Finite Group Databasebeta

← C3×#10593404 #13166765

Subgroups — C3×#10593404 #13166765

Order 1728.

624 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 82 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×D32⋊C4C212yesmaximal
2C3×#63921C211yesmaximal, characteristic
3C3×#63900C211yesmaximal, characteristic, complemented
4C3×#63781C211yesmaximal, characteristic, complemented
5C3×#63737C212yesmaximal, complemented
6#10593404C311yesmaximal, characteristic, direct product factor
7C22×C3×D32C2211yescharacteristic
8C22×C3×#157C2212yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#624C2212yes—
11C2×C3×#624C2211yescharacteristic
12C3×D32⋊C4—24no—
13C32×(C2×D3)⋊C4—21no—
14C32×C4⋊Dic3—21no—
15C3×#3724—21no—
16C3×(C2×D3)⋊Dic3—21no—
17C2×D32⋊C4C2×C312yes—
18C3×D3×C4⋊C4—61no—
19C3×D3×C22⋊C4—61no—
20#63921C2×C311yescharacteristic
21#63900C2×C311yescharacteristic, complemented
22#63781C2×C311yescharacteristic, complemented
23#63737C2×C312yescomplemented
24C22×C3×C3⋊D3C2311yescharacteristic
25C22×C32×D3—21no—
26C2×C3×D32—22no—
27C2×C3×D32—22no—
28C2×C3×D32—21no—
29C2×C3×D32D412yes—
30C2×C3×#157—22no—
31C2×C3×#157—22no—
32C2×C3×#157—22no—
33C2×C33×C4D411yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C3×C4×C3⋊D3—22no—
36C2×C32×Dic3—21no—
37C2×C32×Dic3—22no—
38C3×#624—22no—
39C3×#624—22no—
40C3×#624—22no—
41C3×#397—91nomaximal
42C23×C3×D3—61no—
43C22×D32C22×C311yescharacteristic
44C22×#157C22×C312yes—
45C2×C4×C3⋊D3C22×C311yescharacteristic
46C2×C3×C4×D3—61no—
47C2×C3×C4×D3—61no—
48C2×C3×C4×D3—62no—
49C2×C3×C4×D3—61no—
50C2×#624C22×C312yes—
51C2×#624C22×C311yescharacteristic
52D32⋊C4—24no—
53C32×C4⋊C4—61no—
54C32×C22⋊C4—61no—
55C3×#217—61no—
56C3×(C2×D3)⋊C4—21no—
57C3×(C2×D3)⋊C4—21no—
58C3×(C2×D3)⋊C4—41no—
59C3×(C2×D3)⋊C4—61no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3—41no—
63C3×Dic3⋊C4—62no—
64#3724—21no—
65(C2×D3)⋊Dic3—21no—
66C22×C33C2×D411yescharacteristic
67C2×C3×C3⋊D3C2×D411yescharacteristic
68C2×C3×C3⋊D3#4112yes—
69C2×C3×C3⋊D3#4112yes—
70C2×C3×C3⋊D3#4111yescharacteristic
71C2×C32×D3—22no—
72C2×C32×D3—41no—
73C2×C32×D3—41no—
74C3×D32—24no—
75C3×D32—42no—
76C3×#157—42no—
77C33×C4—21no—
78C3×C3⋊Dic3—21no—
79C32×Dic3—42no—
80C32×Dic3—41no—
81C2×C3×C4⋊C4—91no—
82C2×C3×C22⋊C4—91no—
83C2×C3×C22⋊C4—92no—
84D3×C4⋊C4—61no—
85D3×C22⋊C4—61no—
86C3×#95—91no—
87C3×#95—92no—
88C23×C32—61no—
89C22×C3⋊D3C23×C311yescharacteristic
90C22×C3×D3—21no—
91C22×C3×D3—21no—
92C22×C3×D3—41no—
93C22×C3×D3—62no—
94C22×C3×D3—61no—
95C22×C3×D3—61no—
96C22×C3×D3—62no—
97C22×C3×D3—122no—
98C22×C3×D3—122no—
99C2×D32—21no—
100C2×D32—22no—
101C2×D32—22no—
102C2×D32C3×D412yes—
103C2×#157—22no—
104C2×#157—22no—
105C2×#157—22no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4—62no—
111C2×C32×C4—61no—
112C2×C32×C4C3×D411yescharacteristic
113C2×C3⋊Dic3C3×D411yescharacteristic
114C4×C3⋊D3—22no—
