
C3×#49883