Finite Group Databasebeta

← #13145700

Subgroups — #13145700

Order 1728.

534 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
3C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#10589082—31nomaximal
5#10589081—31nomaximal
6#10589080—31nomaximal
7C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
8C2×C3×C4×C3⋊D3—21no—
9C33×C42C2211yescharacteristic
10C3×#3752—21no—
11C3×C4⋊(C3⋊Dic3)C411yescharacteristic
12C2×C3×C3⋊(C3⋊C8)—21no—
13C2×C3×C3⋊(C3⋊C8)—21no—
14C4×C4⋊(C3⋊D3)D311yescharacteristic
15C3×C4×D12—31no—
16C3×C4×D12—31no—
17C3×C4×D12—61no—
18C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
19C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
20C3×C4⋊(C3⋊C8)—31no—
21C3×C4⋊(C3⋊C8)—31no—
22C3×C4⋊(C3⋊C8)—61no—
23C3×C4×C3⋊C8—31no—
24C3×C4×C3⋊C8—31no—
25C3×C4×C3⋊C8—61no—
26C22×C3×C3⋊D3—21no—
27C2×C33×C4C2×C411yescharacteristic
28C2×C33×C4D411yescharacteristic
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3—21no—
31C3×C4⋊(C3⋊D3)—21no—
32C3×C4⋊(C3⋊D3)C812yescomplemented
33C3×C4×C3⋊D3—41no—
34C3×C4×C3⋊D3—41no—
35C3×C3⋊(C3⋊C8)—22no—
36C3×C3⋊(C3⋊C8)—41no—
37D4⋊(C3⋊C8)—91no—
38#4483—91no—
39#4459—91no—
40C2×C4⋊(C3⋊D3)Dic311yescharacteristic
41C2×C4×C3⋊D3—21no—
42C2×C3×D12—31no—
43C2×C3×D12—31no—
44C2×C3×D12—61no—
45C2×C3×C4×D3—62no—
46C2×C3×C4×D3—61no—
47C2×C3×C4×D3—61no—
48C32×C42—21no—
49C32×C42—21no—
50C32×C42—21no—
51C32×C42—41no—
52C32×C42C2×D311yescharacteristic
53C32×C42C2×D311yescharacteristic
54C32×C42C2×D311yescharacteristic
55#3752—21no—
56C4⋊(C3⋊Dic3)Dic311yescharacteristic
57C2×C3⋊(C3⋊C8)—21no—
58C2×C3⋊(C3⋊C8)—21no—
59C3×(C2×D3)⋊C4—62no—
60C3×(C2×D3)⋊C4—61no—
61C3×(C2×D3)⋊C4—61no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—61no—
65C2×C3×C3⋊C8—62no—
66C2×C3×C3⋊C8—61no—
67C2×C3×C3⋊C8—61no—
68C2×C3×C3⋊C8—62no—
69C2×C3×C3⋊C8—61no—
70C2×C3×C3⋊C8—61no—
71C22×C33C22⋊C411yescharacteristic
72C2×C3×C3⋊D3—22no—
73C2×C3×C3⋊D3—41no—
74C2×C3×C3⋊D3—41no—
75C33×C4—21no—
76C33×C4C2×C811yescharacteristic
77C33×C4M1611yescharacteristic
78C33×C4D811yescharacteristic
79C33×C4SD1611yescharacteristic
80C3×C3⋊Dic3—41no—
81C3×C4×D4—91no—
82C4×D12—31no—
83C4×D12—31no—
84C4×D12—61no—
85C3×C4⋊C8—91no—
86C3×C4×C8—91no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—61no—
90C4×C3⋊C8—31no—
91C4×C3⋊C8—31no—
92C4×C3⋊C8—61no—
93C22×C3⋊D3—21no—
94C22×C3×D3—62no—
95C22×C3×D3—61no—
96C22×C3×D3—61no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—41no—
107C2×C32×C4—41no—
108C2×C32×C4—41no—
109C2×C32×C4C4×D311yescharacteristic
110C2×C32×C4C4×D311yescharacteristic
111C2×C32×C4D1211yescharacteristic
112C2×C32×C4D1211yescharacteristic
113C2×C32×C4C2×Dic311yescharacteristic
114C2×C32×C4#6711yescharacteristic
115C2×C32×C4#6711yescharacteristic
116C2×C32×C4#6711yescharacteristic
117C2×C32×C4#6711yescharacteristic
118C2×C3⋊Dic3—21no—
119C4⋊(C3⋊D3)—21no—
120C4⋊(C3⋊D3)C3⋊C812yescomplemented
121C4×C3⋊D3—41no—
122C4×C3⋊D3—41no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C3×D12—32no—
127C3×D12—32no—
128C3×D12—62no—
129C3×D12—62no—
130C3×D12—61no—
131C3×D12—61no—
132C3×C4×D3—122no—
133C3×C4×D3—122no—
134C3×C4×D3—121no—
135C3×C4×D3—121no—
136C3×C4×D3—121no—
137C3×C4×D3—121no—
138C3⋊(C3⋊C8)—22no—
139C3⋊(C3⋊C8)—41no—
140C3×C3⋊C8—64no—
141C3×C3⋊C8—62no—
142C3×C3⋊C8—62no—
143C3×C3⋊C8—122no—
144C3×C3⋊C8—121no—
