Finite Group Databasebeta

← C3×#10593244 #13165920

Subgroups — C3×#10593244 #13165920

Order 1728.

504 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 30 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×C3⋊#1020C211yesmaximal, characteristic, complemented
2C32×#1020C211yesmaximal, characteristic, complemented
3C3×C3⋊(#120⋊C3)C211yesmaximal, characteristic, complemented
4C3×#1020⋊C2—31nomaximal
5#10593244C311yesmaximal, characteristic, direct product factor
6C3×#5032—31nomaximal
7C32×C22⋊A4C2211yescharacteristic, complemented
8C3×D3×S4—41nomaximal
9C3⋊#1020C2×C311yescharacteristic, complemented
10C3×#1020—21no—
11C3×#1020—31no—
12C3×#1020—61no—
13C3×#1020D311yescharacteristic, complemented
14C3×#1020C2×C311yescharacteristic, complemented
15C3×#1004—31no—
16C32×#120—31no—
17C3×#953—31no—
18C3×#953D311yescharacteristic, complemented
19C3×#120⋊C3—31no—
20C3⋊(#120⋊C3)—21no—
21C3⋊(#120⋊C3)C2×C311yescharacteristic, complemented
22C3×#834—31no—
23C3×#806—31no—
24C3×#806—31no—
25C3×D3×A4—41no—
26C3×C3⋊S4—41no—
27C3×C3⋊S4—81no—
28C32×S4—41no—
29C32×S4—81no—
30#1020⋊C2—31no—
31C3×B(4,𝔽2)—91no—
32#5032—31no—
33C24×C32C2×D311yescharacteristic, complemented
34C3×C22⋊A4—21no—
35C3×C22⋊A4—21no—
36C3×C22⋊A4—21no—
37C3×C22⋊A4—41no—
38C3×C22⋊A4C2×D311yescharacteristic, complemented
39C3×C22⋊A4C22×C311yescharacteristic, complemented
40C2×C3×S4—121no—
41D3×S4—41no—
42C2×C32×D4—31no—
43C2×C32×D4—61no—
44C2×C3×#67—31no—
45C2×C3×#67—31no—
46C2×C3×#67—61no—
47C2×C3×#67—61no—
48C3×#237—61no—
49C3×#237—61no—
50C3×D3×D4—61no—
51C3×D3×D4—61no—
52C3×#235—61no—
53C32×C22⋊C4—31no—
54C32×C22⋊C4—61no—
55C3×#217—31no—
56C3×#217—31no—
57C3×#217—61no—
58C32×A4—41no—
59C32×A4—81no—
60C32×A4—81no—
61C3×D32—161no—
62#1020—31no—
63#1020—61no—
64#1020C3×D311yescharacteristic, complemented
65C3×H(5,𝔽2)—91no—
66#1004—31no—
67C3×#120—31no—
68C3×#120—31no—
69C3×#120—31no—
70C3×#120—61no—
71C3×#120—91no—
72#953—31no—
73#953C3×D311yescharacteristic, complemented
74#120⋊C3—31no—
75#120⋊C3—61no—
76C3×#99—91no—
77C3×#99—91no—
78C3×#99—91no—
79#834—31no—
80#806—31no—
81#806—31no—
82C23×C32—31no—
83C23×C32—31no—
84C23×C32—61no—
85C23×C32—61no—
86C22×C3×D3—31no—
87C22×C3×D3—61no—
88C22×C3×D3S411yescharacteristic, complemented
89C2×C3×A4—121no—
90D3×A4—41no—
91D3×A4—81no—
92C3⋊S4—41no—
93C3⋊S4—81no—
94C3×S4—41no—
95C3×S4—41no—
96C3×S4—81no—
97C3×S4—81no—
98C3×S4—81no—
99C3×S4—82no—
100C3×S4—121no—
101C3×S4—241no—
102C3×S4—241no—
103C3×S4—242no—
104C32×D4—61no—
105C32×D4—61no—
106C32×D4—121no—
107C32×D4—121no—
108C2×C32×C4—31no—
109C2×C32×C4—61no—
110C3×#67—61no—
111C3×#67—61no—
112C3×#67—61no—
113C3×#67—61no—
114C3×#67—61no—
115C3×#67—61no—
116C3×#67—121no—
117C3×#67—121no—
118C2×C3×Dic3—31no—
119C2×C3×Dic3—31no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C3×D12—61no—
123C3×C4×D3—61no—
124C3×C4×D3—61no—
125C3×Dic6—61no—
126B(4,𝔽2)—91noSylow
127C3×C3⋊D3—161no—
128C32×D3—161no—
129C32×D3—161no—
130C24×C3—21no—
131C24×C3D3211yescharacteristic, complemented
132C24×C3C2×C3×D311yescharacteristic, complemented
133C22⋊A4—21no—
134C22⋊A4—21no—
135C22⋊A4—41no—
136C22⋊A4C2×C3×D311yescharacteristic, complemented
137C2×S4—121no—
138C3×#41—181no—
139C3×#41—181no—
140C3×#41—181no—
141C2×C3×D4—31no—
