Finite Group Databasebeta

← C22×C3⋊#5934 #72274501

Subgroups — C22×C3⋊#5934 #72274501

Order 2304.

6036 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 6036 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C2×C3⋊#5934C216yesmaximal, direct product factor
2C22×C3×#997C211yesmaximal, characteristic, complemented
3C22×#5934—33nomaximal
4C22×#5244—31nomaximal
5C23×C3⋊S4—43nomaximal
6C2×C3×#997C2213yescomplemented
7C3⋊#5934C2214yesdirect product factor
8C22×C3×H(5,𝔽2)D311yescharacteristic, complemented
9C22×#997D313yescomplemented
10C2×#5934—318no—
11C22×#937—33no—
12C2×#5244—318no—
13C2×#5244—36no—
14C22×#834—33no—
15C23×C3×A4—43no—
16C22×C3⋊S4—418no—
17C22×C3⋊S4—424no—
18C3×#997C2311yescharacteristic, complemented
19C22×C3×SL(2,𝔽3)—81no—
20C22×B(4,𝔽2)—91noSylow
21C25×D3—31no—
22C23×S4—129no—
23C2×C3×H(5,𝔽2)—39no—
24C2×C3×H(5,𝔽2)—63no—
25C2×C3×H(5,𝔽2)C2×D313yescomplemented
26C22×C3×#41—61no—
27C23×C3×D4—33no—
28C23×#67—63no—
29C2×#997C2×D319yescomplemented
30#5934—312no—
31C22×#217—33no—
32C2×#937—318no—
33C2×#937—318no—
34C2×#937—69no—
35C2×#937—69no—
36C22×(C2×D3)⋊C4—63no—
37#5244—336no—
38#5244—324no—
39#5244—34no—
40C2×#834—318no—
41C2×#834—318no—
42C23×C3⋊D3—161no—
43C22×C3×A4—412no—
44C22×C3×A4—49no—
45C2×C3⋊S4—412no—
46C2×C3⋊S4—472no—
47C2×C3×SL(2,𝔽3)—83no—
48C22×H(5,𝔽2)C3⋊D311yescharacteristic, complemented
49C22×#120—93no—
50C2×B(4,𝔽2)—918no—
51C2×B(4,𝔽2)—96no—
52C22×#99—93no—
53C25×C3—31no—
54C25×C3S413yescomplemented
55C24×D3—36no—
56C24×D3—63no—
57C24×D3—69no—
58C24×D3—128no—
59C23×A4—49no—
60C22×S4—1254no—
61C22×S4—1272no—
62C3×H(5,𝔽2)—39no—
63C3×H(5,𝔽2)—36no—
64C3×H(5,𝔽2)—63no—
65C3×H(5,𝔽2)—63no—
66C3×H(5,𝔽2)—618no—
67C3×H(5,𝔽2)—69no—
68C3×H(5,𝔽2)—62no—
69C3×H(5,𝔽2)C22×D311yescharacteristic, complemented
70C2×C3×#41—63no—
71C2×C3×#41—69no—
72C2×C3×#41—69no—
73C2×C3×#41—63no—
74C22×C3×Q8—21no—
75C22×C3×D4—39no—
76C22×C3×D4—39no—
77C22×C3×D4—39no—
78C22×C3×D4—39no—
79C22×C3×D4—63no—
80C22×C3×D4—618no—
81C23×C3×C4—33no—
82C22×#67—618no—
83C22×#67—618no—
84C22×#67—1224no—
85C23×Dic3—63no—
86#997C22×D313yescomplemented
87C22×SL(2,𝔽3)—83no—
88C2×#217—318no—
89C2×#217—69no—
90#937—336no—
91#937—312no—
92#937—636no—
93#937—618no—
94#937—618no—
95C2×(C2×D3)⋊C4—618no—
96C2×(C2×D3)⋊C4—618no—
97#834—336no—
98#834—312no—
99C23×C32—161no—
100C22×C3⋊D3—168no—
101C22×C3⋊D3—166no—
102C2×C3×A4—418no—
103C2×C3×A4—43no—
104C3⋊S4—424no—
105C3×SL(2,𝔽3)—81no—
106C26—91no—
107C2×H(5,𝔽2)—39no—
108C2×H(5,𝔽2)—63no—
109C2×H(5,𝔽2)C2×C3⋊D313yescomplemented
110C22×#41—61no—
111C23×D4—33no—
112C23×D4—183no—
113C2×#120—918no—
114C2×#120—918no—
115C2×#120—189no—
116C2×#120—189no—
117C22×C22⋊C4—93no—
118C22×C22⋊C4—183no—
119B(4,𝔽2)—936no—
120B(4,𝔽2)—924no—
121B(4,𝔽2)—94no—
122C2×#99—918no—
123C2×#99—918no—
124C24×C3—33no—
125C24×C3—39no—
126C24×C3—33no—
127C24×C3—39no—
128C24×C3—412no—
129C24×C3—124no—
130C24×C3C2×S419yescomplemented
131C23×D3—34no—
132C23×D3—618no—
133C23×D3—624no—
134C23×D3—618no—
135C23×D3—1224no—
136C23×D3—1272no—
137C23×D3—126no—
138C23×D3—129no—
