Finite Group Databasebeta

← C3×#10593408 #13166769

Subgroups — C3×#10593408 #13166769

Order 1728.

656 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3773C212yesmaximal, complemented
2C2×C3×D32⋊C4C212yesmaximal
3C3×#63921C211yesmaximal, characteristic
4C3×#63900C211yesmaximal, characteristic, complemented
5C3×#63781C211yesmaximal, characteristic, complemented
6#10593408C311yesmaximal, characteristic, direct product factor
7C22×C3×D32C2211yescharacteristic
8C22×C3×#157C2212yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#624C2211yescharacteristic
11C2×C3×#624C2212yes—
12C3×#3773—24no—
13C3×D32⋊C4—24no—
14C32×(C2×D3)⋊C4—21no—
15C32×C4⋊Dic3—21no—
16C3×#3724—21no—
17C3×(C2×D3)⋊Dic3—21no—
18C2×#3773C2×C312yescomplemented
19C2×D32⋊C4C2×C312yes—
20C3×D3×C4⋊C4—61no—
21C3×D3×C22⋊C4—61no—
22#63921C2×C311yescharacteristic
23#63900C2×C311yescharacteristic, complemented
24#63781C2×C311yescharacteristic, complemented
25C22×C3×C3⋊D3C2311yescharacteristic
26C22×C32×D3—21no—
27C2×C3×D32—22no—
28C2×C3×D32—22no—
29C2×C3×D32—21no—
30C2×C3×D32Q812yes—
31C2×C3×#157—22no—
32C2×C3×#157—22no—
33C2×C3×#157D414yes—
34C2×C33×C4D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×C4×C3⋊D3—22no—
37C2×C32×Dic3—21no—
38C2×C32×Dic3—22no—
39C3×#624—22no—
40C3×#624—22no—
41C3×#624—22no—
42C3×#399—91nomaximal
43C23×C3×D3—61no—
44C22×D32C22×C311yescharacteristic
45C22×#157C22×C312yes—
46C2×C4×C3⋊D3C22×C311yescharacteristic
47C2×C3×C4×D3—61no—
48C2×C3×C4×D3—61no—
49C2×C3×C4×D3—62no—
50C2×C3×C4×D3—61no—
51C2×#624C22×C312yes—
52C2×#624C22×C311yescharacteristic
53#3773—24no—
54D32⋊C4—24no—
55C32×C4⋊C4—61no—
56C32×C22⋊C4—61no—
57C3×#217—61no—
58C3×(C2×D3)⋊C4—21no—
59C3×(C2×D3)⋊C4—21no—
60C3×(C2×D3)⋊C4—41no—
61C3×(C2×D3)⋊C4—61no—
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3—41no—
65C3×Dic3⋊C4—62no—
66#3724—21no—
67(C2×D3)⋊Dic3—21no—
68C22×C33C2×D411yescharacteristic
69C2×C3×C3⋊D3C2×D412yes—
70C2×C3×C3⋊D3C2×Q811yescharacteristic
71C2×C3×C3⋊D3#4112yes—
72C2×C3×C3⋊D3#4111yescharacteristic
73C2×C32×D3—22no—
74C2×C32×D3—41no—
75C2×C32×D3—41no—
76C3×D32—24no—
77C3×D32—42no—
78C3×#157—24no—
79C33×C4—21no—
80C3×C3⋊Dic3—21no—
81C32×Dic3—42no—
82C32×Dic3—41no—
83C2×C3×C4⋊C4—91no—
84C2×C3×C4⋊C4—92no—
85C2×C3×C22⋊C4—91no—
86C2×C3×C22⋊C4—92no—
87D3×C4⋊C4—61no—
88D3×C22⋊C4—61no—
89C3×#95—91no—
90C23×C32—61no—
91C22×C3⋊D3C23×C311yescharacteristic
92C22×C3×D3—21no—
93C22×C3×D3—21no—
94C22×C3×D3—41no—
95C22×C3×D3—62no—
96C22×C3×D3—61no—
97C22×C3×D3—61no—
98C22×C3×D3—62no—
99C22×C3×D3—122no—
100C22×C3×D3—122no—
101C2×D32—21no—
102C2×D32—22no—
103C2×D32—22no—
104C2×D32C3×Q812yes—
105C2×#157—22no—
106C2×#157—22no—
107C2×#157C3×D414yes—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—41no—
111C2×C32×C4—41no—
112C2×C32×C4—62no—
113C2×C32×C4—61no—
114C2×C32×C4C3×D411yescharacteristic
115C2×C3⋊Dic3C3×D411yescharacteristic
