Finite Group Databasebeta

← C3×#10591417 #13158424

Subgroups — C3×#10591417 #13158424

Order 1728.

704 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 120 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×#64147C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic, complemented
3C32×#832C212yesmaximal, complemented
4C3×#63575C212yesmaximal, complemented
5C3×#63557C211yesmaximal, characteristic, complemented
6C3×#4774—32nomaximal
7#10591417C311yesmaximal, characteristic, direct product factor
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—21no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×#624—21no—
12C3×#3757—21no—
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17#64147C2×C311yescharacteristic
18C3×#938—32no—
19C3×#938—61no—
20C3×D3×C4⋊C4—32no—
21#63900C2×C311yescharacteristic, complemented
22C32×#102—32no—
23C3×#832—22no—
24C3×#832D312yescomplemented
25C3×#832C2×C312yescomplemented
26C3×(C2×D3)⋊C8—32no—
27C3×#818—32no—
28C3×#807—32no—
29#63575C2×C312yescomplemented
30#63557C2×C311yescharacteristic, complemented
31C23×C33—21no—
32C22×C3×C3⋊D3D411yescharacteristic
33C33×D4—22no—
34C2×C33×C4C2311yescharacteristic
35C3×#638—42no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×C4×C3⋊D3—22no—
39C2×C32×Dic3—22no—
40C3×#624—42no—
41C32×C3⋊C8—22no—
42C3×#480—91no—
43#4774—32no—
44C2×C32×D4—22no—
45C2×C32×D4—21no—
46C2×C32×D4—41no—
47C2×C32×D4C2×D312yes—
48C2×C32×D4C22×C311yescharacteristic
49C2×#638—21no—
50C2×C4×C3⋊D3C22×C311yescharacteristic
51C2×C3×#67—62no—
52C2×C3×#67—62no—
53C2×C3×C4×D3—32no—
54C2×C3×C4×D3—62no—
55C2×C3×C4×D3—61no—
56C2×#624—21no—
57C2×C32×C8—32no—
58C32×C4⋊C4—32no—
59#3757—21no—
60C4⋊(C3⋊Dic3)C22×C311yescharacteristic
61C3×#217—62no—
62C3×#217—62no—
63C3×C4⋊Dic3—22no—
64C3×C4⋊Dic3—32no—
65C3×C4⋊Dic3—61no—
66C3×C4⋊Dic3C2×D312yes—
67C3×C4⋊Dic3C22×C312yes—
68C3×Dic3⋊C4—62no—
69C2×C3×C3⋊C8—22no—
70C2×C3×C3⋊C8C2×D312yes—
71C2×C3×C3⋊C8C22×C312yes—
72#3724—21no—
73C22×C33—41no—
74C22×C33—41no—
75C22×C33—41no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3D812yes—
80C33×C4#4112yes—
81C3×C3⋊Dic3—21no—
82C3×C3⋊Dic3—41no—
83C32×Dic3—42no—
84C3×#121—91no—
85C2×C3×C4⋊C4—91no—
86#938—32no—
87#938—61no—
88D3×C4⋊C4—32no—
89C3×#107—92no—
90C3×#102—32no—
91C3×#102—32no—
92C3×#102—62no—
93C3×C22⋊C8—91no—
94#832C3×D312yescomplemented
95(C2×D3)⋊C8—32no—
96#818—32no—
97#807—32no—
98C23×C32—21no—
99C23×C32—22no—
100C23×C32—22no—
101C23×C32—42no—
102C23×C32—42no—
103C22×C3⋊D3C3×D411yescharacteristic
104C22×C3×D3—32no—
105C22×C3×D3—61no—
106C32×D4—22no—
107C32×D4—24no—
108C32×D4—22no—
109C32×D4—22no—
110C32×D4—42no—
111C32×D4—42no—
112C32×D4—44no—
113C2×C32×C4—21no—
114C2×C32×C4—22no—
115C2×C32×C4—41no—
116C2×C32×C4—62no—
117C2×C32×C4C22×D312yes—
118C2×C32×C4C23×C311yescharacteristic
119#638—42no—
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3C3×D411yescharacteristic
122C4×C3⋊D3—22no—
123C3×#67—124no—
