Finite Group Databasebeta

← C3×#10593426 #13166787

Subgroups — C3×#10593426 #13166787

Order 1728.

570 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3774C212yesmaximal, complemented
2C3×#63914C211yesmaximal, characteristic, complemented
3C3×#63902C211yesmaximal, characteristic, complemented
4C3×#63779C211yesmaximal, characteristic, complemented
5C3×#63739C212yesmaximal, complemented
6#10593426C311yesmaximal, characteristic, direct product factor
7C2×C32×D12—21no—
8C2×C3×#625—21no—
9C2×C3×#625C2212yes—
10C2×C3×#625C2211yescharacteristic
11C2×C3×D3×Dic3—21no—
12C2×C3×#622C2212yes—
13C3×#3774—24no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C32×(C2×D3)⋊C4—21no—
16C3×#3724C2211yescharacteristic
17C2×#3774C2×C312yescomplemented
18C3×#940—61no—
19C3×#67⋊C4—61no—
20#63914C2×C311yescharacteristic, complemented
21#63902C2×C311yescharacteristic, complemented
22#63779C2×C311yescharacteristic, complemented
23#63739C2×C312yescomplemented
24C22×C32×D3—21no—
25C22×C32×D3—22no—
26C2×C33×C4D411yescharacteristic
27C2×C3×C3⋊Dic3D411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3—21no—
30C32×D12—42no—
31C3×#625—22no—
32C3×#625—21no—
33C3×#625—22no—
34C3×#625—42no—
35C3×#625D412yescomplemented
36C3×D3×Dic3—42no—
37C3×#622—22no—
38C3×#459—91nomaximal
39C2×C32×D4—61no—
40C2×C3×#67—61no—
41C2×C3×#67—62no—
42C2×C3×#67—62no—
43C22×C3×Dic3—61no—
44C2×C3×D12—21no—
45C2×C3×D12—21no—
46C2×C3×D12—41no—
47C2×C3×C4×D3—61no—
48C2×#625—21no—
49C2×#625C22×C312yes—
50C2×#625C22×C311yescharacteristic
51C2×D3×Dic3—21no—
52C2×#622C22×C312yes—
53#3774—24no—
54C32×C22⋊C4—61no—
55C4×C3⋊Dic3C22×C311yescharacteristic
56C3×(C2×D3)⋊C4—21no—
57C3×(C2×D3)⋊C4—21no—
58C3×(C2×D3)⋊C4—41no—
59C3×Dic3⋊C4—61no—
60C3×C4×Dic3—61no—
61C3×C4×Dic3—61no—
62#3724C22×C311yescharacteristic
63C22×C33C2×D411yescharacteristic
64C2×C32×D3—22no—
65C2×C32×D3—41no—
66C2×C32×D3—41no—
67C2×C32×D3—42no—
68C2×C32×D3—42no—
69C2×C32×D3—42no—
70C33×C4—21no—
71C3×C3⋊Dic3D812yescomplemented
72C3×C3⋊Dic3C2×D411yescharacteristic
73C3×C3⋊Dic3#4111yescharacteristic
74C32×Dic3—41no—
75C2×C3×D8—92no—
76C3×#127—91no—
77C3×C4×D4—91no—
78#940—61no—
79#67⋊C4—61no—
80C3×C4⋊C8—91no—
81C3×#102—92no—
82C23×C32—61no—
83C23×C32—62no—
84C22×C3×D3—21no—
85C22×C3×D3—21no—
86C22×C3×D3—22no—
87C22×C3×D3—22no—
88C22×C3×D3—41no—
89C22×C3×D3—42no—
90C32×D4—122no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4—61no—
96C2×C32×C4C3×D411yescharacteristic
97C2×C3⋊Dic3C3×D411yescharacteristic
98C2×C3⋊Dic3C23×C311yescharacteristic
99C3×#67—62no—
100C3×#67—62no—
101C3×#67—122no—
102C3×#67—122no—
103C3×#67—124no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—41no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—122no—