139C23×D3—1212no—
140C23×D3—121no—
141C23×D3—483no—
142C22×A4—436no—
143C22×A4—427no—
144C2×S4—1236no—
145C2×S4—12216no—
146C3×#41—61no—
147C3×#41—69no—
148C3×#41—69no—
149C3×#41—618no—
150C3×#41—63no—
151C3×#41—618no—
152C3×#41—66no—
153C2×C3×Q8—23no—
154C2×C3×Q8—63no—
155C2×C3×D4—39no—
156C2×C3×D4—318no—
157C2×C3×D4—39no—
158C2×C3×D4—33no—
159C2×C3×D4—39no—
160C2×C3×D4—69no—
161C2×C3×D4—69no—
162C2×C3×D4—618no—
163C2×C3×D4—618no—
164C2×C3×D4—618no—
165C2×C3×D4—618no—
166C2×C3×D4—618no—
167C22×C3×C4—39no—
168C22×C3×C4—39no—
169C22×C3×C4—69no—
170C22×C3×C4—61no—
171C2×#67—636no—
172C2×#67—612no—
173C2×#67—1272no—
174C2×#67—1272no—
175C22×Dic3—618no—
176C22×Dic3—1212no—
177C2×SL(2,𝔽3)—89no—
178#217—312no—
179#217—618no—
180(C2×D3)⋊C4—612no—
181(C2×D3)⋊C4—636no—
182C22×C32—164no—
183C22×C32—163no—
184C2×C3⋊D3—164no—
185C2×C3⋊D3—1624no—
186C3×A4—43no—
187C25—31no—
188C25—96no—
189C25—183no—
190C25—189no—
191C25—368no—
192C25C3⋊S413yescomplemented
193H(5,𝔽2)—39no—
194H(5,𝔽2)—36no—
195H(5,𝔽2)—63no—
196H(5,𝔽2)—63no—
197H(5,𝔽2)—618no—
198H(5,𝔽2)—69no—
199H(5,𝔽2)—62no—
200H(5,𝔽2)C22×C3⋊D311yescharacteristic, complemented
201C2×#41—63no—
202C2×#41—69no—
203C2×#41—69no—
204C2×#41—63no—
205C22×Q8—21no—
206C22×D4—39no—
207C22×D4—39no—
208C22×D4—39no—
209C22×D4—39no—
210C22×D4—63no—
211C22×D4—618no—
212C22×D4—1818no—
213C22×D4—1818no—
214C22×D4—3624no—
215C23×C4—33no—
216C23×C4—183no—
217#120—936no—
218#120—912no—
219#120—1836no—
220#120—1818no—
221#120—1818no—
222C2×C22⋊C4—918no—
223C2×C22⋊C4—1818no—
224C2×C22⋊C4—1818no—
225C2×C22⋊C4—189no—
226#99—936no—
227#99—912no—
228C23×C3—31no—
229C23×C3—39no—
230C23×C3—36no—
231C23×C3—39no—
232C23×C3—39no—
233C23×C3—39no—
234C23×C3—418no—
235C23×C3—63no—
236C23×C3—612no—
237C23×C3—126no—
238C23×C3—1218no—
239C23×C3—1218no—
240C23×C3—163no—
241C23×C3C22×S413yescomplemented
242C23×C3#102011yescharacteristic
243C22×D3—612no—
244C22×D3—672no—
245C22×D3—128no—
246C22×D3—1272no—
247C22×D3—1218no—
248C22×D3—1236no—
249C22×D3—1272no—
250C22×D3—1248no—
251C22×D3—128no—
252C22×D3—126no—
253C22×D3—4818no—
254C22×D3—4824no—
255C2×A4—454no—
256C2×A4—49no—
257S4—1272no—
258C3×Q8—21no—
259C3×Q8—63no—
260C3×Q8—62no—
261C3×D4—63no—
262C3×D4—69no—
263C3×D4—618no—
264C3×D4—618no—
265C2×C3×C4—33no—
266C2×C3×C4—39no—
267C2×C3×C4—69no—
268C2×C3×C4—69no—
269C2×C3×C4—63no—
270C2×C3×C4—63no—
271#67—1224no—
272#67—1272no—
273C2×Dic3—612no—
274C2×Dic3—1236no—
275SL(2,𝔽3)—83no—
276C2×C32—161no—
277C2×C32—166no—
278C3⋊D3—168no—
279C24—33no—
280C24—39no—
281C24—33no—
282C24—39no—
283C24—412no—
284C24—94no—
285C24—124no—
286C24—1818no—
287C24—1818no—
288C24—1824no—
289C24—366no—
290C24—369no—
291C24—3612no—
292C24—3624no—
293C24—3672no—
294C24—361no—
295C24C2×C3⋊S419yescomplemented
296#41—61no—
297#41—69no—
298#41—69no—
299#41—618no—
300#41—63no—
301#41—618no—
302#41—66no—
303C2×Q8—23no—
304C2×Q8—63no—
305C2×D4—39no—
306C2×D4—318no—
307C2×D4—39no—
308C2×D4—33no—
309C2×D4—39no—
310C2×D4—69no—
311C2×D4—69no—
312C2×D4—618no—
313C2×D4—618no—
314C2×D4—618no—