142C2×C3×D4—31no—
143C2×C3×D4—61no—
144C2×C3×D4—61no—
145C2×C3×D4—61no—
146C2×C3×D4—62no—
147C2×C3×D4—91no—
148C2×C3×D4—91no—
149C2×C3×D4—181no—
150C2×C3×D4—181no—
151C2×C3×D4—181no—
152C2×C3×D4—181no—
153C2×#67—31no—
154C2×#67—31no—
155C2×#67—61no—
156C2×#67—61no—
157#237—61no—
158#237—61no—
159D3×D4—61no—
160D3×D4—61no—
161#235—61no—
162C3×C22⋊C4—31no—
163C3×C22⋊C4—31no—
164C3×C22⋊C4—61no—
165C3×C22⋊C4—61no—
166C3×C22⋊C4—61no—
167C3×C22⋊C4—62no—
168C3×C22⋊C4—91no—
169C3×C22⋊C4—91no—
170C3×C22⋊C4—181no—
171#217—31no—
172#217—31no—
173#217—61no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—31no—
177C22×C32—31no—
178C22×C32—61no—
179C22×C32—61no—
180C22×C32—61no—
181C22×C32—121no—
182C22×C32—121no—
183C22×C32C2×S411yescharacteristic
184C2×C3×D3—61no—
185C2×C3×D3—61no—
186C2×C3×D3—121no—
187C2×C3×D3—241no—
188C2×C3×D3—481no—
189C3×A4—41no—
190C3×A4—41no—
191C3×A4—81no—
192C3×A4—81no—
193C3×A4—81no—
194C3×A4—81no—
195C3×A4—81no—
196C3×A4—82no—
197C3×A4—82no—
198C3×A4—81no—
199C3×A4—81no—
200C3×A4—161no—
201C3×A4—161no—
202C3×A4—161no—
203C3×A4—161no—
204C3×A4—161no—
205C3×A4—162no—
206C3×A4—162no—
207D32—161no—
208C32×C4—61no—
209C32×C4—121no—
210C3×Dic3—61no—
211C3×Dic3—61no—
212C3×Dic3—121no—
213H(5,𝔽2)—91no—
214#120—31no—
215#120—31no—
216#120—91no—
217#99—91no—
218#99—91no—
219#99—91no—
220C33—161noSylow
221C23×C3—31no—
222C23×C3—31no—
223C23×C3—31no—
224C23×C3—31no—
225C23×C3—31no—
226C23×C3—61no—
227C23×C3—61no—
228C23×C3—61no—
229C23×C3—61no—
230C23×C3—61no—
231C23×C3—61no—
232C23×C3—62no—
233C23×C3—91no—
234C23×C3—121no—
235C23×C3—181no—
236C22×D3—31no—
237C22×D3—61no—
238C22×D3C3×S411yescharacteristic, complemented
239C2×A4—121no—
240C2×A4—241no—
241S4—41no—
242S4—81no—
243S4—121no—
244S4—241no—
245S4—241no—
246S4—242no—
247C3×Q8—181no—
248C3×D4—61no—
249C3×D4—61no—
250C3×D4—61no—
251C3×D4—61no—
252C3×D4—121no—
253C3×D4—121no—
254C3×D4—121no—
255C3×D4—121no—
256C3×D4—122no—
257C3×D4—121no—
258C3×D4—121no—
259C3×D4—122no—
260C3×D4—181no—
261C3×D4—181no—
262C3×D4—181no—
263C3×D4—181no—
264C3×D4—181no—
265C3×D4—181no—
266C3×D4—181no—
267C3×D4—361no—
268C3×D4—361no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—62no—
275C2×C3×C4—91no—
276C2×C3×C4—91no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4—181no—
281#67—61no—
282#67—61no—
283#67—61no—
284#67—61no—
285#67—61no—
286#67—61no—
287#67—121no—
288#67—121no—
289C2×Dic3—31no—
290C2×Dic3—31no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293D12—61no—
294C4×D3—61no—
295C4×D3—61no—
296Dic6—61no—
297C2×C32—31no—
298C2×C32—61no—
299C2×C32—61no—
300C2×C32—121no—
301C2×C32—481no—
302C3⋊D3—161no—
303C3×D3—41no—
304C3×D3—121no—
305C3×D3—161no—
306C3×D3—161no—
307C3×D3—161no—
308C3×D3—321no—
309C3×D3—321no—
310C3×D3—481no—
311C3×D3—961no—
312C24C3×D3211yescharacteristic, complemented
313#41—181no—
314#41—181no—
315#41—181no—
316C2×D4—31no—
317C2×D4—61no—
318C2×D4—91no—
319C2×D4—91no—
320C2×D4—181no—
321C2×D4—181no—
322C2×D4—181no—
323C2×D4—181no—
324C22⋊C4—31no—
325C22⋊C4—61no—
326C22⋊C4—91no—
327C22⋊C4—91no—
328C22⋊C4—181no—
329C22×C3—21no—
330C22×C3—22no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—41no—
340C22×C3—61no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3—61no—
