Finite Group Databasebeta

← C3×#10593411 #13166773

Subgroups — C3×#10593411 #13166773

Order 1728.

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

normal subgroups only

This group has 676 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×#3797C211yesmaximal, characteristic
2C2×C3×#3773C212yesmaximal
3C3×#63883C211yesmaximal, characteristic
4C3×#63781C211yesmaximal, characteristic
5C3×#63737C212yesmaximal
6#10593411C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C2212yes—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×Dic6—21no—
10C2×C3×#627—21no—
11C2×C3×#624C2211yescharacteristic
12C2×C3×#624C2212yes—
13C2×C3×#624C2211yescharacteristic
14C3×#3797—24no—
15C3×#3773—24no—
16C32×Dic3⋊C4—21no—
17C3×Dic3⋊Dic3—21no—
18C2×C3×D3×Q8—61no—
19C2×#3797C2×C311yescharacteristic
20C2×#3773C2×C312yes—
21C3×D3×C4⋊C4—61no—
22#63883C2×C311yescharacteristic
23#63781C2×C311yescharacteristic
24#63737C2×C312yes—
25C22×C3×C3⋊D3C2311yescharacteristic
26C2×C3×#157—22no—
27C2×C3×#157—22no—
28C2×C3×#157Q814yescomplemented
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3D411yescharacteristic
31C3×C4×C3⋊D3—22no—
32C2×C32×Dic3—22no—
33C2×C32×Dic3—21no—
34C2×C32×Dic3—21no—
35C32×Dic6—24no—
36C3×#627—24no—
37C3×#624—22no—
38C3×#624—21no—
39C3×#624—22no—
40C3×#624D414yes—
41C3×#624Q812yescomplemented
42C3×#395—91nomaximal
43C22×#157C22×C312yes—
44C2×C32×Q8—61no—
45C2×C4×C3⋊D3C22×C311yescharacteristic
46C3×D3×Q8—68no—
47C2×C3×C4×D3—61no—
48C2×C3×C4×D3—62no—
49C2×C3×C4×D3—61no—
50C2×C3×C4×D3—61no—
51C2×C3×C4×D3—61no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—41no—
55C2×C3×Dic6—62no—
56C2×#627—21no—
57C2×#624C22×C311yescharacteristic
58C2×#624C22×C312yes—
59C2×#624C22×C311yescharacteristic
60#3797—24no—
61#3773—24no—
62C32×C4⋊C4—61no—
63C3×C4⋊Dic3—61no—
64C3×Dic3⋊C4—21no—
65C3×Dic3⋊C4—21no—
66C3×Dic3⋊C4—41no—
67C3×Dic3⋊C4—61no—
68Dic3⋊Dic3—21no—
69C22×C33C2×D411yescharacteristic
70C2×C3×C3⋊D3C2×D412yes—
71C2×C3×C3⋊D3C2×Q812yes—
72C2×C3×C3⋊D3C2×Q811yescharacteristic
73C2×C3×C3⋊D3#4111yescharacteristic
74C3×#157—24no—
75C33×C4—21no—
76C3×C3⋊Dic3—21no—
77C32×Dic3—24no—
78C32×Dic3—22no—
79C32×Dic3—41no—
80C22×C3×Q8—91no—
81C2×D3×Q8—61no—
82C2×C3×C4⋊C4—91no—
83C2×C3×C4⋊C4—92no—
84D3×C4⋊C4—61no—
85C3×#95—91no—
86C3×#95—92no—
87C22×C3⋊D3C23×C311yescharacteristic
88C22×C3×D3—61no—
89C22×C3×D3—61no—
90C2×#157—22no—
91C2×#157—22no—
92C2×#157C3×Q814yescomplemented
93C32×Q8—64no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4—41no—
98C2×C32×C4—61no—
99C2×C32×C4—62no—
100C2×C32×C4—61no—
101C2×C32×C4C3×D411yescharacteristic
102C2×C3⋊Dic3C3×D411yescharacteristic
