Finite Group Databasebeta

← #10592371⋊C3 #13141360

Subgroups — #10592371⋊C3 #13141360

Order 1728.

909 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C2×#3752⋊C3C212yesmaximal, complemented
3C2×#3752⋊C3C212yesmaximal, complemented
4C2×#3750⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
6#10592371C311yesmaximal, characteristic, complemented
7C3×#4672—31nomaximal
8C23×(C3⋊D3)⋊C3C2212yes—
9C22×C4×H(3,𝔽3)C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)—24no—
14C2×C4⋊((C3⋊D3)⋊C3)—22no—
15#3752⋊C3—24no—
16#3752⋊C3—24no—
17#3750⋊C3—22no—
18C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
19C22×C3×D12—31no—
20C2×#3752C2×C312yescomplemented
21C2×#3752C2×C312yescomplemented
22C2×#3750C2×C311yescharacteristic, complemented
23C2×C3×(C2×D3)⋊C4—32no—
24C2×C3×(C2×D3)⋊C4—32no—
25C2×C3×Dic3⋊C4—31no—
26C32×#95—21no—
27C32×#95D311yescharacteristic, complemented
28C32×#95C2×C311yescharacteristic, complemented
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×(C3⋊D3)⋊C3—22no—
31C22×(C3⋊D3)⋊C3—22no—
32C22×(C3⋊D3)⋊C3—22no—
33C22×(C3⋊D3)⋊C3—22no—
34C22×(C3⋊D3)⋊C3—22no—
35C22×(C3⋊D3)⋊C3—22no—
36C22×(C3⋊D3)⋊C3D414yes—
37C2×C4×H(3,𝔽3)—22no—
38C2×C4×H(3,𝔽3)—22no—
39C2×C4×H(3,𝔽3)—22no—
40C2×C4×H(3,𝔽3)—22no—
41C2×C4×H(3,𝔽3)D412yes—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—22no—
46C2×(C3⋊Dic3)⋊C3D412yescomplemented
47C4⋊((C3⋊D3)⋊C3)—28no—
48C3×#394—91no—
49#4672—31no—
50#4672—91no—
51C23×C3⋊D3C22×C312yes—
52C23×C3×D3—32no—
53C22×C32×C4—22no—
54C22×C32×C4—21no—
55C22×C32×C4C2×D312yes—
56C22×C32×C4C2×D311yescharacteristic
57C22×C32×C4C22×C312yes—
58C22×C32×C4C22×C311yescharacteristic
59C22×C3⋊Dic3C22×C311yescharacteristic
60C22×C3⋊Dic3C22×C311yescharacteristic
61C2×C4⋊(C3⋊D3)—24no—
62C2×C4⋊(C3⋊D3)—22no—
63C22×C3×Dic3—31no—
64C22×C3×Dic3—31no—
65C2×C3×D12—64no—
66C2×C3×D12—62no—
67#3752—24no—
68#3752—24no—
69#3750—22no—
70C3×(C2×D3)⋊C4—64no—
71C3×(C2×D3)⋊C4—64no—
72C3×Dic3⋊C4—62no—
73C22×H(3,𝔽3)C2×D412yes—
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C22×H(3,𝔽3)#4111yescharacteristic
77C22×H(3,𝔽3)#4112yes—
78C2×(C3⋊D3)⋊C3—24no—
79C2×(C3⋊D3)⋊C3—24no—
80C2×(C3⋊D3)⋊C3—24no—
81C2×(C3⋊D3)⋊C3—42no—
82C2×(C3⋊D3)⋊C3—42no—
83C2×(C3⋊D3)⋊C3—42no—
84C2×(C3⋊D3)⋊C3—42no—
85C4×H(3,𝔽3)—22no—
86C4×H(3,𝔽3)—42no—
87(C3⋊Dic3)⋊C3—22no—
88(C3⋊Dic3)⋊C3—41no—
89C22×C3×D4—91no—
90C22×D12—31no—
91C22×D12—91no—
92C2×C3×C4⋊C4—91no—
93C2×C3×C22⋊C4—92no—
94C2×C3×C22⋊C4—92no—
95C2×(C2×D3)⋊C4—32no—
96C2×(C2×D3)⋊C4—32no—
97C2×(C2×D3)⋊C4—92no—
98C2×(C2×D3)⋊C4—92no—
99C2×Dic3⋊C4—31no—
100C2×Dic3⋊C4—91no—
101C3×#95—31no—
102C3×#95—31no—
103C3×#95—61no—
104C3×#95C3×D311yescharacteristic
105C23×C32—21no—
106C23×C32C22×D311yescharacteristic
107C23×C32C23×C311yescharacteristic
108C22×C3⋊D3—22no—
109C22×C3⋊D3—22no—
110C22×C3⋊D3—22no—
111C22×C3⋊D3—22no—
112C22×C3⋊D3—22no—
113C22×C3⋊D3—22no—
114C22×C3⋊D3C3×D414yes—
115C22×C3×D3—34no—
116C22×C3×D3—62no—
117C22×C3×D3—62no—
118C22×C3×D3—62no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—24no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—42no—
133C2×C32×C4—42no—
134C2×C32×C4—42no—
135C2×C32×C4D1212yes—
136C2×C32×C4C3×D412yes—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3C3×D412yescomplemented
142C4⋊(C3⋊D3)—28no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—62no—
148C3×D12—68no—