124C3×#67—124no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—42no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C3×C4×D3—62no—
133C3×C4×D3—62no—
134C3×C4×D3—122no—
135C3×C4×D3—124no—
136#624—42no—
137C32×C8—62no—
138C3×C3⋊C8—22no—
139C3×C3⋊C8—22no—
140C3×C3⋊C8—42no—
141#480—91noSylow
142C2×C33—81no—
143C2×C33#12311yescharacteristic
144C2×C33C2×D811yescharacteristic
145C2×C33#13711yescharacteristic
146C3×C3⋊D3—22no—
147C2×C3×D4—22no—
148C2×C3×D4—21no—
149C2×C3×D4—41no—
150C2×C3×D4—181no—
151C2×C3×D4D3211yescharacteristic
152C2×C3×D4C2×C3×D312yes—
153C22×C3×C4—91no—
154C22×C3×C4—181no—
155C2×#67—62no—
156C2×#67—62no—
157C2×C4×D3—32no—
158C2×C4×D3—61no—
159C2×C4×D3—62no—
160C2×C3×C8—32no—
161C2×C3×C8—32no—
162C2×C3×C8—62no—
163C3×C4⋊C4—32no—
164C3×C4⋊C4—32no—
165C3×C4⋊C4—62no—
166C3×C4⋊C4—91no—
167C3×C4⋊C4—181no—
168C3×C22⋊C4—181no—
169#217—62no—
170#217—62no—
171C4⋊Dic3—32no—
172C4⋊Dic3—61no—
173C4⋊Dic3C2×C3×D312yes—
174Dic3⋊C4—62no—
175C2×C3⋊C8C2×C3×D312yes—
176C22×C32—21no—
177C22×C32—22no—
178C22×C32—41no—
179C22×C32—44no—
180C22×C32—42no—
181C22×C32—44no—
182C22×C32—42no—
183C22×C32—41no—
184C22×C32—41no—
185C22×C32—41no—
186C22×C32—42no—
187C22×C32—42no—
188C22×C32—42no—
189C22×C32—42no—
190C22×C32—82no—
191C22×C32—82no—
192C22×C32—82no—
193C22×C32—82no—
194C22×C32D3×D412yes—
195C22×C32C2×C3×D411yescharacteristic
196C2×C3⋊D3—21no—
197C2×C3⋊D3—21no—
198C2×C3⋊D3C3×D812yes—
199C2×C3×D3—34no—
200C2×C3×D3—62no—
201C2×C3×D3—62no—
202C2×C3×D3—62no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C32×C4—22no—
206C32×C4—22no—
207C32×C4—22no—
208C32×C4—42no—
209C32×C4—122no—
210C32×C4#23512yes—
211C32×C4#23712yes—
212C32×C4C3×#4112yes—
213C3⋊Dic3—21no—
214C3⋊Dic3—41no—
215C3×Dic3—42no—
216C3×Dic3—42no—
217C3×Dic3—62no—
218C3×Dic3—82no—
219C3×Dic3—121no—
220C3×Dic3—122no—
221C3×Dic3—122no—
222#121—91no—
223C2×C4⋊C4—91no—
224#107—92no—
225#102—32no—
226C22⋊C8—91no—
227C33#48011yescharacteristic, complemented, Sylow
228C23×C3—22no—
229C23×C3—21no—
230C23×C3—22no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—91no—
234C22×D3—32no—
235C22×D3—61no—
236C3×D4—24no—
237C3×D4—22no—
238C3×D4—22no—
239C3×D4—22no—
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—44no—
243C3×D4—362no—
244C2×C3×C4—22no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—62no—
248C2×C3×C4—62no—
249C2×C3×C4—91no—
250C2×C3×C4—122no—
251C2×C3×C4—182no—
252C2×C3×C4—181no—
253C2×C3×C4—362no—
254C2×C3×C4C2×D3211yescharacteristic
255C2×C3×C4C22×C3×D312yes—
256#67—124no—
257#67—124no—
258C2×Dic3—22no—
259C2×Dic3—32no—
260C2×Dic3—61no—
261C2×Dic3—62no—
262C2×Dic3—62no—
263C4×D3—62no—
264C4×D3—62no—
265C4×D3—122no—
266C4×D3—124no—
267C3×C8—62no—
268C3×C8—62no—
269C3×C8—122no—
270C3⋊C8—22no—
271C2×C32—22no—
272C2×C32—22no—
273C2×C32—22no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—82no—
281C2×C32—84no—
282C2×C32—84no—
283C2×C32—82no—
284C2×C32—81no—
285C2×C32#88312yes—
