Finite Group Databasebeta

← C3×#10593525 #13166891

Subgroups — C3×#10593525 #13166891

Order 1728.

674 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C22×C3×#627C211yesmaximal, characteristic, complemented
2C2×C3×#3775C212yesmaximal, complemented
3C3×#63980C211yesmaximal, characteristic
4C3×C22⋊#622C211yesmaximal, characteristic, complemented
5C3×#63740C212yesmaximal, complemented
6#10593525C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C32×#67—21no—
9C22×C32×Dic3—21no—
10C2×C3×#627—21no—
11C2×C3×#627—24no—
12C2×C3×#627C2212yes—
13C2×C3×#625C2211yescharacteristic
14C2×C3×#622C2212yes—
15C3×#3775—24no—
16C3×#3724C2211yescharacteristic
17C3×(C2×D3)⋊Dic3—21no—
18C22×C3×Dic6—61no—
19C22×#627C2×C311yescharacteristic, complemented
20C2×#3775C2×C312yescomplemented
21C3×#939—61no—
22#63980C2×C311yescharacteristic
23C22⋊#622C2×C311yescharacteristic, complemented
24#63740C2×C312yescomplemented
25C23×C33D411yescharacteristic
26C22×C32×D3—21no—
27C2×C3×C3⋊Dic3—22no—
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C32×#67—42no—
31C2×C32×Dic3—24no—
32C2×C32×Dic3—22no—
33C2×C32×Dic3—21no—
34C3×#627—24no—
35C3×#627—22no—
36C3×#627D414yescomplemented
37C3×#625—22no—
38C3×#622—22no—
39C3×#448—91nomaximal
40C2×C32×D4—61no—
41C22×C32×C4—61no—
42C22×C3⋊Dic3C22×C311yescharacteristic
43C2×C3×#67—21no—
44C2×C3×#67—21no—
45C2×C3×#67—41no—
46C2×C3×#67—61no—
47C22×C3×Dic3—21no—
48C22×C3×Dic3—21no—
49C22×C3×Dic3—41no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3—61no—
52C2×C3×Dic6—62no—
53C2×C3×Dic6—68no—
54C2×C3×Dic6—62no—
55C2×#627—21no—
56C2×#627—24no—
57C2×#627C22×C312yes—
58C2×#625C22×C311yescharacteristic
59C2×#622C22×C312yes—
60#3775—24no—
61C3×#217—61no—
62C3×(C2×D3)⋊C4—61no—
63C3×Dic3⋊C4—61no—
64#3724C22×C311yescharacteristic
65(C2×D3)⋊Dic3—21no—
66C22×C33—21no—
67C22×C33—21no—
68C22×C33SD1612yescomplemented
69C22×C33C2×D411yescharacteristic
70C2×C32×D3—41no—
71C2×C32×D3—41no—
72C2×C32×D3—41no—
73C3×C3⋊Dic3—21no—
74C3×C3⋊Dic3C2×D412yes—
75C32×Dic3—24no—
76C32×Dic3—41no—
77C22×C3×Q8—91no—
78C22×Dic6—61no—
79C2×C3×SD16—92no—
80C3×#121—91no—
81#939—61no—
82C3×#103—92no—
83C3×C22⋊C8—91no—
84C23×C32—21no—
85C23×C32—21no—
86C23×C32—41no—
87C23×C32—41no—
88C23×C32—61no—
89C23×C32C3×D411yescharacteristic
90C22×C3×D3—21no—
91C22×C3×D3—21no—
92C22×C3×D3—41no—
93C32×D4—122no—
94C2×C32×C4—64no—
95C2×C32×C4—62no—
96C2×C32×C4—61no—
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3C3×D411yescharacteristic
99C2×C3⋊Dic3C23×C311yescharacteristic
100C3×#67—42no—
101C3×#67—44no—
102C3×#67—42no—
103C3×#67—122no—
104C2×C3×Dic3—24no—
105C2×C3×Dic3—24no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—42no—
111C2×C3×Dic3—44no—
112C2×C3×Dic3—41no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—64no—
118C2×C3×Dic3—122no—
119C3×Dic6—64no—
120C3×Dic6—64no—
121C3×Dic6—68no—
122#627—24no—
123#627—22no—
124#627C3×D414yescomplemented
125#625—22no—
126#622—22no—
127#448—91noSylow
128C2×C33—22no—
129C2×C33#12011yescharacteristic
130C2×C33C2×SD1611yescharacteristic
131C2×C33#13711yescharacteristic
132C32×D3—81no—
133C2×C3×Q8—92no—
134C2×C3×Q8—181no—
135C2×C3×Q8—184no—
136C2×C3×D4—61no—
137C2×C3×D4—61no—
138C2×C3×D4—91no—
139C2×C3×D4—121no—
140C22×C3×C4—61no—
141C22×C3×C4—61no—
142C22×C3×C4—91no—
143C22×C3×C4—121no—
144C2×#67—21no—
145C2×#67—61no—
146C22×Dic3—21no—
147C22×Dic3—61no—
148C22×Dic3—61no—
149C2×Dic6—62no—
150C2×Dic6—62no—
151C2×Dic6—68no—
152C3×SD16—184no—
153C2×C3×C8—92no—
154C3×C4⋊C4—91no—
155C3×C22⋊C4—181no—
156#217—61no—
157(C2×D3)⋊C4—61no—
158Dic3⋊C4—61no—
159C22×C32—22no—
160C22×C32—22no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—21no—
165C22×C32—22no—
166C22×C32—22no—
167C22×C32—42no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32—42no—
171C22×C32—41no—
172C22×C32—41no—
173C22×C32—42no—
174C22×C32—42no—
175C22×C32—81no—
176C22×C32—81no—
177C22×C32—121no—
178C22×C32—121no—
179C22×C32—121no—
180C22×C32C3×SD1612yescomplemented
181C22×C32C2×C3×D411yescharacteristic
182C2×C3×D3—41no—
183C2×C3×D3—41no—
184C2×C3×D3—41no—
185C2×C3×D3—42no—
186C2×C3×D3—41no—
187C2×C3×D3—41no—