103C4×C3⋊D3—22no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—22no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—41no—
111C2×C3×Dic3—42no—
112C2×C3×Dic3—41no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C3×C4×D3—64no—
116C3×C4×D3—62no—
117C3×C4×D3—62no—
118C3×C4×D3—64no—
119C3×C4×D3—64no—
120C3×C4×D3—122no—
121C3×C4×D3—122no—
122C3×Dic6—24no—
123C3×Dic6—24no—
124C3×Dic6—44no—
125C3×Dic6—68no—
126#627—24no—
127#624—22no—
128#624—21no—
129#624—22no—
130#624C3×D414yes—
131#624C3×Q812yescomplemented
132#395—91noSylow
133C2×C33#12011yescharacteristic
134C2×C33C22⋊Q811yescharacteristic
135C2×C33C4⋊Q811yescharacteristic
136C3×C3⋊D3C22⋊Q812yes—
137C3×C3⋊D3C4⋊Q812yes—
138C2×C3×Q8—61no—
139C2×C3×Q8—61no—
140C2×C3×Q8—121no—
141C2×C3×Q8—181no—
142C2×C3×Q8—184no—
143C22×C3×C4—91no—
144C22×C3×C4—92no—
145C22×C3×C4—91no—
146C22×C3×C4—92no—
147C22×C3×C4—91no—
148D3×Q8—68no—
149C2×C4×D3—61no—
150C2×C4×D3—62no—
151C2×C4×D3—61no—
152C2×C4×D3—61no—
153C2×C4×D3—61no—
154C2×Dic6—21no—
155C2×Dic6—62no—
156C3×C4⋊C4—61no—
157C3×C4⋊C4—61no—
158C3×C4⋊C4—121no—
159C3×C4⋊C4—181no—
160C3×C4⋊C4—184no—
161C4⋊Dic3—61no—
162Dic3⋊C4—21no—
163Dic3⋊C4—61no—
164C22×C32—21no—
165C22×C32—21no—
166C22×C32—41no—
167C22×C32—41no—
168C22×C32C2×C3×D411yescharacteristic
169C2×C3⋊D3C2×C3×D412yes—
170C2×C3⋊D3C2×C3×Q812yes—
171C2×C3⋊D3C2×C3×Q811yescharacteristic
172C2×C3⋊D3C3×#4111yescharacteristic
173C2×C3×D3—62no—
174C2×C3×D3—62no—
175C2×C3×D3—61no—
176C2×C3×D3—61no—
177C2×C3×D3—62no—
178C2×C3×D3—62no—
179C2×C3×D3—61no—
180C2×C3×D3—61no—
181#157—24no—
182C32×C4—22no—
183C32×C4—21no—
184C32×C4—41no—
185C32×C4—42no—
186C32×C4—64no—
187C32×C4—62no—
188C32×C4—81no—
189C32×C4—121no—
190C3⋊Dic3—21no—
191C3×Dic3—24no—
192C3×Dic3—22no—
193C3×Dic3—24no—
194C3×Dic3—22no—
195C3×Dic3—44no—
196C3×Dic3—42no—
197C3×Dic3—41no—
198C3×Dic3—41no—
199C3×Dic3—62no—
200C3×Dic3—81no—
201C3×Dic3—121no—
202C22×Q8—91no—
203C2×C4⋊C4—91no—
204C2×C4⋊C4—92no—
205#95—91no—
206#95—92no—
207C33#39511yescharacteristic, complemented, Sylow
208C23×C3—91no—
209C22×D3—61no—
210C22×D3—61no—
211C3×Q8—64no—
212C3×Q8—64no—
213C3×Q8—124no—
214C3×Q8—184no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—61no—
220C2×C3×C4—62no—
221C2×C3×C4—61no—
222C2×C3×C4—62no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—94no—
226C2×C3×C4—92no—
227C2×C3×C4—94no—
228C2×C3×C4—91no—
229C2×C3×C4—121no—
230C2×C3×C4—122no—
231C2×C3×C4—121no—
232C2×C3×C4—182no—
233C2×C3×C4—181no—
234C2×C3×C4—182no—
235C2×C3×C4—182no—
236C2×C3×C4—182no—
237C2×C3×C4—182no—
238C2×C3×C4GO+(4,𝔽2)11yescharacteristic
239C2×Dic3—21no—
240C2×Dic3—22no—
