Finite Group Databasebeta

← C2×C3×#63731 #13166726

Subgroups — C2×C3×#63731 #13166726

Order 1728.

648 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 98 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×C3×#3798C211yesmaximal, characteristic
2C2×C3×C4×#157C211yesmaximal, characteristic, complemented
3C2×C3×#3712C211yesmaximal, characteristic, complemented
4C3×#63731C214yesmaximal, direct product factor
5C2×#63731C311yesmaximal, characteristic, direct product factor
6C22×C3×#157—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×C4×D3—21no—
9C2×C3×#627C411yescharacteristic, complemented
10C2×C3×#626—21no—
11C2×C3×#625C411yescharacteristic, complemented
12C3×#3798—21no—
13C3×#3798—21no—
14C3×#3798C412yescomplemented
15C3×#3798C2212yes—
16C3×C4×#157—21no—
17C3×C4×#157C2212yescomplemented
18C2×C32×C3⋊C8—21no—
19C3×#3712—21no—
20C3×#3712C2212yescomplemented
21C2×C3×#235—61no—
22C2×#3798C2×C311yescharacteristic
23C2×C4×#157C2×C311yescharacteristic, complemented
24C2×C3×ΓL(1,𝔽25)—61no—
25C2×#3712C2×C311yescharacteristic, complemented
26#63731C2×C314yesdirect product factor
27C22×C3×C3⋊D3D411yescharacteristic
28C22×C32×D3—21no—
29C2×C3×#157—24no—
30C2×C3×#157—22no—
31C2×C33×C4D411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C3×C4×C3⋊D3C2×C411yescharacteristic
34C3×C4×C3⋊D3D411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C3×C4×C3⋊D3C2311yescharacteristic
37C2×C32×Dic3—21no—
38C32×C4×D3—22no—
39C32×C4×D3—22no—
40C3×#627—21no—
41C3×#627C2×C412yescomplemented
42C3×#626—22no—
43C3×#626—22no—
44C3×#625—21no—
45C3×#625C2×C412yescomplemented
46C32×C3⋊C8—22no—
47C2×C3×#104—91nomaximal
48C22×#157—21no—
49C22×C32×C4—61no—
50C2×C4×C3⋊D3C22×C311yescharacteristic
51C2×C3×#67—61no—
52C2×C3×#67—61no—
53C3×#235—62no—
54C3×#235—62no—
55C3×#235—62no—
56C3×#235—62no—
57C2×C3×D12—61no—
58C2×C3×C4×D3—21no—
59C2×C3×C4×D3—21no—
60C2×C3×C4×D3—41no—
61C2×C3×C4×D3—61no—
62C2×C3×C4×D3—61no—
63C2×C3×Dic6—61no—
64C2×#627C3×C411yescharacteristic, complemented
65C2×#626—21no—
66C2×#625C3×C411yescharacteristic, complemented
67#3798—21no—
68#3798—21no—
69#3798C3×C412yescomplemented
70#3798C22×C312yes—
71C4×#157—21no—
72C4×#157C22×C312yescomplemented
73C2×C32×C8—61no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8—41no—
77C3×ΓL(1,𝔽25)—62no—
78C3×ΓL(1,𝔽25)—62no—
79#3712—21no—
80#3712C22×C312yescomplemented
81C22×C33C22⋊C411yescharacteristic
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3C22⋊C411yescharacteristic
84C2×C3×C3⋊D3C2×D411yescharacteristic
85C2×C32×D3—22no—
86C2×C32×D3—42no—
87C3×#157—24no—
88C33×C4C22⋊C411yescharacteristic
89C33×C4C2×D411yescharacteristic
90C3×C3⋊Dic3C22⋊C411yescharacteristic
91C3×C3⋊Dic3C22×C411yescharacteristic
92C32×Dic3—22no—
93C2×C3×#41—91no—
94C2×#235—61no—
95C2×C3×M16—91no—
96C2×C3×C42—91no—
97C2×ΓL(1,𝔽25)—61no—
98C3×#104—94no—
99C23×C32—61no—
100C22×C3⋊D3C3×D411yescharacteristic
101C22×C3×D3—21no—
102C22×C3×D3—21no—
103C22×C3×D3—41no—
104C22×C3×D3—61no—
105C22×C3×D3—61no—
106C2×#157—24no—
107C2×#157—22no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—41no—
111C2×C32×C4—41no—
112C2×C32×C4—62no—
113C2×C32×C4—62no—
114C2×C32×C4—61no—
115C2×C32×C4C3×D411yescharacteristic
116C2×C3⋊Dic3C2×C3×C411yescharacteristic
117C4×C3⋊D3C2×C3×C411yescharacteristic
118C4×C3⋊D3C3×D411yescharacteristic