112C3×D12—42no—
113C3×D12—42no—
114C3×D12—44no—
115C3×C4×D3—122no—
116#625—21no—
117#625—22no—
118#625—22no—
119#625—42no—
120#625C3×D412yescomplemented
121D3×Dic3—42no—
122#622—22no—
123#459—91noSylow
124C2×C33#12111yescharacteristic
125C2×C33C2×D811yescharacteristic
126C2×C33AGL(1,ℤ/8)11yescharacteristic
127C32×D3—42no—
128C32×D3—82no—
129C2×C3×D4—61no—
130C2×C3×D4—61no—
131C2×C3×D4—91no—
132C2×C3×D4—92no—
133C2×C3×D4—121no—
134C2×C3×D4—181no—
135C22×C3×C4—181no—
136C2×#67—61no—
137C2×#67—62no—
138C2×#67—62no—
139C22×Dic3—61no—
140C2×D12—21no—
141C2×C4×D3—61no—
142C3×D8—184no—
143C2×C3×C8—92no—
144C3×C4⋊C4—91no—
145C3×C22⋊C4—61no—
146C3×C22⋊C4—61no—
147C3×C22⋊C4—121no—
148C3×C42—91no—
149(C2×D3)⋊C4—21no—
150Dic3⋊C4—61no—
151C4×Dic3—61no—
152C4×Dic3—61no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—41no—
156C22×C32—41no—
157C22×C32—62no—
158C22×C32—122no—
159C22×C32—121no—
160C22×C32—121no—
161C22×C32—122no—
162C22×C32—122no—
163C22×C32C2×C3×D411yescharacteristic
164C2×C3×D3—22no—
165C2×C3×D3—22no—
166C2×C3×D3—42no—
167C2×C3×D3—41no—
168C2×C3×D3—41no—
169C2×C3×D3—41no—
170C2×C3×D3—41no—
171C2×C3×D3—42no—
172C2×C3×D3—42no—
173C2×C3×D3—42no—
174C2×C3×D3—44no—
175C2×C3×D3—42no—
176C2×C3×D3—42no—
177C2×C3×D3—42no—
178C2×C3×D3—81no—
179C2×C3×D3—81no—
180C2×C3×D3—82no—
181C2×C3×D3—82no—
182C32×C4—22no—
183C32×C4—21no—
184C32×C4—41no—
185C32×C4—42no—
186C32×C4—81no—
187C32×C4—121no—
188C3⋊Dic3C3×D812yescomplemented
189C3⋊Dic3C2×C3×D411yescharacteristic
190C3⋊Dic3C3×#4111yescharacteristic
191C3×Dic3—41no—
192C3×Dic3—41no—
193C3×Dic3—62no—
194C3×Dic3—62no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—61no—
199C3×Dic3—81no—
200C2×D8—92no—
201#127—91no—
202C4×D4—91no—
203C4⋊C8—91no—
204#102—92no—
205C33#45911yescharacteristic, complemented, Sylow
206C23×C3—61no—
207C23×C3—62no—
208C23×C3—61no—
209C23×C3—62no—
210C23×C3—121no—
211C23×C3—122no—
212C22×D3—21no—
213C22×D3—22no—
214C3×D4—92no—
215C3×D4—122no—
216C3×D4—122no—
217C3×D4—124no—
218C3×D4—181no—
219C3×D4—182no—
220C3×D4—182no—
221C3×D4—362no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—91no—
229C2×C3×C4—91no—
230C2×C3×C4—121no—
231C2×C3×C4—362no—
232C2×C3×C4GO+(4,𝔽2)11yescharacteristic
233#67—62no—
234#67—62no—
235#67—122no—
236#67—122no—
237#67—124no—
238C2×Dic3—21no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—122no—
244D12—42no—
245C4×D3—122no—
246C3×C8—182no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—122no—
260C2×C32—242no—
261C2×C32C3×#12111yescharacteristic
262C2×C32C2×C3×D811yescharacteristic
263C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
264C3×D3—42no—