115C2×C3×Dic3—22no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—42no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C3×C4×D3—64no—
124C3×C4×D3—122no—
125C3×C4×D3—122no—
126C3×C4×D3—122no—
127C3×C4×D3—122no—
128#624—22no—
129#624—22no—
130#624—22no—
131#397—91noSylow
132C2×C33#12111yescharacteristic
133C2×C33#12311yescharacteristic
134C2×C33#12411yescharacteristic
135C3×C3⋊D3#12312yes—
136C3×C3⋊D3#12612yes—
137C32×D3—42no—
138C24×C3—91no—
139C23×D3—61no—
140C22×C3×C4—91no—
141C22×C3×C4—92no—
142C22×C3×C4—91no—
143C22×C3×C4—92no—
144C2×C4×D3—61no—
145C2×C4×D3—61no—
146C2×C4×D3—62no—
147C2×C4×D3—61no—
148C3×C4⋊C4—61no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4—121no—
151C3×C4⋊C4—181no—
152C3×C22⋊C4—61no—
153C3×C22⋊C4—61no—
154C3×C22⋊C4—121no—
155C3×C22⋊C4—181no—
156C3×C22⋊C4—184no—
157#217—61no—
158(C2×D3)⋊C4—21no—
159(C2×D3)⋊C4—61no—
160C4⋊Dic3—21no—
161Dic3⋊C4—62no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—41no—
165C22×C32—41no—
166C22×C32—62no—
167C22×C32—121no—
168C22×C32—121no—
169C22×C32C2×C3×D411yescharacteristic
170C2×C3⋊D3C2×C3×D411yescharacteristic
171C2×C3⋊D3C3×#4112yes—
172C2×C3⋊D3C3×#4112yes—
173C2×C3⋊D3C3×#4111yescharacteristic
174C2×C3×D3—22no—
175C2×C3×D3—22no—
176C2×C3×D3—42no—
177C2×C3×D3—41no—
178C2×C3×D3—41no—
179C2×C3×D3—41no—
180C2×C3×D3—41no—
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—62no—
184C2×C3×D3—61no—
185C2×C3×D3—62no—
186C2×C3×D3—62no—
187C2×C3×D3—62no—
188C2×C3×D3—61no—
189C2×C3×D3—81no—
190C2×C3×D3—81no—
191C2×C3×D3—124no—
192C2×C3×D3—124no—
193D32—24no—
194D32—42no—
195#157—42no—
196C32×C4—22no—
197C32×C4—21no—
198C32×C4—41no—
199C32×C4—42no—
200C32×C4—81no—
201C32×C4—121no—
202C32×C4—122no—
203C3⋊Dic3—21no—
204C3×Dic3—41no—
205C3×Dic3—42no—
206C3×Dic3—42no—
207C3×Dic3—41no—
208C3×Dic3—62no—
209C3×Dic3—82no—
210C3×Dic3—81no—
211C3×Dic3—121no—
212C2×C4⋊C4—91no—
213C2×C22⋊C4—91no—
214C2×C22⋊C4—92no—
215#95—91no—
216#95—92no—
217C33#39711yescharacteristic, complemented, Sylow
218C23×C3—61no—
219C23×C3—61no—
220C23×C3—92no—
221C23×C3—91no—
222C23×C3—121no—
223C23×C3—181no—
224C23×C3—182no—
225C23×C3—182no—
226C22×D3—21no—
227C22×D3—62no—
228C22×D3—61no—
229C22×D3—61no—
230C22×D3—62no—
231C22×D3—122no—
232C22×D3—122no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—62no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—62no—
241C2×C3×C4—91no—
242C2×C3×C4—121no—
243C2×C3×C4—122no—
244C2×C3×C4—182no—
245C2×C3×C4—182no—
246C2×C3×C4—182no—
247C2×C3×C4—182no—
248C2×C3×C4—182no—
249C2×C3×C4—182no—
250C2×C3×C4—182no—
251C2×C3×C4GO+(4,𝔽2)11yescharacteristic
252C2×Dic3—22no—
253C2×Dic3—21no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C4×D3—64no—
257C4×D3—122no—
258C4×D3—122no—
259C4×D3—122no—
260C4×D3—122no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—41no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32—122no—