119C4×C3⋊D3C3×D411yescharacteristic
120C4×C3⋊D3C23×C311yescharacteristic
121C3×#67—62no—
122C3×#67—62no—
123C3×#67—62no—
124C3×#67—62no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—41no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C3×D12—62no—
131C3×D12—62no—
132C3×C4×D3—22no—
133C3×C4×D3—22no—
134C3×C4×D3—22no—
135C3×C4×D3—22no—
136C3×C4×D3—42no—
137C3×C4×D3—42no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—61no—
145C3×C4×D3—61no—
146C3×Dic6—62no—
147C3×Dic6—62no—
148#627—21no—
149#627C2×C3×C412yescomplemented
150#626—22no—
151#626—22no—
152#625—21no—
153#625C2×C3×C412yescomplemented
154C32×C8—62no—
155C3×C3⋊C8—22no—
156C3×C3⋊C8—22no—
157C3×C3⋊C8—42no—
158C2×#104—91noSylow
159C2×C33#10412yescomplemented
160C2×C33C2×C22⋊C411yescharacteristic
161C3×C3⋊D3—21no—
162C3×C3⋊D3—21no—
163C32×D3—42no—
164C3×#41—92no—
165C3×#41—92no—
166C3×#41—181no—
167C3×#41—181no—
168C2×C3×Q8—91no—
169C2×C3×D4—91no—
170C2×C3×D4—181no—
171C22×C3×C4—61no—
172C22×C3×C4—61no—
173C22×C3×C4—91no—
174C22×C3×C4—121no—
175C22×C3×C4—181no—
176C2×#67—61no—
177C2×#67—61no—
178#235—62no—
179#235—62no—
180#235—62no—
181#235—62no—
182C2×D12—61no—
183C2×C4×D3—21no—
184C2×C4×D3—61no—
185C2×C4×D3—61no—
186C2×Dic6—61no—
187C3×M16—92no—
188C3×M16—181no—
189C2×C3×C8—61no—
190C2×C3×C8—61no—
191C2×C3×C8—121no—
192C3×C42—92no—
193C3×C42—181no—
194C2×C3⋊C8—21no—
195ΓL(1,𝔽25)—62no—
196ΓL(1,𝔽25)—62no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—62no—
202C22×C32—122no—
203C22×C32C3×C22⋊C411yescharacteristic
204C2×C3⋊D3—22no—
205C2×C3⋊D3C3×C22⋊C411yescharacteristic
206C2×C3⋊D3C2×C3×D411yescharacteristic
207C2×C3×D3—22no—
208C2×C3×D3—22no—
209C2×C3×D3—42no—
210C2×C3×D3—42no—
211C2×C3×D3—44no—
212C2×C3×D3—42no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C2×C3×D3—61no—
217C2×C3×D3—122no—
218C2×C3×D3—122no—
219#157—24no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—41no—
227C32×C4—41no—
228C32×C4—62no—
229C32×C4C3×C22⋊C411yescharacteristic
230C32×C4C2×C3×D411yescharacteristic
231C3⋊Dic3C3×C22⋊C411yescharacteristic
232C3⋊Dic3C22×C3×C411yescharacteristic
233C3×Dic3—22no—
234C3×Dic3—22no—
235C3×Dic3—42no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—61no—
240C2×#41—91no—
241C2×M16—91no—
242C2×C42—91no—
243#104—94no—
244C33C2×#10411yescharacteristic, complemented, Sylow
245C23×C3—61no—
246C23×C3—61no—
247C23×C3—91no—
248C23×C3—121no—
249C22×D3—21no—
250C22×D3—61no—
251C22×D3—61no—
252C3×Q8—92no—
253C3×Q8—181no—
254C3×D4—92no—
255C3×D4—182no—
256C3×D4—182no—
257C3×D4—181no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—62no—
267C2×C3×C4—62no—
268C2×C3×C4—91no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—122no—
274C2×C3×C4—122no—
275C2×C3×C4—121no—
276C2×C3×C4—182no—
277C2×C3×C4—184no—
278C2×C3×C4GO+(4,𝔽2)11yescharacteristic
279#67—62no—
280#67—62no—
281#67—62no—
282#67—62no—
283C2×Dic3—21no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286D12—62no—
287D12—62no—
288C4×D3—22no—
289C4×D3—22no—
290C4×D3—61no—
291C4×D3—61no—
292C4×D3—61no—
293C4×D3—61no—
294C4×D3—61no—
295C4×D3—61no—
296C4×D3—61no—
297C4×D3—61no—
298Dic6—62no—
299Dic6—62no—
300C3×C8—62no—
301C3×C8—62no—
302C3×C8—122no—
303C3⋊C8—22no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—42no—
309C2×C32—42no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32—122no—
