Finite Group Databasebeta

← C3×#10593494 #13166861

Subgroups — C3×#10593494 #13166861

Order 1728.

470 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C3×#63980C211yesmaximal, characteristic
2C3×#63957C211yesmaximal, characteristic, complemented
3C3×C22⋊#622C211yesmaximal, characteristic, complemented
4C3×#63743C211yesmaximal, characteristic, complemented
5C3×#63742C211yesmaximal, characteristic, complemented
6C3×#63739C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic, complemented
8#10593494C311yesmaximal, characteristic, direct product factor
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C32×#67—21no—
11C2×C3×#625C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C32×#217—21no—
15C3×#3724C2211yescharacteristic
16C3×#3724C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×(C2×D3)⋊Dic3—21no—
19C3×Dic32—21no—
20C3×#939—61no—
21C3×#877—61no—
22#63980C2×C311yescharacteristic
23#63957C2×C311yescharacteristic, complemented
24C22⋊#622C2×C311yescharacteristic, complemented
25#63743C2×C311yescharacteristic, complemented
26#63742C2×C311yescharacteristic, complemented
27#63739C2×C311yescharacteristic, complemented
28#63739C2×C311yescharacteristic, complemented
29C23×C33D411yescharacteristic
30C22×C32×D3—21no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C32×#67—42no—
36C2×C32×Dic3—21no—
37C2×C32×Dic3—21no—
38C2×C32×Dic3—21no—
39C3×#625—21no—
40C3×#625—21no—
41C3×#622—21no—
42C3×#622—21no—
43C3×#482—91nomaximal
44C2×C32×D4—61no—
45C22×C3⋊Dic3C22×C311yescharacteristic
46C2×C3×#67—21no—
47C2×C3×#67—21no—
48C2×C3×#67—41no—
49C2×C3×#67—61no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3—61no—
52C2×#625C22×C311yescharacteristic
53C2×#622C22×C311yescharacteristic
54C2×#622C22×C311yescharacteristic
55C32×C22⋊C4—61no—
56C3×#217—21no—
57C3×#217—21no—
58C3×#217—41no—
59C3×#217—61no—
60C3×(C2×D3)⋊C4—61no—
61C3×Dic3⋊C4—61no—
62C3×Dic3⋊C4—61no—
63C3×Dic3⋊C4—61no—
64C3×C4×Dic3—61no—
65C3×C4×Dic3—61no—
66#3724C22×C311yescharacteristic
67#3724C22×C311yescharacteristic
68#3724C22×C311yescharacteristic
69(C2×D3)⋊Dic3—21no—
70Dic32—21no—
71C22×C33—21no—
72C22×C33—21no—
73C22×C33—21no—
74C22×C33C2×D411yescharacteristic
75C2×C32×D3—41no—
76C2×C32×D3—41no—
77C2×C32×D3—41no—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3#4111yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81C32×Dic3—41no—
82C32×Dic3—41no—
83C32×Dic3—41no—
84C3×#121—91no—
85C3×#117—91no—
86#939—61no—
87#877—61no—
88C3×#107—91no—
89C3×#106—91no—
90C3×#102—91no—
91C3×#102—91no—
92C3×C22⋊C8—91no—
93C23×C32—21no—
94C23×C32—21no—
95C23×C32—41no—
96C23×C32—41no—
97C23×C32—61no—
98C23×C32C3×D411yescharacteristic
99C22×C3×D3—21no—
100C22×C3×D3—21no—
101C22×C3×D3—41no—
102C32×D4—122no—
103C2×C32×C4—61no—
104C2×C32×C4—61no—
105C2×C32×C4—61no—
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3C3×D411yescharacteristic
109C2×C3⋊Dic3C23×C311yescharacteristic
110C3×#67—42no—
111C3×#67—42no—
112C3×#67—44no—
113C3×#67—121no—
114C3×#67—121no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—41no—
123C2×C3×Dic3—41no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—121no—
131C2×C3×Dic3—121no—
132#625—21no—
133#625—21no—
134#622—21no—
135#622—21no—
136#482—91noSylow
137C2×C33—41no—
138C2×C33#12311yescharacteristic
139C2×C33#13511yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C32×D3—81no—
142C2×C3×D4—61no—
143C2×C3×D4—61no—
144C2×C3×D4—91no—
145C2×C3×D4—121no—
146C22×C3×C4—91no—
147C2×#67—21no—
148C2×#67—61no—
149C22×Dic3—61no—
150C22×Dic3—61no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C3×C4⋊C4—91no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—91no—
156C3×C22⋊C4—61no—
157C3×C22⋊C4—61no—
158C3×C22⋊C4—121no—
159C3×C22⋊C4—181no—
160C3×C42—181no—
161#217—21no—
162#217—61no—
163(C2×D3)⋊C4—61no—
164Dic3⋊C4—61no—
165Dic3⋊C4—61no—
166Dic3⋊C4—61no—
167C4×Dic3—61no—
168C4×Dic3—61no—
169C22×C32—22no—
170C22×C32—22no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32—41no—
181C22×C32—41no—
182C22×C32—42no—
183C22×C32—42no—
184C22×C32—81no—
185C22×C32—81no—
186C22×C32—81no—
187C22×C32—81no—
188C22×C32—121no—
189C22×C32—121no—
190C22×C32—121no—
191C22×C32C2×C3×D411yescharacteristic
192C2×C3×D3—41no—
193C2×C3×D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—42no—