315C2×D4—618no—
316C2×D4—618no—
317C2×D4—1836no—
318C2×D4—1812no—
319C2×D4—3672no—
320C2×D4—3672no—
321C22×C4—39no—
322C22×C4—39no—
323C22×C4—69no—
324C22×C4—61no—
325C22×C4—1818no—
326C22×C4—3612no—
327C22⋊C4—912no—
328C22⋊C4—1812no—
329C22⋊C4—1836no—
330C22⋊C4—1818no—
331C22×C3—33no—
332C22×C3—39no—
333C22×C3—43no—
334C22×C3—618no—
335C22×C3—618no—
336C22×C3—129no—
337C22×C3—121no—
338C22×C3—129no—
339C22×C3—126no—
340C22×C3—169no—
341C22×C3—1612no—
342C22×C3—243no—
343C22×C3#593414yescomplemented
344C22×C3C2×#102013yes—
345C2×D3—624no—
346C2×D3—1212no—
347C2×D3—1272no—
348C2×D3—124no—
349C2×D3—1224no—
350C2×D3—4872no—
351C2×D3—4812no—
352A4—49no—
353C3×C4—61no—
354C3×C4—63no—
355Dic3—1212no—
356C32—161noSylow
357C23—31no—
358C23—39no—
359C23—36no—
360C23—39no—
361C23—39no—
362C23—39no—
363C23—418no—
364C23—63no—
365C23—612no—
366C23—126no—
367C23—1218no—
368C23—1218no—
369C23—1812no—
370C23—1872no—
371C23—3618no—
372C23—3636no—
373C23—3672no—
374C23—368no—
375C23—3672no—
376C23—3648no—
377C23—366no—
378C23—368no—
379C23C22×C3⋊S413yescomplemented
380C23C3⋊#102011yescharacteristic
381Q8—21no—
382Q8—63no—
383Q8—62no—
384D4—63no—
385D4—69no—
386D4—618no—
387D4—618no—
388D4—3624no—
389D4—3672no—
390C2×C4—33no—
391C2×C4—39no—
392C2×C4—69no—
393C2×C4—69no—
394C2×C4—63no—
395C2×C4—63no—
396C2×C4—1812no—
397C2×C4—3636no—
398C2×C3—63no—
399C2×C3—69no—
400C2×C3—1618no—
401C2×C3—163no—
402C2×C3C2×#593416yescomplemented
403C2×C3C22×#102011yescharacteristic
404D3—128no—
405D3—4824no—
406C22—33no—
407C22—39no—
408C22—43no—
409C22—618no—
410C22—618no—
411C22—129no—
412C22—121no—
413C22—129no—
414C22—126no—
415C22—1824no—
416C22—243no—
417C22—3612no—
418C22—3672no—
419C22—3624no—
420C22—364no—
421C22C3⋊#593414yesdirect product factor
422C22C2×C3⋊#102013yes—
423C4—61no—
424C4—63no—
425C4—3612no—
426C3—163no—
427C3C22×#593411yescharacteristic, complemented
428C2—63no—
429C2—69no—
430C2—368no—
431C2C2×C3⋊#593416yesdirect product factor
432C2C22×C3⋊#102011yescharacteristic
subgroupG/H
center, hypercenterC23C3⋊#1020
commutator, cosocle, hypocenterC3×#997C23
FrattiniC2C22×C3⋊#1020
FittingC22×C3×H(5,𝔽2)D3
socleC23×C3#1020
LCSG ▷ C3×#997
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊#5934C2direct (N × H)
2. C22×C3×#997C2semidirect
6. C2×C3×#997C22semidirect
7. C3⋊#5934C22direct (N × H)
8. C22×C3×H(5,𝔽2)D3semidirect
9. C22×#997D3semidirect
18. C3×#997C23semidirect
25. C2×C3×H(5,𝔽2)C2×D3semidirect
29. C2×#997C2×D3semidirect
48. C22×H(5,𝔽2)C3⋊D3semidirect
54. C25×C3S4semidirect
69. C3×H(5,𝔽2)C22×D3semidirect
86. #997C22×D3semidirect
109. C2×H(5,𝔽2)C2×C3⋊D3semidirect
130. C24×C3C2×S4semidirect
192. C25C3⋊S4semidirect
200. H(5,𝔽2)C22×C3⋊D3semidirect
241. C23×C3C22×S4semidirect
295. C24C2×C3⋊S4semidirect
343. C22×C3#5934semidirect
379. C23C22×C3⋊S4semidirect
402. C2×C3C2×#5934semidirect
421. C22C3⋊#5934direct (N × H)
427. C3C22×#5934semidirect
431. C2C2×C3⋊#5934direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊#59342
C22×C3×#9972
C22×#59343
C22×#52443
C23×C3⋊S44