Finite Group Databasebeta

← C3×#11715850 #72230797

Subgroups — C3×#11715850 #72230797

Order 2304.

2439 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 2439 conjugacy classes of subgroups — too many to draw. Only its 40 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×C22⋊#997C211yesmaximal, characteristic, complemented
2#11715850C311yesmaximal, characteristic, direct product factor
3C3×#32898—31nomaximal
4C2×C3×#1020—43nomaximal
5C3×#5934—44nomaximal
6C22×C3×H(5,𝔽2)D311yescharacteristic, complemented
7C22⋊#997—21no—
8C22⋊#997C2×C311yescharacteristic, complemented
9C2×C3×B(4,𝔽2)—34no—
10C3×#2403—33no—
11C3×#1790—34no—
12C3×#1781—33no—
13C2×C3×C22⋊A4—43no—
14C3×#1020—46no—
15C3×#997—44no—
16C3×#997—86no—
17C3×C22⋊SL(2,𝔽3)—81no—
18#32898—31noSylow
19C2×#1020—43no—
20C2×C3×H(5,𝔽2)—34no—
21C2×C3×H(5,𝔽2)—66no—
22C22×C3×#41—61no—
23C23×C3×D4—31no—
24C23×C3×D4—33no—
25#5934—44no—
26C2×C3×#120—36no—
27C2×C3×#120—63no—
28C3×B(4,𝔽2)—612no—
29C3×B(4,𝔽2)—64no—
30C2×C3×#99—312no—
31C3×#392—66no—
32C3×#386—36no—
33C3×#386—63no—
34C3×#379—33no—
35C3×C22⋊A4—43no—
36C2×C3×S4—161no—
37C2×C3×S4—166no—
38C22×H(5,𝔽2)C3×D311yescharacteristic, complemented
39C2×B(4,𝔽2)—34no—
40#2403—33no—
41#1790—34no—
42#1781—33no—
43C25×C3—31no—
44C25×C3—31no—
45C25×C3S413yescomplemented
46C2×C22⋊A4—43no—
47C2×C22⋊A4—83no—
48#1020—46no—
49C3×H(5,𝔽2)—26no—
50C3×H(5,𝔽2)—312no—
51C3×H(5,𝔽2)—66no—
52C3×H(5,𝔽2)—612no—
53C3×H(5,𝔽2)—612no—
54C3×H(5,𝔽2)—64no—
55C3×H(5,𝔽2)S414yescomplemented
56C2×C3×#41—612no—
57C2×C3×#41—612no—
58C22×C3×Q8—21no—
59C22×C3×D4—36no—
60C22×C3×D4—36no—
61C22×C3×D4—612no—
62C22×C3×D4—612no—
63C22×C3×D4—63no—
64C22×C3×D4—66no—
65C22×C3×D4—63no—
66C22×C3×D4—63no—
67C22×C3×D4—66no—
68C22×C3×D4—122no—
69C22×C3×D4—122no—
70C23×C3×C4—33no—
71C23×C3×C4—31no—
72#997—44no—
73#997—86no—
74#997—812no—
75#997—84no—
76C22⋊SL(2,𝔽3)—81no—
77C22⋊SL(2,𝔽3)—82no—
78C3×#120—612no—
79C3×#120—612no—
80C3×#120—126no—
81C3×#120—126no—
82C2×C3×C22⋊C4—36no—
83C2×C3×C22⋊C4—66no—
84C2×C3×C22⋊C4—63no—
85C2×C3×C22⋊C4—66no—
86C2×C3×C22⋊C4—63no—
87C3×#99—612no—
88C3×#99—612no—
89C3×#95—66no—
90C2×C3×A4—161no—
91C2×C3×A4—166no—
92C2×C3×A4—323no—
93C3×S4—162no—
94C3×S4—1612no—
95C3×SL(2,𝔽3)—324no—
96C2×H(5,𝔽2)—34no—
97C2×H(5,𝔽2)—66no—
98C22×#41—61no—
99C23×D4—33no—
100C23×D4—31no—
101C2×#120—36no—
102C2×#120—63no—
103B(4,𝔽2)—612no—
104B(4,𝔽2)—64no—
105C2×#99—312no—
106#392—66no—
107#386—36no—
108#386—63no—
109#379—33no—
110C24×C3—31no—
111C24×C3—33no—
112C24×C3—33no—
113C24×C3—36no—
114C24×C3—43no—
115C24×C3—63no—
116C24×C3—64no—
117C24×C3—63no—
118C24×C3—64no—
119C24×C3—121no—
120C24×C3—121no—
121C24×C3—122no—
122C24×C3—123no—
123C24×C3—241no—
124C24×C3—241no—
125C22⋊A4—43no—
126C22⋊A4—83no—
127C2×S4—166no—
128C2×S4—161no—
129C3×#41—64no—
130C3×#41—624no—
131C3×#41—612no—
132C3×#41—612no—
133C3×#41—64no—
134C3×#41—68no—
135C2×C3×Q8—64no—
136C2×C3×D4—312no—
137C2×C3×D4—312no—
138C2×C3×D4—612no—
139C2×C3×D4—612no—
140C2×C3×D4—612no—
141C2×C3×D4—612no—
142C2×C3×D4—612no—
