Finite Group Databasebeta

← C3×#10593530 #13166899

Subgroups — C3×#10593530 #13166899

Order 1728.

1224 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 98 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×GO+(4,𝔽2)C211yesmaximal, characteristic, complemented
2C2×C3×#3811C212yesmaximal, complemented
3C2×C3×C22⋊#157C211yesmaximal, characteristic, complemented
4C2×C3×D32⋊C4C212yesmaximal, complemented
5C3×#63737C211yesmaximal, characteristic, complemented
6#10593530C311yesmaximal, characteristic, direct product factor
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C3×D32C2212yes—
9C22×C3×#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C2×C3×GO+(4,𝔽2)—24no—
12C2×C3×GO+(4,𝔽2)—22no—
13C2×C32×#67—22no—
14C3×#3811—24no—
15C3×#3811—24no—
16C2×C3×#626—22no—
17C2×C3×#624C2212yes—
18C3×C22⋊#157—22no—
19C3×D32⋊C4—24no—
20C22×GO+(4,𝔽2)C2×C311yescharacteristic, complemented
21C2×C3×D3×D4—62no—
22C2×#3811C2×C312yescomplemented
23C2×C22⋊#157C2×C311yescharacteristic, complemented
24C2×D32⋊C4C2×C312yescomplemented
25#63737C2×C311yescharacteristic, complemented
26C23×C33D411yescharacteristic, complemented
27C22×C3×C3⋊D3—22no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3—21no—
30C22×C3×C3⋊D3—21no—
31C22×C3×C3⋊D3D411yescharacteristic, complemented
32C22×C3×C3⋊D3C2311yescharacteristic
33C22×C32×D3—22no—
34C2×C3×D32—22no—
35C2×C3×D32—22no—
36C2×C3×D32—22no—
37C2×C3×D32—22no—
38C2×C3×D32—22no—
39C2×C3×D32D414yescomplemented
40C2×C3×#157—22no—
41C2×C3×#157—22no—
42C2×C3×#157—21no—
43C2×C3×#157D412yescomplemented
44C3×GO+(4,𝔽2)—28no—
45C32×#67—28no—
46C2×C32×Dic3—22no—
47C3×#626—28no—
48C3×#624—22no—
49C3×#624D414yescomplemented
50C3×#392—91nomaximal
51C23×C3⋊D3C22×C311yescharacteristic
52C23×C3×D3—62no—
53C23×C3×D3—62no—
54C22×D32C22×C312yes—
55C22×#157C22×C311yescharacteristic
56C22×#157C22×C311yescharacteristic
57C2×GO+(4,𝔽2)—24no—
58C2×GO+(4,𝔽2)—22no—
59C2×C32×D4—62no—
60C2×C3×#67—22no—
61C2×C3×#67—22no—
62C2×C3×#67—42no—
63C2×C3×#67—62no—
64C3×D3×D4—68no—
65C3×D3×D4—68no—
66C2×C3×D12—62no—
67C2×C3×C4×D3—62no—
68#3811—24no—
69#3811—24no—
70C2×#626—22no—
71C2×#624C22×C312yes—
72C22⋊#157—22no—
73D32⋊C4—24no—
74C22×C33—22no—
75C22×C33—21no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—22no—
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—42no—
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3C2×D412yes—
83C2×C3×C3⋊D3C2×D411yescharacteristic
84C2×C3×C3⋊D3C2×D412yes—
85C2×C3×C3⋊D3#4111yescharacteristic
86C2×C32×D3—24no—
87C2×C32×D3—42no—
88C2×C32×D3—42no—
89C3×D32—24no—
90C3×D32—24no—
91C3×D32—42no—
92C3×D32—42no—
93C3×#157—22no—
94C3×#157—41no—
95C32×Dic3—24no—