286C2×C32D3×D812yes—
287C2×C32#91512yes—
288C2×C32C3×#12311yescharacteristic
289C2×C32C2×C3×D811yescharacteristic
290C2×C32C3×#13711yescharacteristic
291C3⋊D3—22no—
292C3×D3—64no—
293C3×D3—122no—
294C2×D4—181no—
295C2×D4C3×D3211yescharacteristic
296C22×C4—91no—
297C22×C4—181no—
298C2×C8—32no—
299C4⋊C4—32no—
300C4⋊C4—91no—
301C4⋊C4—181no—
302C22⋊C4—181no—
303C22×C3—21no—
304C22×C3—22no—
305C22×C3—42no—
306C22×C3—44no—
307C22×C3—42no—
308C22×C3—42no—
309C22×C3—44no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—82no—
318C22×C3—82no—
319C22×C3—82no—
320C22×C3—82no—
321C22×C3—92no—
322C22×C3—181no—
323C22×C3—181no—
324C22×C3#381111yescharacteristic
325C22×C3C3×D3×D412yes—
326C2×D3—34no—
327C2×D3—62no—
328C2×D3—62no—
329C2×D3—62no—
330C2×D3—121no—
331C2×D3—121no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—42no—
336C3×C4—122no—
337C3×C4—122no—
338C3×C4—181no—
339C3×C4—242no—
340C3×C4—361no—
341C3×C4#380412yes—
342C3×C4C3×#23512yes—
343C3×C4C3×#23712yes—
344Dic3—42no—
345Dic3—62no—
346Dic3—121no—
347Dic3—122no—
348Dic3—122no—
349C32—22no—
350C32—21no—
351C32—41no—
352C32#477412yescomplemented
353C32C3×#48011yescharacteristic, complemented
354C23—21no—
355C23—91no—
356D4—22no—
357D4—362no—
358C2×C4—62no—
359C2×C4—91no—
360C2×C4—182no—
361C2×C4—181no—
362C2×C4—362no—
363C2×C4C2×C3×D3211yescharacteristic
364C8—62no—
365C2×C3—21no—
366C2×C3—22no—
367C2×C3—21no—
368C2×C3—22no—
369C2×C3—21no—
370C2×C3—22no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—82no—
375C2×C3—84no—
376C2×C3—84no—
377C2×C3—82no—
378C2×C3—81no—
379C2×C3—182no—
380C2×C3#6392611yescharacteristic
381C2×C3#6393111yescharacteristic
382C2×C3#6397511yescharacteristic
383C2×C3C3×#88312yes—
384C2×C3C3×D3×D812yes—
385C2×C3C3×#91512yes—
386D3—64no—
387D3—122no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—92no—
392C22—181no—
393C22—181no—
394C22C3×#381111yescharacteristic
395C4—122no—
396C4—181no—
397C4—361no—
398C4C3×#380412yes—
399C3—21no—
400C3—22no—
401C3—41no—
402C3#1059141711yescharacteristic, direct product factor
403C3C3×#477412yescomplemented
404C2—81no—
405C2—182no—
406C2C3×#6392611yescharacteristic
407C2C3×#6393111yescharacteristic
408C2C3×#6397511yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64147C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#480
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#63900C2semidirect
3. C32×#832C2semidirect
4. C3×#63575C2semidirect
5. C3×#63557C2semidirect
7. #10591417C3direct (N × H)
21. #63900C2×C3semidirect
24. C3×#832D3semidirect
25. C3×#832C2×C3semidirect
29. #63575C2×C3semidirect
30. #63557C2×C3semidirect
94. #832C3×D3semidirect
227. C33#480semidirect
352. C32#4774semidirect
353. C32C3×#480semidirect
402. C3#10591417direct (N × H)
403. C3C3×#4774semidirect

Maximal subgroups

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

subgroupindex
C3×#641472
C3×#639002
C32×#8322
C3×#635752
C3×#635572
C3×#47743
#105914173