143C2×C3×D4—66no—
144C2×C3×D4—624no—
145C2×C3×D4—612no—
146C2×C3×D4—612no—
147C2×C3×D4—66no—
148C2×C3×D4—66no—
149C2×C3×D4—66no—
150C2×C3×D4—126no—
151C2×C3×D4—123no—
152C2×C3×D4—123no—
153C2×C3×D4—126no—
154C2×C3×D4—123no—
155C2×C3×D4—121no—
156C2×C3×D4—126no—
157C2×C3×D4—1212no—
158C2×C3×D4—126no—
159C2×C3×D4—246no—
160C22×C3×C4—36no—
161C22×C3×C4—612no—
162C22×C3×C4—63no—
163C22×C3×C4—61no—
164C22×C3×C4—63no—
165C22×C3×C4—66no—
166C22×C3×C4—122no—
167C3×C22⋊C4—612no—
168C3×C22⋊C4—1212no—
169C3×C22⋊C4—1212no—
170C3×C22⋊C4—123no—
171C3×C22⋊C4—123no—
172C3×C22⋊C4—243no—
173C2×C3×D3—641no—
174C3×A4—166no—
175C3×A4—161no—
176C3×A4—323no—
177C25—31no—
178C25—31no—
179C25C3×S413yescomplemented
180H(5,𝔽2)—26no—
181H(5,𝔽2)—312no—
182H(5,𝔽2)—66no—
183H(5,𝔽2)—612no—
184H(5,𝔽2)—612no—
185H(5,𝔽2)—64no—
186H(5,𝔽2)C3×S414yescomplemented
187C2×#41—612no—
188C2×#41—612no—
189C22×Q8—21no—
190C22×D4—36no—
191C22×D4—36no—
192C22×D4—612no—
193C22×D4—612no—
194C22×D4—63no—
195C22×D4—66no—
196C22×D4—63no—
197C22×D4—63no—
198C22×D4—66no—
199C22×D4—122no—
200C22×D4—122no—
201C23×C4—33no—
202C23×C4—31no—
203#120—612no—
204#120—612no—
205#120—126no—
206#120—126no—
207C2×C22⋊C4—36no—
208C2×C22⋊C4—66no—
209C2×C22⋊C4—63no—
210C2×C22⋊C4—66no—
211C2×C22⋊C4—63no—
212#99—612no—
213#99—612no—
214#95—66no—
215C23×C3—23no—
216C23×C3—33no—
217C23×C3—34no—
218C23×C3—33no—
219C23×C3—36no—
220C23×C3—66no—
221C23×C3—63no—
222C23×C3—63no—
223C23×C3—66no—
224C23×C3—63no—
225C23×C3—66no—
226C23×C3—123no—
227C23×C3—126no—
228C23×C3—121no—
229C23×C3—126no—
230C23×C3—126no—
231C23×C3—121no—
232C23×C3—121no—
233C23×C3—123no—
234C23×C3—123no—
235C23×C3—123no—
236C23×C3—126no—
237C23×C3—243no—
238C23×C3—243no—
239C23×C3—244no—
240C23×C3—244no—
241C23×C3—248no—
242C23×C3#102011yescharacteristic
243C23×C3#102016yescomplemented
244C2×A4—161no—
245C2×A4—166no—
246C2×A4—323no—
247C2×A4—326no—
248C2×A4—321no—
249C2×A4—326no—
250S4—162no—
251S4—1612no—
252C3×Q8—24no—
253C3×Q8—64no—
254C3×D4—612no—
255C3×D4—624no—
256C3×D4—612no—
257C3×D4—122no—
258C3×D4—126no—
259C3×D4—126no—
260C3×D4—122no—
261C3×D4—2412no—
262C3×D4—2412no—
263C2×C3×C4—312no—
264C2×C3×C4—66no—
265C2×C3×C4—66no—
266C2×C3×C4—612no—
267C2×C3×C4—66no—
268C2×C3×C4—121no—
269C2×C3×C4—123no—
270C2×C3×C4—126no—
271C2×C3×C4—126no—
272C2×C3×C4—126no—
273SL(2,𝔽3)—324no—
274SL(2,𝔽3)—328no—
275C2×C32—641no—
276C3×D3—642no—
277C24—31no—
278C24—33no—
279C24—33no—
280C24—36no—
281C24—43no—
282C24—63no—
283C24—64no—
284C24—63no—
285C24—64no—
286C24—121no—
287C24—121no—
288C24—122no—
289C24—123no—
290C24—241no—
291C24—241no—
292#41—64no—
293#41—624no—
294#41—612no—
295#41—612no—
296#41—64no—
297#41—68no—
298C2×Q8—64no—
299C2×D4—312no—
300C2×D4—312no—
301C2×D4—612no—
302C2×D4—612no—
303C2×D4—612no—
304C2×D4—612no—
305C2×D4—612no—
306C2×D4—66no—
307C2×D4—624no—
308C2×D4—612no—
309C2×D4—612no—
310C2×D4—66no—
311C2×D4—66no—
312C2×D4—66no—
313C2×D4—126no—
314C2×D4—123no—
315C2×D4—123no—
316C2×D4—126no—
317C2×D4—123no—
318C2×D4—121no—
319C2×D4—126no—