96C22×C3×D4—92no—
97C22×C3×D4—91no—
98C2×D3×D4—62no—
99C2×C3×C22⋊C4—91no—
100C2×C3×C22⋊C4—92no—
101C3×#95—91no—
102C23×C32—22no—
103C23×C32—42no—
104C23×C32—62no—
105C23×C32C3×D411yescharacteristic, complemented
106C22×C3⋊D3—22no—
107C22×C3⋊D3—22no—
108C22×C3⋊D3—21no—
109C22×C3⋊D3—21no—
110C22×C3⋊D3C3×D411yescharacteristic, complemented
111C22×C3⋊D3C23×C311yescharacteristic
112C22×C3×D3—22no—
113C22×C3×D3—22no—
114C22×C3×D3—42no—
115C22×C3×D3—64no—
116C22×C3×D3—62no—
117C22×C3×D3—62no—
118C22×C3×D3—64no—
119C22×C3×D3—64no—
120C22×C3×D3—64no—
121C22×C3×D3—122no—
122C22×C3×D3—122no—
123C22×C3×D3—122no—
124C22×C3×D3—122no—
125C22×C3×D3—122no—
126C22×C3×D3—122no—
127C22×C3×D3—122no—
128C22×C3×D3—122no—
129C2×D32—22no—
130C2×D32—22no—
131C2×D32—22no—
132C2×D32—22no—
133C2×D32—22no—
134C2×D32C3×D414yescomplemented
135C2×#157—22no—
136C2×#157—22no—
137C2×#157—21no—
138C2×#157C3×D412yescomplemented
139GO+(4,𝔽2)—28no—
140C32×D4—68no—
141C2×C32×C4—62no—
142C3×#67—28no—
143C3×#67—28no—
144C3×#67—48no—
145C3×#67—68no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—42no—
149C3×D12—68no—
150C3×C4×D3—64no—
151C3×C4×D3—64no—
152#626—28no—
153#624—22no—
154#624C3×D414yescomplemented
155#392—91noSylow
156C2×C33—41no—
157C2×C33#12012yes—
158C2×C33#12111yescharacteristic
159C3×C3⋊D3—41no—
160C3×C3⋊D3#12011yescharacteristic
161C3×C3⋊D3#12112yes—
162C3×C3⋊D3#12711yescharacteristic
163C32×D3—44no—
164C24×C3—92no—
165C24×C3—91no—
166C23×D3—62no—
167C23×D3—62no—
168C2×C3×D4—62no—
169C2×C3×D4—62no—
170C2×C3×D4—122no—
171C2×C3×D4—182no—
172C2×C3×D4—184no—
173C2×C3×D4—182no—
174C2×C3×D4—184no—
175C2×C3×D4—184no—
176C22×C3×C4—92no—
177C22×C3×C4—91no—
178C22×C3×C4—91no—
179C2×#67—22no—
180C2×#67—62no—
181D3×D4—68no—
182D3×D4—68no—
183C2×D12—62no—
184C2×C4×D3—62no—
185C3×C22⋊C4—184no—
186C3×C22⋊C4—182no—
187C22×C32—24no—
188C22×C32—22no—
189C22×C32—22no—
190C22×C32—21no—
191C22×C32—42no—
192C22×C32—42no—
193C22×C32—42no—
194C22×C32—44no—
195C22×C32—64no—
196C22×C32—82no—
197C22×C32—82no—
198C22×C32—122no—
199C22×C32—122no—
200C22×C32C2×C3×D411yescharacteristic
201C2×C3⋊D3—22no—
202C2×C3⋊D3—21no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—42no—
205C2×C3⋊D3—41no—
206C2×C3⋊D3C2×C3×D412yes—
207C2×C3⋊D3C2×C3×D411yescharacteristic
208C2×C3⋊D3C2×C3×D412yes—
209C2×C3⋊D3C3×#4111yescharacteristic
210C2×C3×D3—24no—
211C2×C3×D3—24no—
212C2×C3×D3—44no—
213C2×C3×D3—42no—
214C2×C3×D3—42no—
215C2×C3×D3—42no—
216C2×C3×D3—42no—
217C2×C3×D3—62no—
218C2×C3×D3—62no—
219C2×C3×D3—62no—
220C2×C3×D3—62no—
221C2×C3×D3—62no—
222C2×C3×D3—62no—
223C2×C3×D3—64no—
224C2×C3×D3—82no—
225C2×C3×D3—82no—
226C2×C3×D3—124no—
227C2×C3×D3—124no—
228C2×C3×D3—122no—