116C4×C3⋊D3—22no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—42no—
122C2×C3×Dic3—41no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C3×C4×D3—64no—
126C3×C4×D3—122no—
127C3×C4×D3—122no—
128C3×C4×D3—122no—
129C3×C4×D3—122no—
130#624—22no—
131#624—22no—
132#624—22no—
133#399—91noSylow
134C2×C33C22⋊Q811yescharacteristic
135C2×C33#12311yescharacteristic
136C2×C33#12711yescharacteristic
137C3×C3⋊D3C22⋊Q812yes—
138C3×C3⋊D3#12312yes—
139C32×D3—42no—
140C24×C3—91no—
141C23×D3—61no—
142C22×C3×C4—91no—
143C22×C3×C4—92no—
144C22×C3×C4—92no—
145C22×C3×C4—91no—
146C2×C4×D3—61no—
147C2×C4×D3—61no—
148C2×C4×D3—62no—
149C2×C4×D3—61no—
150C3×C4⋊C4—61no—
151C3×C4⋊C4—61no—
152C3×C4⋊C4—121no—
153C3×C4⋊C4—181no—
154C3×C4⋊C4—184no—
155C3×C22⋊C4—61no—
156C3×C22⋊C4—61no—
157C3×C22⋊C4—121no—
158C3×C22⋊C4—181no—
159C3×C22⋊C4—184no—
160#217—61no—
161(C2×D3)⋊C4—21no—
162(C2×D3)⋊C4—61no—
163C4⋊Dic3—21no—
164Dic3⋊C4—62no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—41no—
168C22×C32—41no—
169C22×C32—62no—
170C22×C32—121no—
171C22×C32—121no—
172C22×C32C2×C3×D411yescharacteristic
173C2×C3⋊D3C2×C3×D412yes—
174C2×C3⋊D3C2×C3×Q811yescharacteristic
175C2×C3⋊D3C3×#4112yes—
176C2×C3⋊D3C3×#4111yescharacteristic
177C2×C3×D3—22no—
178C2×C3×D3—22no—
179C2×C3×D3—42no—
180C2×C3×D3—41no—
181C2×C3×D3—41no—
182C2×C3×D3—41no—
183C2×C3×D3—41no—
184C2×C3×D3—62no—
185C2×C3×D3—62no—
186C2×C3×D3—61no—
187C2×C3×D3—61no—
188C2×C3×D3—62no—
189C2×C3×D3—62no—
190C2×C3×D3—61no—
191C2×C3×D3—61no—
192C2×C3×D3—81no—
193C2×C3×D3—81no—
194C2×C3×D3—124no—
195C2×C3×D3—124no—
196D32—24no—
197D32—42no—
198#157—24no—
199C32×C4—22no—
200C32×C4—21no—
201C32×C4—41no—
202C32×C4—42no—
203C32×C4—81no—
204C32×C4—122no—
205C32×C4—121no—
206C3⋊Dic3—21no—
207C3×Dic3—42no—
208C3×Dic3—41no—
209C3×Dic3—42no—
210C3×Dic3—41no—
211C3×Dic3—62no—
212C3×Dic3—82no—
213C3×Dic3—81no—
214C3×Dic3—121no—
215C2×C4⋊C4—91no—
216C2×C4⋊C4—92no—
217C2×C22⋊C4—91no—
218C2×C22⋊C4—92no—
219#95—91no—
220C33#39911yescharacteristic, complemented, Sylow
221C23×C3—61no—
222C23×C3—61no—
223C23×C3—92no—
224C23×C3—91no—
225C23×C3—121no—
226C23×C3—181no—
227C23×C3—182no—
228C23×C3—182no—
229C22×D3—21no—
230C22×D3—62no—
231C22×D3—61no—
232C22×D3—61no—
233C22×D3—62no—
234C22×D3—122no—
235C22×D3—122no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—62no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—62no—
244C2×C3×C4—94no—
245C2×C3×C4—91no—
246C2×C3×C4—121no—
247C2×C3×C4—122no—
248C2×C3×C4—182no—
249C2×C3×C4—182no—
250C2×C3×C4—182no—
251C2×C3×C4—182no—
252C2×C3×C4—182no—
253C2×C3×C4—182no—
254C2×C3×C4GO+(4,𝔽2)11yescharacteristic
255C2×Dic3—22no—
256C2×Dic3—21no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C4×D3—64no—
260C4×D3—122no—
261C4×D3—122no—
262C4×D3—122no—
263C4×D3—122no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—41no—
276C2×C32—122no—
277C2×C32C3×C22⋊Q811yescharacteristic
278C2×C32C3×#12311yescharacteristic