149#394—91noSylow
150C2×H(3,𝔽3)#12011yescharacteristic
151C2×H(3,𝔽3)#12112yes—
152C2×H(3,𝔽3)#12111yescharacteristic
153C2×H(3,𝔽3)#12312yes—
154C2×H(3,𝔽3)#12411yescharacteristic
155(C3⋊D3)⋊C3—44no—
156C24×C3—92no—
157C23×D3—32no—
158C23×D3—92no—
159C2×C3×D4—182no—
160C2×C3×D4—184no—
161C22×C3×C4—31no—
162C22×C3×C4—31no—
163C22×C3×C4—32no—
164C22×C3×C4—32no—
165C22×C3×C4—62no—
166C22×C3×C4—61no—
167C22×C3×C4—91no—
168C22×C3×C4—91no—
169C22×C3×C4C2×C3×D312yes—
170C22×C3×C4C2×C3×D311yescharacteristic
171C22×Dic3—31no—
172C22×Dic3—31no—
173C22×Dic3—91no—
174C22×Dic3—91no—
175C2×D12—64no—
176C2×D12—62no—
177C2×D12—182no—
178C2×D12—184no—
179C3×C4⋊C4—182no—
180C3×C22⋊C4—184no—
181C3×C22⋊C4—184no—
182(C2×D3)⋊C4—64no—
183(C2×D3)⋊C4—64no—
184(C2×D3)⋊C4—184no—
185(C2×D3)⋊C4—184no—
186Dic3⋊C4—62no—
187Dic3⋊C4—182no—
188C22×C32—21no—
189C22×C32—22no—
190C22×C32—22no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32C2×D1211yescharacteristic
194C22×C32#23512yes—
195C22×C32D3×D412yes—
196C22×C32D3×D411yescharacteristic
197C22×C32#23911yescharacteristic
198C22×C32C2×C3×D411yescharacteristic
199C22×C32C2×C3×D412yes—
200C22×C32C2×C3×D411yescharacteristic
201C22×C32C3×#4112yes—
202C22×C32C3×#4111yescharacteristic
203C2×C3⋊D3—24no—
204C2×C3⋊D3—24no—
205C2×C3⋊D3—24no—
206C2×C3⋊D3—42no—
207C2×C3⋊D3—42no—
208C2×C3⋊D3—42no—
209C2×C3⋊D3—42no—
210C2×C3×D3—64no—
211C2×C3×D3—64no—
212C2×C3×D3—64no—
213C2×C3×D3—122no—
214C2×C3×D3—122no—
215C2×C3×D3—122no—
216C2×C3×D3—122no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—42no—
220C32×C4—44no—
221C32×C4—42no—
222C32×C4—42no—
223C3⋊Dic3—22no—
224C3⋊Dic3—41no—
225C3×Dic3—62no—
226C3×Dic3—121no—
227C22×D4—91no—
228C2×C4⋊C4—91no—
229C2×C22⋊C4—92no—
230C2×C22⋊C4—92no—
231#95(C3⋊D3)⋊C311yescharacteristic, complemented
232H(3,𝔽3)#39411yescharacteristic, complemented, Sylow
233C23×C3—31no—
234C23×C3—31no—
235C23×C3—61no—
236C23×C3—94no—
237C23×C3—182no—
238C23×C3—182no—
239C23×C3—182no—
240C23×C3—182no—
241C23×C3—182no—
242C23×C3—182no—
243C23×C3C22×C3×D311yescharacteristic
244C22×D3—34no—
245C22×D3—62no—
246C22×D3—62no—
247C22×D3—62no—
248C22×D3—62no—
249C22×D3—62no—
250C22×D3—62no—
251C22×D3—94no—
252C22×D3—182no—
253C22×D3—182no—
254C22×D3—182no—
255C22×D3—182no—
256C22×D3—182no—
257C22×D3—182no—
258C3×D4—188no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—32no—
264C2×C3×C4—32no—
265C2×C3×C4—62no—
266C2×C3×C4—62no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—62no—
272C2×C3×C4—62no—
273C2×C3×C4—64no—
274C2×C3×C4—62no—
275C2×C3×C4—92no—
276C2×C3×C4—122no—
277C2×C3×C4—122no—
278C2×C3×C4—122no—
279C2×C3×C4—181no—
280C2×C3×C4—181no—
281C2×C3×C4—182no—
282C2×C3×C4—181no—
283C2×C3×C4C3×D1212yes—
284C2×Dic3—32no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—61no—
288C2×Dic3—62no—
289C2×Dic3—92no—
290C2×Dic3—181no—
291C2×Dic3—181no—
292C2×Dic3—181no—
293C2×Dic3—182no—
294D12—68no—
295D12—188no—
296C2×C32—21no—
297C2×C32—22no—
298C2×C32—22no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32#87511yescharacteristic
302C2×C32#88211yescharacteristic
303C2×C32#88412yes—
304C2×C32#89412yes—
305C2×C32#89511yescharacteristic
306C2×C32C3×#12011yescharacteristic
307C2×C32C3×#12112yes—
308C2×C32C3×#12111yescharacteristic
309C2×C32C3×#12312yes—
310C2×C32C3×#12411yescharacteristic
311C3⋊D3—44no—
312C3×D3—124no—