196C2×C3×D3—41no—
197C2×C3×D3—41no—
198C2×C3×D3—41no—
199C2×C3×D3—81no—
200C2×C3×D3—81no—
201C32×C4—121no—
202C32×C4—121no—
203C32×C4—121no—
204C3⋊Dic3—21no—
205C3⋊Dic3C3×#4111yescharacteristic
206C3⋊Dic3C3×#4111yescharacteristic
207C3×Dic3—41no—
208C3×Dic3—41no—
209C3×Dic3—41no—
210C3×Dic3—42no—
211C3×Dic3—41no—
212C3×Dic3—41no—
213C3×Dic3—41no—
214C3×Dic3—61no—
215C3×Dic3—61no—
216C3×Dic3—61no—
217C3×Dic3—61no—
218C3×Dic3—62no—
219C3×Dic3—81no—
220C3×Dic3—81no—
221C3×Dic3—121no—
222#121—91no—
223#117—91no—
224#107—91no—
225#106—91no—
226#102—91no—
227#102—91no—
228C22⋊C8—91no—
229C33#48211yescharacteristic, complemented, Sylow
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—61no—
235C23×C3—61no—
236C23×C3—121no—
237C23×C3GO+(4,𝔽2)11yescharacteristic
238C22×D3—21no—
239C3×D4—122no—
240C3×D4—122no—
241C3×D4—124no—
242C3×D4—181no—
243C3×D4—181no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—91no—
251C2×C3×C4—91no—
252C2×C3×C4—121no—
253C2×C3×C4—121no—
254C2×C3×C4—121no—
255C2×C3×C4—181no—
256C2×C3×C4—181no—
257#67—42no—
258#67—121no—
259#67—121no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3—62no—
268C2×Dic3—62no—
269C2×Dic3—121no—
270C2×Dic3—121no—
271C3×C8—181no—
272C3×C8—181no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32—41no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—42no—
287C2×C32—42no—
288C2×C32—82no—
289C2×C32—82no—
290C2×C32—241no—
291C2×C32C3×#12311yescharacteristic
292C2×C32C3×#13511yescharacteristic
293C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
294C3×D3—81no—
295C3×D3—82no—
296C3×D3—81no—
297C2×D4—61no—
298C2×D4—91no—
299C22×C4—91no—
300C2×C8—91no—
301C2×C8—91no—
302C4⋊C4—91no—
303C4⋊C4—91no—
304C4⋊C4—91no—
305C22⋊C4—61no—
306C22⋊C4—181no—
307C42—181no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—42no—
320C22×C3—42no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3—121no—
330C22×C3—122no—
331C22×C3—121no—
332C22×C3—121no—
333C22×C3—121no—
334C22×C3—241no—
335C22×C3—241no—
336C22×C3C2×GO+(4,𝔽2)11yescharacteristic
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—41no—
340C3×C4—91no—
341C3×C4—91no—
342C3×C4—121no—
343C3×C4—121no—
344C3×C4—121no—
345C3×C4—122no—
346C3×C4—121no—
347C3×C4—121no—
348C3×C4—121no—
349C3×C4—181no—
350C3×C4—241no—
351C3×C4—241no—
352Dic3—41no—
353Dic3—41no—
354Dic3—41no—
355Dic3—61no—
356Dic3—61no—
357Dic3—61no—
358Dic3—61no—
359Dic3—62no—
360Dic3—121no—
361C32—21no—
362C32—21no—
363C32—41no—
364C32—41no—
365C32C3×#48211yescharacteristic, complemented
366C23—61no—
367C23C3×GO+(4,𝔽2)11yescharacteristic
368D4—122no—
369D4—181no—
370D4—181no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—91no—
375C2×C4—91no—
376C2×C4—181no—
377C2×C4—181no—
378C8—181no—
379C8—181no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—42no—
387C2×C3—42no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—41no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—82no—
396C2×C3—82no—
397C2×C3—241no—
398C2×C3—242no—
399C2×C3—241no—
400C2×C3#6423311yescharacteristic
401C2×C3#6423611yescharacteristic
402C2×C3#6423711yescharacteristic
403D3—81no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—121no—
408C22—121no—
409C22—121no—
410C22C2×C3×GO+(4,𝔽2)11yescharacteristic
411C4—91no—
412C4—91no—
413C4—121no—
414C4—121no—
415C4—121no—
416C4—181no—
417C3—21no—
418C3—21no—
419C3—41no—
420C3—41no—
421C3#1059349411yescharacteristic, direct product factor
422C2—41no—
423C2—241no—
424C2C3×#6423311yescharacteristic
425C2C3×#6423611yescharacteristic
426C2C3×#6423711yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC3×#63980C2
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#482
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C3×#63957C2semidirect
3. C3×C22⋊#622C2semidirect
4. C3×#63743C2semidirect
5. C3×#63742C2semidirect
6. C3×#63739C2semidirect
7. C3×#63739C2semidirect
8. #10593494C3direct (N × H)
23. #63957C2×C3semidirect
24. C22⋊#622C2×C3semidirect
25. #63743C2×C3semidirect
26. #63742C2×C3semidirect
27. #63739C2×C3semidirect
28. #63739C2×C3semidirect
229. C33#482semidirect
365. C32C3×#482semidirect
421. C3#10593494direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639802
C3×#639572
C3×C22⋊#6222
C3×#637432
C3×#637422
C3×#637392
#105934943
C3×#4829