320C2×D4—1212no—
321C2×D4—126no—
322C2×D4—246no—
323C22×C4—36no—
324C22×C4—612no—
325C22×C4—63no—
326C22×C4—61no—
327C22×C4—63no—
328C22×C4—66no—
329C22×C4—122no—
330C22⋊C4—612no—
331C22⋊C4—1212no—
332C22⋊C4—1212no—
333C22⋊C4—123no—
334C22⋊C4—123no—
335C22⋊C4—243no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—36no—
339C22×C3—46no—
340C22×C3—63no—
341C22×C3—63no—
342C22×C3—63no—
343C22×C3—63no—
344C22×C3—63no—
345C22×C3—83no—
346C22×C3—126no—
347C22×C3—126no—
348C22×C3—122no—
349C22×C3—124no—
350C22×C3—126no—
351C22×C3—1212no—
352C22×C3—241no—
353C22×C3—243no—
354C22×C3—246no—
355C22×C3—246no—
356C22×C3—241no—
357C22×C3#593411yescharacteristic, complemented
358C2×D3—641no—
359A4—166no—
360A4—161no—
361A4—323no—
362A4—326no—
363A4—326no—
364A4—321no—
365C3×C4—64no—
366C3×C4—122no—
367C3×C4—246no—
368C32—641noSylow
369C23—23no—
370C23—33no—
371C23—34no—
372C23—33no—
373C23—36no—
374C23—66no—
375C23—63no—
376C23—63no—
377C23—66no—
378C23—63no—
379C23—66no—
380C23—123no—
381C23—126no—
382C23—121no—
383C23—126no—
384C23—126no—
385C23—121no—
386C23—121no—
387C23—123no—
388C23—123no—
389C23—123no—
390C23—126no—
391C23—243no—
392C23—243no—
393C23—244no—
394C23—244no—
395C23—248no—
396C23C3×#102011yescharacteristic
397C23C3×#102016yescomplemented
398Q8—24no—
399Q8—64no—
400D4—612no—
401D4—624no—
402D4—612no—
403D4—122no—
404D4—126no—
405D4—126no—
406D4—122no—
407D4—2412no—
408D4—2412no—
409C2×C4—312no—
410C2×C4—66no—
411C2×C4—66no—
412C2×C4—612no—
413C2×C4—66no—
414C2×C4—121no—
415C2×C4—123no—
416C2×C4—126no—
417C2×C4—126no—
418C2×C4—126no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—66no—
422C2×C3—123no—
423C2×C3—242no—
424C2×C3—641no—
425C2×C3—1281no—
426C2×C3#8758811yescharacteristic
427D3—642no—
428C22—31no—
429C22—31no—
430C22—36no—
431C22—46no—
432C22—63no—
433C22—63no—
434C22—63no—
435C22—63no—
436C22—63no—
437C22—83no—
438C22—126no—
439C22—126no—
440C22—122no—
441C22—124no—
442C22—126no—
443C22—1212no—
444C22—241no—
445C22—243no—
446C22—246no—
447C22—246no—
448C22—241no—
449C22C3×#593411yescharacteristic, complemented
450C4—64no—
451C4—122no—
452C4—246no—
453C3—641no—
454C3—1281no—
455C3#1171585011yescharacteristic, direct product factor
456C2—31no—
457C2—31no—
458C2—66no—
459C2—123no—
460C2—242no—
461C2C3×#8758811yescharacteristic
subgroupG/H
center, hypercenterC2×C3#87588
commutator, cosocle, hypocenterC22⋊#997C2×C3
FrattiniC2C3×#87588
FittingC22×C3×H(5,𝔽2)D3
socleC23×C3#1020
LCSG ▷ C22⋊#997
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C3×C22⋊#997C2semidirect
2. #11715850C3direct (N × H)
6. C22×C3×H(5,𝔽2)D3semidirect
8. C22⋊#997C2×C3semidirect
38. C22×H(5,𝔽2)C3×D3semidirect
45. C25×C3S4semidirect
55. C3×H(5,𝔽2)S4semidirect
179. C25C3×S4semidirect
186. H(5,𝔽2)C3×S4semidirect
243. C23×C3#1020semidirect
357. C22×C3#5934semidirect
397. C23C3×#1020semidirect
449. C22C3×#5934semidirect
455. C3#11715850direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C22⋊#9972
#117158503
C3×#328983
C2×C3×#10204
C3×#59344