188C2×C3×D3—41no—
189C2×C3×D3—81no—
190C2×C3×D3—81no—
191C32×C4—64no—
192C32×C4—121no—
193C3⋊Dic3—21no—
194C3⋊Dic3C2×C3×D412yes—
195C3×Dic3—24no—
196C3×Dic3—24no—
197C3×Dic3—44no—
198C3×Dic3—41no—
199C3×Dic3—42no—
200C3×Dic3—41no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—121no—
205C22×Q8—91no—
206C2×SD16—92no—
207#121—91no—
208#103—92no—
209C22⋊C8—91no—
210C33#44811yescharacteristic, complemented, Sylow
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—61no—
216C23×C3—61no—
217C23×C3—121no—
218C23×C3GO+(4,𝔽2)11yescharacteristic
219C22×D3—21no—
220C3×Q8—94no—
221C3×Q8—182no—
222C3×Q8—184no—
223C3×D4—122no—
224C3×D4—124no—
225C3×D4—122no—
226C3×D4—182no—
227C2×C3×C4—64no—
228C2×C3×C4—62no—
229C2×C3×C4—64no—
230C2×C3×C4—62no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—91no—
234C2×C3×C4—91no—
235C2×C3×C4—124no—
236C2×C3×C4—122no—
237C2×C3×C4—121no—
238C2×C3×C4—182no—
239#67—42no—
240#67—122no—
241C2×Dic3—24no—
242C2×Dic3—22no—
243C2×Dic3—21no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—64no—
249C2×Dic3—122no—
250Dic6—64no—
251Dic6—64no—
252Dic6—68no—
253C3×C8—182no—
254C2×C32—24no—
255C2×C32—22no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—42no—
269C2×C32—44no—
270C2×C32—82no—
271C2×C32—241no—
272C2×C32C3×#12011yescharacteristic
273C2×C32C2×C3×SD1611yescharacteristic
274C2×C32C3×#13711yescharacteristic
275C3×D3—81no—
276C3×D3—82no—
277C3×D3—81no—
278C2×Q8—92no—
279C2×Q8—181no—
280C2×Q8—184no—
281C2×D4—61no—
282C2×D4—91no—
283C22×C4—61no—
284C22×C4—91no—
285SD16—184no—
286C2×C8—92no—
287C4⋊C4—91no—
288C22⋊C4—181no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—42no—
300C22×C3—42no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—42no—
304C22×C3—42no—
305C22×C3—81no—
306C22×C3—81no—
307C22×C3—121no—
308C22×C3—121no—
309C22×C3—121no—
310C22×C3—122no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—121no—
314C22×C3—241no—
315C22×C3—241no—
316C22×C3#377512yescomplemented
317C22×C3C2×GO+(4,𝔽2)11yescharacteristic
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—41no—
321C3×C4—64no—
322C3×C4—64no—
323C3×C4—92no—
324C3×C4—124no—
325C3×C4—121no—
326C3×C4—122no—
327C3×C4—121no—
328C3×C4—181no—
329Dic3—24no—
330Dic3—41no—
331Dic3—62no—
332Dic3—62no—
333Dic3—62no—
334Dic3—121no—
335C32—21no—
336C32—21no—
337C32—41no—
338C32—41no—
339C32C3×#44811yescharacteristic, complemented
340C23—61no—
341C23C3×GO+(4,𝔽2)11yescharacteristic
342Q8—94no—
343Q8—182no—
344Q8—184no—
345D4—122no—
346D4—182no—
347C2×C4—64no—
348C2×C4—62no—
349C2×C4—61no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—182no—
353C8—182no—
354C2×C3—24no—
355C2×C3—22no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—42no—
363C2×C3—44no—
364C2×C3—41no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—82no—
371C2×C3—241no—
372C2×C3—242no—
373C2×C3—241no—
374C2×C3C2×#377511yescharacteristic
375C2×C3#6423911yescharacteristic
376C2×C3#6424111yescharacteristic
377D3—81no—
378C22—21no—
379C22—21no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22C3×#377512yescomplemented
384C22C2×C3×GO+(4,𝔽2)11yescharacteristic
385C4—64no—
386C4—92no—
387C4—121no—
388C4—181no—
389C3—21no—
390C3—21no—
391C3—41no—
392C3—41no—
393C3#1059352511yescharacteristic, direct product factor
394C2—22no—
395C2—241no—
396C2C2×C3×#377511yescharacteristic
397C2C3×#6423911yescharacteristic
398C2C3×#6424111yescharacteristic
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×#448
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
1. C22×C3×#627C2semidirect
2. C2×C3×#3775C2semidirect
4. C3×C22⋊#622C2semidirect
5. C3×#63740C2semidirect
6. #10593525C3direct (N × H)
19. C22×#627C2×C3semidirect
20. C2×#3775C2×C3semidirect
23. C22⋊#622C2×C3semidirect
24. #63740C2×C3semidirect
36. C3×#627D4semidirect
68. C22×C33SD16semidirect
124. #627C3×D4semidirect
180. C22×C32C3×SD16semidirect
210. C33#448semidirect
316. C22×C3#3775semidirect
339. C32C3×#448semidirect
383. C22C3×#3775semidirect
393. C3#10593525direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6272
C2×C3×#37752
C3×#639802
C3×C22⋊#6222
C3×#637402
#105935253
C3×#4489