229C2×C3×D3—122no—
230C2×C3×D3—124no—
231C2×C3×D3—124no—
232C2×C3×D3—124no—
233C2×C3×D3—124no—
234C2×C3×D3—124no—
235C2×C3×D3—124no—
236D32—24no—
237D32—24no—
238D32—42no—
239D32—42no—
240#157—22no—
241#157—41no—
242C32×C4—64no—
243C3×Dic3—24no—
244C3×Dic3—24no—
245C3×Dic3—44no—
246C22×D4—92no—
247C22×D4—91no—
248C2×C22⋊C4—91no—
249C2×C22⋊C4—92no—
250#95—91no—
251C33#39211yescharacteristic, complemented, Sylow
252C23×C3—22no—
253C23×C3—42no—
254C23×C3—62no—
255C23×C3—62no—
256C23×C3—94no—
257C23×C3—91no—
258C23×C3—91no—
259C23×C3—122no—
260C23×C3—182no—
261C23×C3—182no—
262C23×C3—181no—
263C23×C3—181no—
264C23×C3—182no—
265C23×C3—182no—
266C23×C3—182no—
267C23×C3—182no—
268C23×C3—182no—
269C23×C3GO+(4,𝔽2)11yescharacteristic, complemented
270C22×D3—22no—
271C22×D3—64no—
272C22×D3—62no—
273C22×D3—62no—
274C22×D3—64no—
275C22×D3—64no—
276C22×D3—64no—
277C22×D3—122no—
278C22×D3—122no—
279C22×D3—122no—
280C22×D3—122no—
281C22×D3—122no—
282C22×D3—122no—
283C22×D3—122no—
284C22×D3—122no—
285C3×D4—68no—
286C3×D4—68no—
287C3×D4—128no—
288C3×D4—188no—
289C3×D4—188no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—94no—
293C2×C3×C4—92no—
294C2×C3×C4—122no—
295C2×C3×C4—182no—
296C2×C3×C4—182no—
297C2×C3×C4—182no—
298C2×C3×C4—181no—
299#67—28no—
300#67—68no—
301C2×Dic3—22no—
302D12—68no—
303C4×D3—64no—
304C4×D3—64no—
305C2×C32—22no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—42no—
309C2×C32—42no—
310C2×C32—42no—
311C2×C32—41no—
312C2×C32—44no—
313C2×C32—84no—
314C2×C32—124no—
315C2×C32C3×#12012yes—
316C2×C32C3×#12111yescharacteristic
317C3⋊D3—41no—
318C3⋊D3C3×#12011yescharacteristic
319C3⋊D3C3×#12112yes—
320C3⋊D3C3×#12711yescharacteristic
321C3×D3—44no—
322C3×D3—44no—
323C3×D3—62no—
324C3×D3—62no—
325C3×D3—62no—
326C3×D3—62no—
327C3×D3—84no—
328C3×D3—124no—
329C24—92no—
330C24—91no—
331C2×D4—62no—
332C2×D4—182no—
333C2×D4—184no—
334C2×D4—182no—
335C2×D4—184no—
336C2×D4—184no—
337C22×C4—92no—
338C22×C4—91no—
339C22×C4—91no—
340C22⋊C4—184no—
341C22⋊C4—182no—
342C22×C3—24no—
343C22×C3—22no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—42no—
347C22×C3—42no—
348C22×C3—44no—
349C22×C3—42no—
350C22×C3—64no—
351C22×C3—64no—
352C22×C3—82no—
353C22×C3—82no—
354C22×C3—92no—
355C22×C3—92no—
356C22×C3—91no—
357C22×C3—91no—
358C22×C3—124no—
359C22×C3—122no—
360C22×C3—122no—
361C22×C3—122no—
362C22×C3—122no—
363C22×C3—184no—
364C22×C3—184no—
365C22×C3—182no—
366C22×C3—181no—
367C22×C3—242no—
368C22×C3—242no—
369C22×C3—361no—
370C22×C3—361no—
371C22×C3—362no—
372C22×C3—362no—
373C22×C3—362no—
374C22×C3C2×GO+(4,𝔽2)11yescharacteristic
375C2×D3—24no—
376C2×D3—42no—
377C2×D3—42no—
378C2×D3—62no—