265C3×D3—42no—
266C3×D3—82no—
267C3×D3—82no—
268C3×D3—84no—
269C3×D3—82no—
270C2×D4—61no—
271C2×D4—91no—
272C2×D4—92no—
273C2×D4—181no—
274C22×C4—181no—
275D8—184no—
276C2×C8—92no—
277C4⋊C4—91no—
278C22⋊C4—61no—
279C42—91no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—62no—
285C22×C3—62no—
286C22×C3—122no—
287C22×C3—121no—
288C22×C3—121no—
289C22×C3—122no—
290C22×C3—122no—
291C22×C3—122no—
292C22×C3—124no—
293C22×C3—121no—
294C22×C3—121no—
295C22×C3—122no—
296C22×C3—122no—
297C22×C3—122no—
298C22×C3—241no—
299C22×C3—241no—
300C22×C3—242no—
301C22×C3—242no—
302C22×C3C2×GO+(4,𝔽2)11yescharacteristic
303C2×D3—22no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—42no—
307C2×D3—42no—
308C2×D3—42no—
309C3×C4—22no—
310C3×C4—21no—
311C3×C4—41no—
312C3×C4—42no—
313C3×C4—81no—
314C3×C4—92no—
315C3×C4—91no—
316C3×C4—91no—
317C3×C4—121no—
318C3×C4—121no—
319C3×C4—241no—
320Dic3—41no—
321Dic3—62no—
322Dic3—61no—
323Dic3—61no—
324Dic3—62no—
325Dic3—61no—
326Dic3—61no—
327C32—21no—
328C32—21no—
329C32—41no—
330C32—41no—
331C32C3×#45911yescharacteristic, complemented
332C23—61no—
333C23—62no—
334D4—92no—
335D4—122no—
336D4—181no—
337D4—182no—
338D4—182no—
339D4—362no—
340C2×C4—61no—
341C2×C4—91no—
342C2×C4—91no—
343C2×C4—362no—
344C2×C4C3×GO+(4,𝔽2)11yescharacteristic
345C8—182no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—122no—
359C2×C3—122no—
360C2×C3—242no—
361C2×C3—242no—
362C2×C3—244no—
363C2×C3—242no—
364C2×C3#6422511yescharacteristic
365C2×C3#6423011yescharacteristic
366C2×C3#6423611yescharacteristic
367D3—42no—
368D3—82no—
369C22—62no—
370C22—121no—
371C22—121no—
372C22—122no—
373C22—122no—
374C22—122no—
375C22C2×C3×GO+(4,𝔽2)11yescharacteristic
376C4—21no—
377C4—92no—
378C4—91no—
379C4—91no—
380C4—121no—
381C3—21no—
382C3—21no—
383C3—41no—
384C3—41no—
385C3#1059342611yescharacteristic, direct product factor
386C2—122no—
387C2—242no—
388C2C3×#6422511yescharacteristic
389C2C3×#6423011yescharacteristic
390C2C3×#6423611yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#459
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3774C2semidirect
2. C3×#63914C2semidirect
3. C3×#63902C2semidirect
4. C3×#63779C2semidirect
5. C3×#63739C2semidirect
6. #10593426C3direct (N × H)
17. C2×#3774C2×C3semidirect
20. #63914C2×C3semidirect
21. #63902C2×C3semidirect
22. #63779C2×C3semidirect
23. #63739C2×C3semidirect
35. C3×#625D4semidirect
71. C3×C3⋊Dic3D8semidirect
120. #625C3×D4semidirect
188. C3⋊Dic3C3×D8semidirect
205. C33#459semidirect
331. C32C3×#459semidirect
385. C3#10593426direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37742
C3×#639142
C3×#639022
C3×#637792
C3×#637392
#105934263
C3×#4599