274C2×C32C3×#12111yescharacteristic
275C2×C32C3×#12311yescharacteristic
276C2×C32C3×#12411yescharacteristic
277C3⋊D3C3×#12312yes—
278C3⋊D3C3×#12612yes—
279C3×D3—42no—
280C3×D3—42no—
281C3×D3—62no—
282C3×D3—62no—
283C3×D3—62no—
284C3×D3—62no—
285C3×D3—82no—
286C24—91no—
287C22×C4—91no—
288C22×C4—92no—
289C22×C4—91no—
290C22×C4—92no—
291C4⋊C4—61no—
292C4⋊C4—181no—
293C22⋊C4—61no—
294C22⋊C4—181no—
295C22⋊C4—184no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—62no—
301C22×C3—62no—
302C22×C3—92no—
303C22×C3—91no—
304C22×C3—91no—
305C22×C3—92no—
306C22×C3—122no—
307C22×C3—121no—
308C22×C3—121no—
309C22×C3—121no—
310C22×C3—121no—
311C22×C3—184no—
312C22×C3—241no—
313C22×C3—241no—
314C22×C3—362no—
315C22×C3C2×GO+(4,𝔽2)11yescharacteristic
316C2×D3—22no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—62no—
320C2×D3—61no—
321C2×D3—61no—
322C2×D3—62no—
323C2×D3—62no—
324C2×D3—61no—
325C2×D3—61no—
326C2×D3—62no—
327C2×D3—124no—
328C2×D3—124no—
329C3×C4—22no—
330C3×C4—21no—
331C3×C4—41no—
332C3×C4—42no—
333C3×C4—81no—
334C3×C4—121no—
335C3×C4—122no—
336C3×C4—121no—
337C3×C4—122no—
338C3×C4—181no—
339C3×C4—241no—
340C3×C4—242no—
341C3×C4—362no—
342Dic3—41no—
343Dic3—42no—
344Dic3—62no—
345Dic3—121no—
346C32—21no—
347C32—21no—
348C32—41no—
349C32—41no—
350C32C3×#39711yescharacteristic, complemented
351C23—61no—
352C23—92no—
353C23—91no—
354C23—181no—
355C23—182no—
356C23—182no—
357C2×C4—61no—
358C2×C4—62no—
359C2×C4—91no—
360C2×C4—182no—
361C2×C4—182no—
362C2×C4—182no—
363C2×C4—182no—
364C2×C4—182no—
365C2×C4—182no—
366C2×C4—182no—
367C2×C4C3×GO+(4,𝔽2)11yescharacteristic
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—92no—
381C2×C3—92no—
382C2×C3—122no—
383C2×C3—122no—
384C2×C3—242no—
385C2×C3#6422111yescharacteristic
386C2×C3#6423011yescharacteristic
387C2×C3#6423311yescharacteristic
388D3—42no—
389D3—62no—
390D3—62no—
391D3—62no—
392D3—62no—
393C22—62no—
394C22—92no—
395C22—92no—
396C22—91no—
397C22—91no—
398C22—121no—
399C22—121no—
400C22—184no—
401C22—362no—
402C22C2×C3×GO+(4,𝔽2)11yescharacteristic
403C4—21no—
404C4—121no—
405C4—122no—
406C4—181no—
407C4—362no—
408C3—21no—
409C3—21no—
410C3—41no—
411C3—41no—
412C3#1059340411yescharacteristic, direct product factor
413C2—92no—
414C2—92no—
415C2—122no—
416C2C3×#6422111yescharacteristic
417C2C3×#6423011yescharacteristic
418C2C3×#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×#397
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
3. C3×#63900C2semidirect
4. C3×#63781C2semidirect
5. C3×#63737C2semidirect
6. #10593404C3direct (N × H)
21. #63900C2×C3semidirect
22. #63781C2×C3semidirect
23. #63737C2×C3semidirect
217. C33#397semidirect
350. C32C3×#397semidirect
412. C3#10593404direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D32⋊C42
C3×#639212
C3×#639002
C3×#637812
C3×#637372
#105934043
C3×#3979