279C2×C32C3×#12711yescharacteristic
280C3⋊D3C3×C22⋊Q812yes—
281C3⋊D3C3×#12312yes—
282C3×D3—42no—
283C3×D3—42no—
284C3×D3—62no—
285C3×D3—62no—
286C3×D3—62no—
287C3×D3—62no—
288C3×D3—82no—
289C24—91no—
290C22×C4—91no—
291C22×C4—92no—
292C22×C4—92no—
293C22×C4—91no—
294C4⋊C4—61no—
295C4⋊C4—181no—
296C4⋊C4—184no—
297C22⋊C4—61no—
298C22⋊C4—181no—
299C22⋊C4—184no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—62no—
305C22×C3—62no—
306C22×C3—91no—
307C22×C3—91no—
308C22×C3—92no—
309C22×C3—92no—
310C22×C3—122no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—121no—
314C22×C3—121no—
315C22×C3—184no—
316C22×C3—241no—
317C22×C3—241no—
318C22×C3—362no—
319C22×C3C2×GO+(4,𝔽2)11yescharacteristic
320C2×D3—22no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—62no—
324C2×D3—62no—
325C2×D3—62no—
326C2×D3—62no—
327C2×D3—61no—
328C2×D3—61no—
329C2×D3—61no—
330C2×D3—61no—
331C2×D3—124no—
332C2×D3—124no—
333C3×C4—22no—
334C3×C4—21no—
335C3×C4—41no—
336C3×C4—42no—
337C3×C4—81no—
338C3×C4—122no—
339C3×C4—121no—
340C3×C4—121no—
341C3×C4—122no—
342C3×C4—184no—
343C3×C4—181no—
344C3×C4—242no—
345C3×C4—241no—
346Dic3—42no—
347Dic3—41no—
348Dic3—62no—
349Dic3—121no—
350C32—21no—
351C32—21no—
352C32—41no—
353C32—41no—
354C32C3×#39911yescharacteristic, complemented
355C23—61no—
356C23—92no—
357C23—91no—
358C23—181no—
359C23—182no—
360C23—182no—
361C2×C4—61no—
362C2×C4—62no—
363C2×C4—94no—
364C2×C4—91no—
365C2×C4—182no—
366C2×C4—182no—
367C2×C4—182no—
368C2×C4—182no—
369C2×C4—182no—
370C2×C4—182no—
371C2×C4C3×GO+(4,𝔽2)11yescharacteristic
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—92no—
385C2×C3—92no—
386C2×C3—122no—
387C2×C3—122no—
388C2×C3—242no—
389C2×C3D32⋊Q811yescharacteristic
390C2×C3#6423111yescharacteristic
391C2×C3#6423311yescharacteristic
392D3—42no—
393D3—62no—
394D3—62no—
395D3—62no—
396D3—62no—
397C22—62no—
398C22—92no—
399C22—91no—
400C22—91no—
401C22—92no—
402C22—121no—
403C22—121no—
404C22—184no—
405C22—362no—
406C22C2×C3×GO+(4,𝔽2)11yescharacteristic
407C4—21no—
408C4—121no—
409C4—122no—
410C4—184no—
411C4—181no—
412C3—21no—
413C3—21no—
414C3—41no—
415C3—41no—
416C3#1059340811yescharacteristic, direct product factor
417C2—92no—
418C2—92no—
419C2—122no—
420C2C3×D32⋊Q811yescharacteristic
421C2C3×#6423111yescharacteristic
422C2C3×#6423311yescharacteristic
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×D32C22
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#399
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
1. C2×C3×#3773C2semidirect
4. C3×#63900C2semidirect
5. C3×#63781C2semidirect
6. #10593408C3direct (N × H)
18. C2×#3773C2×C3semidirect
23. #63900C2×C3semidirect
24. #63781C2×C3semidirect
220. C33#399semidirect
354. C32C3×#399semidirect
416. C3#10593408direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37732
C2×C3×D32⋊C42
C3×#639212
C3×#639002
C3×#637812
#105934083
C3×#3999