313C24—92no—
314C2×D4—184no—
315C2×D4—182no—
316C22×C4—91no—
317C22×C4—91no—
318C22×C4C2×(C3⋊D3)⋊C312yes—
319C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
320C4⋊C4—182no—
321C22⋊C4—184no—
322C22⋊C4—184no—
323C22×C3—31no—
324C22×C3—31no—
325C22×C3—32no—
326C22×C3—32no—
327C22×C3—31no—
328C22×C3—32no—
329C22×C3—31no—
330C22×C3—32no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—61no—
334C22×C3—62no—
335C22×C3—62no—
336C22×C3—61no—
337C22×C3—61no—
338C22×C3—184no—
339C22×C3—184no—
340C22×C3—184no—
341C22×C3—362no—
342C22×C3—362no—
343C22×C3—362no—
344C22×C3—362no—
345C22×C3C2×C3×D1211yescharacteristic
346C22×C3C3×#23512yes—
347C22×C3C3×D3×D412yes—
348C22×C3C3×D3×D411yescharacteristic
349C22×C3C3×#23911yescharacteristic
350C2×D3—64no—
351C2×D3—64no—
352C2×D3—64no—
353C2×D3—122no—
354C2×D3—122no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—184no—
358C2×D3—184no—
359C2×D3—184no—
360C2×D3—362no—
361C2×D3—362no—
362C2×D3—362no—
363C2×D3—362no—
364C3×C4—22no—
365C3×C4—42no—
366C3×C4—62no—
367C3×C4—62no—
368C3×C4—122no—
369C3×C4—122no—
370C3×C4—122no—
371C3×C4—124no—
372C3×C4—182no—
373C3×C4—361no—
374Dic3—62no—
375Dic3—121no—
376Dic3—182no—
377Dic3—361no—
378C32—21no—
379C32#467211yescharacteristic, complemented
380C32C3×#39411yescharacteristic, complemented
381C23—94no—
382C23—182no—
383C23—182no—
384C23—182no—
385C23—182no—
386C23—182no—
387C23—182no—
388C23C22×(C3⋊D3)⋊C311yescharacteristic
389D4—188no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—92no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4—181no—
398C2×C4—182no—
399C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—32no—
405C2×C3—32no—
406C2×C3—32no—
407C2×C3—32no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—61no—
411C2×C3—61no—
412C2×C3—62no—
413C2×C3—62no—
414C2×C3—61no—
415C2×C3—364no—
416C2×C3C3×#87511yescharacteristic
417C2×C3C3×#88211yescharacteristic
418C2×C3C3×#88412yes—
419C2×C3C3×#89412yes—
420C2×C3C3×#89511yescharacteristic
421D3—124no—
422D3—364no—
423C22—184no—
424C22—184no—
425C22—184no—
426C22—362no—
427C22—362no—
428C22—362no—
429C22—362no—
430C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
431C22#3828⋊C312yes—
432C22D4×(C3⋊D3)⋊C312yes—
433C22D4×(C3⋊D3)⋊C311yescharacteristic
434C22#8872211yescharacteristic
435C4—22no—
436C4—42no—
437C4—182no—
438C4—361no—
439C3—31no—
440C3—31no—
441C3—61no—
442C3C3×#467211yescharacteristic
443C2—364no—
444C2#64084⋊C311yescharacteristic
445C2#64091⋊C311yescharacteristic
446C2#64093⋊C312yes—
447C2#64103⋊C312yes—
448C2#64104⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingH(3,𝔽3)×#95C2
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenter#95(C3⋊D3)⋊C3
hypocenterC32C3×#394
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
2. C2×#3752⋊C3C2semidirect
3. C2×#3752⋊C3C2semidirect
4. C2×#3750⋊C3C2semidirect
5. H(3,𝔽3)×#95C2semidirect
6. #10592371C3semidirect
20. C2×#3752C2×C3semidirect
21. C2×#3752C2×C3semidirect
22. C2×#3750C2×C3semidirect
27. C32×#95D3semidirect
28. C32×#95C2×C3semidirect
46. C2×(C3⋊Dic3)⋊C3D4semidirect
141. C2×C3⋊Dic3C3×D4semidirect
231. #95(C3⋊D3)⋊C3semidirect
232. H(3,𝔽3)#394semidirect
379. C32#4672semidirect
380. C32C3×#394semidirect

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×#3752⋊C32
C2×#3750⋊C32
H(3,𝔽3)×#952
#105923713
C3×#46723