313C2×C32C3×#10412yescomplemented
314C2×C32C2×C3×C22⋊C411yescharacteristic
315C3⋊D3—21no—
316C3⋊D3—21no—
317C3×D3—42no—
318C3×D3—44no—
319C3×D3—42no—
320C3×D3—121no—
321C3×D3—121no—
322C3×D3—121no—
323C3×D3—121no—
324#41—92no—
325#41—92no—
326#41—181no—
327#41—181no—
328C2×Q8—91no—
329C2×D4—91no—
330C2×D4—181no—
331C22×C4—61no—
332C22×C4—91no—
333C22×C4—181no—
334M16—92no—
335M16—181no—
336C2×C8—61no—
337C42—92no—
338C42—181no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—62no—
344C22×C3—62no—
345C22×C3—91no—
346C22×C3—91no—
347C22×C3—122no—
348C22×C3—122no—
349C22×C3—124no—
350C22×C3—122no—
351C22×C3—182no—
352C22×C3D32⋊C411yescharacteristic
353C2×D3—22no—
354C2×D3—42no—
355C2×D3—61no—
356C2×D3—61no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—122no—
360C2×D3—122no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—62no—
370C3×C4—62no—
371C3×C4—91no—
372C3×C4—91no—
373C3×C4—122no—
374C3×C4—184no—
375C3×C4D32⋊C411yescharacteristic
376C3×C4C2×GO+(4,𝔽2)11yescharacteristic
377Dic3—22no—
378Dic3—61no—
379Dic3—61no—
380Dic3—61no—
381Dic3—61no—
382C32—21no—
383C32—21no—
384C32—41no—
385C32—41no—
386C32C2×C3×#10411yescharacteristic, complemented
387C23—61no—
388C23—91no—
389Q8—92no—
390Q8—181no—
391D4—92no—
392D4—182no—
393D4—182no—
394D4—181no—
395C2×C4—61no—
396C2×C4—62no—
397C2×C4—62no—
398C2×C4—91no—
399C2×C4—91no—
400C2×C4—91no—
401C2×C4—91no—
402C2×C4—91no—
403C2×C4—182no—
404C2×C4—184no—
405C2×C4C3×GO+(4,𝔽2)11yescharacteristic
406C8—62no—
407C2×C3—22no—
408C2×C3—22no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—42no—
412C2×C3—42no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—122no—
416C2×C3—124no—
417C2×C3—122no—
418C2×C3—181no—
419C2×C3—181no—
420C2×C3#6373112yesdirect product factor
421C2×C3C2×D32⋊C411yescharacteristic
422D3—42no—
423D3—121no—
424D3—121no—
425D3—121no—
426D3—121no—
427C22—62no—
428C22—91no—
429C22—91no—
430C22—122no—
431C22—182no—
432C22C3×D32⋊C411yescharacteristic
433C4—62no—
434C4—91no—
435C4—91no—
436C4—184no—
437C4C3×D32⋊C411yescharacteristic
438C4C2×C3×GO+(4,𝔽2)11yescharacteristic
439C3—21no—
440C3—21no—
441C3—41no—
442C3—41no—
443C3C2×#6373111yescharacteristic, direct product factor
444C2—122no—
445C2—181no—
446C2—181no—
447C2C3×#6373112yesdirect product factor
448C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC2×C3×#3798C2
cosocleC4×C3⋊D3C23×C3
hypocenterC32C2×C3×#104
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×#157C2semidirect
3. C2×C3×#3712C2semidirect
4. C3×#63731C2direct (N × H)
5. C2×#63731C3direct (N × H)
9. C2×C3×#627C4semidirect
11. C2×C3×#625C4semidirect
14. C3×#3798C4semidirect
17. C3×C4×#157C22semidirect
20. C3×#3712C22semidirect
23. C2×C4×#157C2×C3semidirect
25. C2×#3712C2×C3semidirect
26. #63731C2×C3direct (N × H)
41. C3×#627C2×C4semidirect
45. C3×#625C2×C4semidirect
64. C2×#627C3×C4semidirect
66. C2×#625C3×C4semidirect
69. #3798C3×C4semidirect
72. C4×#157C22×C3semidirect
80. #3712C22×C3semidirect
149. #627C2×C3×C4semidirect
153. #625C2×C3×C4semidirect
159. C2×C33#104semidirect
244. C33C2×#104semidirect
313. C2×C32C3×#104semidirect
386. C32C2×C3×#104semidirect
420. C2×C3#63731direct (N × H)
443. C3C2×#63731direct (N × H)
447. C2C3×#63731direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37982
C2×C3×C4×#1572
C2×C3×#37122
C3×#637312
C2×#637313
C2×C3×#1049