145C3×C3⋊C8—121no—
146D4⋊C8—271noSylow
147C2×C33C22⋊C811yescharacteristic
148C2×C33#10211yescharacteristic
149C2×C33#10411yescharacteristic
150C3×C3⋊D3—42no—
151C2×C3×D4—91no—
152C22×C3×C4—181no—
153C2×D12—31no—
154C2×D12—31no—
155C2×D12—61no—
156C2×C4×D3—62no—
157C2×C4×D3—61no—
158C2×C4×D3—61no—
159C2×C3×C8—181no—
160C2×C3×C8—181no—
161C3×C4⋊C4—91no—
162C3×C22⋊C4—181no—
163C3×C42—21no—
164C3×C42—21no—
165C3×C42—21no—
166C3×C42—41no—
167C3×C42D3211yescharacteristic
168C3×C42D3211yescharacteristic
169C3×C42D3211yescharacteristic
170(C2×D3)⋊C4—62no—
171(C2×D3)⋊C4—61no—
172(C2×D3)⋊C4—61no—
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—61no—
176C2×C3⋊C8—62no—
177C2×C3⋊C8—61no—
178C2×C3⋊C8—61no—
179C2×C3⋊C8—62no—
180C2×C3⋊C8—61no—
181C2×C3⋊C8—61no—
182C22×C32—21no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—41no—
186C22×C32(C2×D3)⋊C411yescharacteristic
187C22×C32(C2×D3)⋊C411yescharacteristic
188C22×C32#21711yescharacteristic
189C2×C3⋊D3—22no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C2×C3×D3—64no—
193C2×C3×D3—62no—
194C2×C3×D3—62no—
195C2×C3×D3—122no—
196C2×C3×D3—122no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C32×C4—21no—
202C32×C4—22no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—21no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4—41no—
218C32×C4—41no—
219C32×C4—41no—
220C32×C4—41no—
221C32×C4—41no—
222C32×C4—41no—
223C32×C4—42no—
224C32×C4D3×C811yescharacteristic
225C32×C4D3×C811yescharacteristic
226C32×C4ΓL(1,𝔽25)11yescharacteristic
227C32×C4ΓL(1,𝔽25)11yescharacteristic
228C32×C4#20411yescharacteristic
229C32×C4D2411yescharacteristic
230C32×C4C2×C3⋊C811yescharacteristic
231C32×C4#20811yescharacteristic
232C32×C4#21311yescharacteristic
233C32×C4#21311yescharacteristic
234C32×C4#21411yescharacteristic
235C32×C4#21511yescharacteristic
236C3⋊Dic3—41no—
237C3×Dic3—122no—
238C3×Dic3—121no—
239C3×Dic3—121no—
240C4×D4—91no—
241C4⋊C8—91no—
242C4×C8—91no—
243C33D4⋊C811yescharacteristic, complemented, Sylow
244C23×C3—181no—
245C22×D3—62no—
246C22×D3—61no—
247C22×D3—61no—
248C3×D4—92no—
249C3×D4—181no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—181no—
263C2×C3×C4—361no—
264C2×C3×C4—361no—
265C2×C3×C4D3×Dic311yescharacteristic
266C2×C3×C4D3×Dic311yescharacteristic
267C2×C3×C4#62411yescharacteristic
268C2×C3×C4#62511yescharacteristic
269C2×C3×C4#62511yescharacteristic
270C2×C3×C4#62611yescharacteristic
271C2×C3×C4#62611yescharacteristic
272C2×C3×C4#62611yescharacteristic
273C2×C3×C4#62611yescharacteristic
274C2×Dic3—62no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277D12—32no—
278D12—32no—
279D12—62no—
280D12—62no—
281D12—61no—
282D12—61no—
283C4×D3—122no—
284C4×D3—122no—
285C4×D3—121no—
286C4×D3—121no—
287C4×D3—121no—
288C4×D3—121no—
289C3×C8—182no—
290C3×C8—361no—
291C3⋊C8—64no—
292C3⋊C8—62no—
293C3⋊C8—62no—
294C3⋊C8—122no—
295C3⋊C8—121no—
296C3⋊C8—121no—
297C2×C32—21no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—41no—
307C2×C32—41no—
308C2×C32—41no—
309C2×C32#80511yescharacteristic
310C2×C32#80911yescharacteristic
311C2×C32(C2×D3)⋊C811yescharacteristic
312C2×C32(C2×D3)⋊C811yescharacteristic
313C2×C32#82111yescharacteristic
314C2×C32#82511yescharacteristic
315C2×C32#83011yescharacteristic
316C2×C32#83211yescharacteristic
317C2×C32#83711yescharacteristic