344C22×C3—61no—
345C22×C3—61no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—62no—
349C22×C3—121no—
350C22×C3—121no—
351C22×C3—121no—
352C22×C3—121no—
353C22×C3—122no—
354C22×C3—121no—
355C22×C3—121no—
356C22×C3—122no—
357C22×C3—181no—
358C22×C3—181no—
359C22×C3—361no—
360C22×C3—721no—
361C22×C3D3×S411yescharacteristic
362C22×C3C2×C3×S411yescharacteristic
363C2×D3—61no—
364C2×D3—61no—
365C2×D3—121no—
366C2×D3—241no—
367C2×D3—481no—
368A4—41no—
369A4—81no—
370A4—81no—
371A4—82no—
372A4—81no—
373A4—81no—
374A4—161no—
375A4—161no—
376A4—161no—
377A4—162no—
378A4—162no—
379A4—161no—
380C3×C4—61no—
381C3×C4—61no—
382C3×C4—121no—
383C3×C4—122no—
384C3×C4—121no—
385C3×C4—121no—
386C3×C4—181no—
387C3×C4—181no—
388C3×C4—361no—
389Dic3—61no—
390Dic3—61no—
391Dic3—121no—
392C32—161no—
393C32—161no—
394C32—321no—
395C32—321no—
396C32—321no—
397C32—641no—
398C32#1020⋊C211yescharacteristic, complemented
399C23—31no—
400C23—31no—
401C23—31no—
402C23—61no—
403C23—61no—
404C23—91no—
405C23—181no—
406Q8—181no—
407D4—61no—
408D4—61no—
409D4—121no—
410D4—121no—
411D4—181no—
412D4—181no—
413D4—181no—
414D4—181no—
415D4—181no—
416D4—181no—
417D4—181no—
418D4—361no—
419D4—361no—
420C2×C4—31no—
421C2×C4—61no—
422C2×C4—91no—
423C2×C4—91no—
424C2×C4—181no—
425C2×C4—181no—
426C2×C4—181no—
427C2×C4—181no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—61no—
431C2×C3—62no—
432C2×C3—61no—
433C2×C3—61no—
434C2×C3—61no—
435C2×C3—61no—
436C2×C3—121no—
437C2×C3—121no—
438C2×C3—121no—
439C2×C3—121no—
440C2×C3—241no—
441C2×C3—361no—
442C2×C3—481no—
443C2×C3—961no—
444D3—41no—
445D3—121no—
446D3—161no—
447D3—481no—
448D3—961no—
449C22—21no—
450C22—21no—
451C22—31no—
452C22—31no—
453C22—61no—
454C22—61no—
455C22—61no—
456C22—121no—
457C22—121no—
458C22—181no—
459C22—181no—
460C22—361no—
461C22—721no—
462C22C3×D3×S411yescharacteristic
463C4—61no—
464C4—121no—
465C4—181no—
466C4—181no—
467C4—361no—
468C3—21no—
469C3—161no—
470C3—321no—
471C3—321no—
472C3—641no—
473C3#1059324411yescharacteristic, direct product factor
474C3C3×#1020⋊C211yescharacteristic, complemented
475C2—31no—
476C2—61no—
477C2—61no—
478C2—121no—
479C2—121no—
480C2—361no—
subgroupG/H
center, hypercenterC3#10593244
commutator, cosocle, hypocenterC3×C22⋊A4C22×C3
FrattiniC22C3×D3×S4
FittingC24×C32C2×D3
socleC22×C32C2×S4
LCSG ▷ C3×C22⋊A4
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×C3⋊#1020C2semidirect
2. C32×#1020C2semidirect
3. C3×C3⋊(#120⋊C3)C2semidirect
5. #10593244C3direct (N × H)
7. C32×C22⋊A4C22semidirect
9. C3⋊#1020C2×C3semidirect
13. C3×#1020D3semidirect
14. C3×#1020C2×C3semidirect
18. C3×#953D3semidirect
21. C3⋊(#120⋊C3)C2×C3semidirect
33. C24×C32C2×D3semidirect
38. C3×C22⋊A4C2×D3semidirect
39. C3×C22⋊A4C22×C3semidirect
64. #1020C3×D3semidirect
73. #953C3×D3semidirect
88. C22×C3×D3S4semidirect
131. C24×C3D32semidirect
132. C24×C3C2×C3×D3semidirect
136. C22⋊A4C2×C3×D3semidirect
238. C22×D3C3×S4semidirect
312. C24C3×D32semidirect
398. C32#1020⋊C2semidirect
473. C3#10593244direct (N × H)
474. C3C3×#1020⋊C2semidirect

Maximal subgroups

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

subgroupindex
C3×C3⋊#10202
C32×#10202
C3×C3⋊(#120⋊C3)2
C3×#1020⋊C23
#105932443
C3×#50323
C3×D3×S44