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—62no—
382C2×D3—62no—
383C2×D3—62no—
384C2×D3—64no—
385C2×D3—124no—
386C2×D3—124no—
387C2×D3—122no—
388C2×D3—122no—
389C2×D3—124no—
390C2×D3—124no—
391C2×D3—124no—
392C2×D3—124no—
393C2×D3—124no—
394C2×D3—124no—
395C3×C4—64no—
396C3×C4—64no—
397C3×C4—124no—
398C3×C4—182no—
399C3×C4—361no—
400Dic3—24no—
401C32—22no—
402C32—42no—
403C32C3×#39211yescharacteristic, complemented
404C23—62no—
405C23—94no—
406C23—91no—
407C23—91no—
408C23—181no—
409C23—182no—
410C23—182no—
411C23—181no—
412C23—182no—
413C23—182no—
414C23—182no—
415C23—182no—
416C23—182no—
417C23C3×GO+(4,𝔽2)11yescharacteristic, complemented
418D4—68no—
419D4—188no—
420D4—188no—
421C2×C4—62no—
422C2×C4—94no—
423C2×C4—92no—
424C2×C4—182no—
425C2×C4—182no—
426C2×C4—182no—
427C2×C4—181no—
428C2×C3—22no—
429C2×C3—22no—
430C2×C3—22no—
431C2×C3—44no—
432C2×C3—41no—
433C2×C3—42no—
434C2×C3—42no—
435C2×C3—42no—
436C2×C3—84no—
437C2×C3—91no—
438C2×C3—91no—
439C2×C3—92no—
440C2×C3—124no—
441C2×C3—124no—
442C2×C3—244no—
443C2×C3—361no—
444C2×C3#6424112yes—
445C2×C3#6424211yescharacteristic
446D3—44no—
447D3—62no—
448D3—62no—
449D3—62no—
450D3—62no—
451D3—124no—
452C22—22no—
453C22—21no—
454C22—64no—
455C22—92no—
456C22—92no—
457C22—91no—
458C22—91no—
459C22—122no—
460C22—122no—
461C22—184no—
462C22—184no—
463C22—182no—
464C22—181no—
465C22—361no—
466C22—361no—
467C22—362no—
468C22—362no—
469C22—362no—
470C22C2×C3×GO+(4,𝔽2)11yescharacteristic
471C4—64no—
472C4—182no—
473C4—361no—
474C3—22no—
475C3—42no—
476C3#1059353011yescharacteristic, direct product factor
477C2—41no—
478C2—91no—
479C2—91no—
480C2—92no—
481C2—124no—
482C2—361no—
483C2C3×#6424112yes—
484C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#392
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×GO+(4,𝔽2)C2semidirect
2. C2×C3×#3811C2semidirect
3. C2×C3×C22⋊#157C2semidirect
4. C2×C3×D32⋊C4C2semidirect
5. C3×#63737C2semidirect
6. #10593530C3direct (N × H)
20. C22×GO+(4,𝔽2)C2×C3semidirect
22. C2×#3811C2×C3semidirect
23. C2×C22⋊#157C2×C3semidirect
24. C2×D32⋊C4C2×C3semidirect
25. #63737C2×C3semidirect
26. C23×C33D4semidirect
31. C22×C3×C3⋊D3D4semidirect
39. C2×C3×D32D4semidirect
43. C2×C3×#157D4semidirect
49. C3×#624D4semidirect
105. C23×C32C3×D4semidirect
110. C22×C3⋊D3C3×D4semidirect
134. C2×D32C3×D4semidirect
138. C2×#157C3×D4semidirect
154. #624C3×D4semidirect
251. C33#392semidirect
269. C23×C3GO+(4,𝔽2)semidirect
403. C32C3×#392semidirect
417. C23C3×GO+(4,𝔽2)semidirect
476. C3#10593530direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×GO+(4,𝔽2)2
C2×C3×#38112
C2×C3×C22⋊#1572
C2×C3×D32⋊C42
C3×#637372
#105935303
C3×#3929