318C3⋊D3—42no—
319C3×D3—124no—
320C3×D3—122no—
321C3×D3—122no—
322C2×D4—91no—
323C22×C4—181no—
324C2×C8—181no—
325C2×C8—181no—
326C4⋊C4—91no—
327C22⋊C4—181no—
328C42#111311yescharacteristic
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—41no—
333C22×C3—182no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3(C2×D3)⋊Dic311yescharacteristic
337C22×C3(C2×D3)⋊Dic311yescharacteristic
338C22×C3#372211yescharacteristic
339C2×D3—64no—
340C2×D3—62no—
341C2×D3—62no—
342C2×D3—122no—
343C2×D3—122no—
344C2×D3—121no—
345C2×D3—121no—
346C2×D3—121no—
347C2×D3—121no—
348C3×C4—22no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—42no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4—361no—
372C3×C4D3×C3⋊C811yescharacteristic
373C3×C4D3×C3⋊C811yescharacteristic
374C3×C4#371011yescharacteristic
375C3×C4#371111yescharacteristic
376C3×C4#371111yescharacteristic
377C3×C4#371211yescharacteristic
378C3×C4#371311yescharacteristic
379C3×C4#371411yescharacteristic
380C3×C4#371411yescharacteristic
381C3×C4#371511yescharacteristic
382C3×C4#371611yescharacteristic
383C3×C4#371711yescharacteristic
384Dic3—122no—
385Dic3—121no—
386Dic3—121no—
387C32—21no—
388C32—21no—
389C32—21no—
390C32—41no—
391C32#445911yescharacteristic, complemented
392C32#448311yescharacteristic, complemented
393C32D4⋊(C3⋊C8)11yescharacteristic, complemented
394C23—181no—
395D4—92no—
396D4—181no—
397C2×C4—181no—
398C2×C4—361no—
399C2×C4—361no—
400C2×C4#629411yescharacteristic
401C2×C4#629511yescharacteristic
402C2×C4#629511yescharacteristic
403C8—182no—
404C8—361no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—41no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—362no—
418C2×C3(C2×D3)⋊(C3⋊C8)11yescharacteristic
419C2×C3(C2×D3)⋊(C3⋊C8)11yescharacteristic
420C2×C3#6355711yescharacteristic
421C2×C3#6356211yescharacteristic
422C2×C3#6356311yescharacteristic
423C2×C3#6356411yescharacteristic
424C2×C3#6356811yescharacteristic
425C2×C3#6356911yescharacteristic
426C2×C3#6357011yescharacteristic
427D3—124no—
428D3—122no—
429D3—122no—
430C22—182no—
431C22—361no—
432C22—361no—
433C22#8881411yescharacteristic
434C4—21no—
435C4—361no—
436C4#8880811yescharacteristic
437C4#8880911yescharacteristic
438C4#8881011yescharacteristic
439C4#8881111yescharacteristic
440C3—21no—
441C3—21no—
442C3—21no—
443C3—41no—
444C3#1058908011yescharacteristic, complemented
445C3#1058908111yescharacteristic, complemented
446C3#1058908211yescharacteristic, complemented
447C2—362no—
448C2#1172207211yescharacteristic
449C2#1172207511yescharacteristic
450C2#1172207711yescharacteristic
subgroupG/H
centerC2×C4#6295
commutatorC33×C4C2×C8
Frattini, hypercenterC42#1113
Fitting, cosocleC33×C42C22
socleC22×C33C22⋊C4
prime radicalC2×C3×C4⋊(C3⋊D3)C4
hypocenterC33D4⋊C8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
3. C3×C4×C3⋊(C3⋊C8)C2semidirect
18. C4⋊(C3⋊(C3⋊C8))D3semidirect
19. C4×C3⋊(C3⋊C8)D3semidirect
32. C3×C4⋊(C3⋊D3)C8semidirect
120. C4⋊(C3⋊D3)C3⋊C8semidirect
243. C33D4⋊C8semidirect
391. C32#4459semidirect
392. C32#4483semidirect
393. C32D4⋊(C3⋊C8)semidirect
444. C3#10589080semidirect
445. C3#10589081semidirect
446. C3#10589082semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊(C3⋊D3)2
C3×C4⋊(C3⋊(C3⋊C8))2
C3×C4×C3⋊(C3⋊C8)2
#105890823
#105890813
#105890803