241C2×Dic3—21no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C4×D3—64no—
245C4×D3—62no—
246C4×D3—62no—
247C4×D3—64no—
248C4×D3—64no—
249C4×D3—122no—
250C4×D3—122no—
251Dic6—24no—
252Dic6—68no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32C3×#12011yescharacteristic
266C2×C32C3×C22⋊Q811yescharacteristic
267C2×C32C3×C4⋊Q811yescharacteristic
268C3⋊D3C3×C22⋊Q812yes—
269C3⋊D3C3×C4⋊Q812yes—
270C3×D3—62no—
271C3×D3—62no—
272C3×D3—62no—
273C3×D3—62no—
274C2×Q8—61no—
275C2×Q8—181no—
276C2×Q8—184no—
277C22×C4—91no—
278C22×C4—92no—
279C22×C4—91no—
280C22×C4—92no—
281C22×C4—91no—
282C4⋊C4—61no—
283C4⋊C4—181no—
284C4⋊C4—184no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—91no—
290C22×C3—91no—
291C22×C3—92no—
292C22×C3—92no—
293C22×C3C2×GO+(4,𝔽2)11yescharacteristic
294C2×D3—62no—
295C2×D3—61no—
296C2×D3—61no—
297C2×D3—62no—
298C2×D3—61no—
299C2×D3—62no—
300C2×D3—62no—
301C2×D3—61no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—41no—
305C3×C4—42no—
306C3×C4—64no—
307C3×C4—62no—
308C3×C4—62no—
309C3×C4—64no—
310C3×C4—81no—
311C3×C4—124no—
312C3×C4—122no—
313C3×C4—121no—
314C3×C4—121no—
315C3×C4—184no—
316C3×C4—181no—
317C3×C4—241no—
318Dic3—24no—
319Dic3—22no—
320Dic3—41no—
321Dic3—62no—
322Dic3—121no—
323C32—21no—
324C32—21no—
325C32—41no—
326C32—41no—
327C32C3×#39511yescharacteristic, complemented
328C23—91no—
329Q8—64no—
330Q8—184no—
331C2×C4—61no—
332C2×C4—62no—
333C2×C4—61no—
334C2×C4—94no—
335C2×C4—92no—
336C2×C4—94no—
337C2×C4—91no—
338C2×C4—182no—
339C2×C4—182no—
340C2×C4—181no—
341C2×C4—182no—
342C2×C4—182no—
343C2×C4—182no—
344C2×C4C3×GO+(4,𝔽2)11yescharacteristic
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—92no—
358C2×C3—92no—
359C2×C3D32⋊Q811yescharacteristic
360C2×C3#6422211yescharacteristic
361C2×C3#6424111yescharacteristic
362D3—62no—
363D3—62no—
364D3—62no—
365D3—62no—
366C22—92no—
367C22—91no—
368C22—91no—
369C22—92no—
370C22C2×C3×GO+(4,𝔽2)11yescharacteristic
371C4—21no—
372C4—64no—
373C4—62no—
374C4—121no—
375C4—184no—
376C4—181no—
377C3—21no—
378C3—21no—
379C3—41no—
380C3—41no—
381C3#1059341111yescharacteristic, direct product factor
382C2—92no—
383C2—92no—
384C2C3×D32⋊Q811yescharacteristic
385C2C3×#6422211yescharacteristic
386C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#395
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
6. #10593411C3direct (N × H)
28. C2×C3×#157Q8semidirect
41. C3×#624Q8semidirect
92. C2×#157C3×Q8semidirect
131. #624C3×Q8semidirect
207. C33#395semidirect
327. C32C3×#395semidirect
381. C3#10593411direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C2×C3×#37732
C3×#638832
C3×#637812
C3×#637372
#105934113
C3×#3959