Finite Group Databasebeta

← C3×#84110 #11865337

Subgroups — C3×#84110 #11865337

Order 1152.

2049 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 2049 conjugacy classes of subgroups — too many to draw. Only its 322 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×C32×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C3×#5804C213yesmaximal, complemented
3C3×#5244C214yesmaximal, complemented
4C3×#5243C214yesmaximal, complemented
5C2×C3×#834C213yesmaximal, complemented
6C3×#3026—31nomaximal
7#84110C311yesmaximal, characteristic, direct product factor
8C32×H(5,𝔽2)—26no—
9C32×H(5,𝔽2)C2214yes—
10C2×C32×#41—23no—
11C22×C32×D4—23no—
12C22×C32×D4C2213yes—
13C22×C3×#67C2211yescharacteristic, complemented
14C3×#953—23no—
15C2×C3×#217C2213yescomplemented
16C3×#939—26no—
17C3×#937C2216yescomplemented
18C3×#935C2216yescomplemented
19C3×#929—23no—
20C3×#834C22112yescomplemented
21C2×C3×H(5,𝔽2)—21no—
22C2×C3×H(5,𝔽2)D311yescharacteristic, complemented
23C2×C3×H(5,𝔽2)C2×C311yescharacteristic, complemented
24C3×#534—33no—
25#5804C2×C313yescomplemented
26C3×#458—34no—
27C3×B(4,𝔽2)—34no—
28C2×C3×#99—33no—
29#5244C2×C314yescomplemented
30#5243C2×C314yescomplemented
31C2×#834C2×C313yescomplemented
32C24×C32—21no—
33C24×C32D413yescomplemented
34C24×C32C2311yescharacteristic
35C23×C3×D3C2311yescharacteristic, complemented
36C32×#41—212no—
37C32×#41—212no—
38C2×C32×Q8—21no—
39C2×C32×D4—23no—
40C2×C32×D4—212no—
41C2×C32×D4—26no—
42C2×C32×D4—212no—
43C2×C32×D4—26no—
44C2×C32×D4D416yescomplemented
45C2×C32×D4C2316yes—
46C22×C32×C4—23no—
47C22×C32×C4D413yescomplemented
48C2×C3×#67—23no—
49C2×C3×#67—23no—
50C2×C3×#67—26no—
51C22×C3×Dic3—23no—
52C22×C3×Dic3C2311yescharacteristic, complemented
53C3×#217—23no—
54C3×#217—26no—
55C3×#217C2316yescomplemented
56C3×(C2×D3)⋊C4—26no—
57C3×Dic3⋊C4—26no—
58#3026—31noSylow
59C3×H(5,𝔽2)—212no—
60C3×H(5,𝔽2)—26no—
61C3×H(5,𝔽2)—26no—
62C3×H(5,𝔽2)—24no—
63C3×H(5,𝔽2)C2×D314yes—
64C3×H(5,𝔽2)C22×C314yes—
65C2×C3×#41—26no—
66C2×C3×#41—23no—
67C2×C3×#41—23no—
68C22×C3×D4—26no—
69C22×C3×D4—23no—
70C22×C3×D4—23no—
71C22×C3×D4—23no—
72C22×C3×D4—31no—
73C22×C3×D4C2×D313yes—
74C22×C3×D4C22×C313yes—
75C22×#67C22×C311yescharacteristic, complemented
76C3×#123—36no—
77C3×#123—63no—
78C3×#121—66no—
79C3×#120—36no—
80C3×#120—63no—
81C2×C3×C22⋊C4—33no—
82#953—23no—
83C2×#217C22×C313yescomplemented
84#939—26no—
85#937C22×C316yescomplemented
86#935C22×C316yescomplemented
87#929—23no—
88C3×#99—312no—
89#834C22×C3112yescomplemented
90C23×C32—24no—
91C23×C32—23no—
92C23×C32—23no—
93C23×C32—81no—
94C23×C32—83no—
95C23×C32—82no—
96C23×C32C2×D416yes—
97C23×C32C2×D413yes—
98C23×C32C2×D413yes—
99C23×C32C2411yescharacteristic
100C22×C3×D3—23no—
101C22×C3×D3—81no—
102C32×Q8—24no—
103C32×D4—212no—
104C32×D4—212no—
105C32×D4—212no—
106C2×C32×C4—212no—
107C2×C32×C4—23no—
108C2×C32×C4—23no—
109C2×C32×C4C2×D416yes—
110C3×#67—41no—
111C3×#67—43no—
112C3×#67—86no—
113C2×C3×Dic3—23no—
114C2×C3×Dic3—26no—
115C2×C3×Dic3—83no—
116C2×H(5,𝔽2)C3×D311yescharacteristic, complemented
117#534—33no—
118#458—34no—
119B(4,𝔽2)—34no—
120C2×#99—33no—
121C24×C3—22no—
122C24×C3—21no—
123C24×C3—21no—
124C24×C3—23no—
125C24×C3—21no—
126C24×C3—31no—
127C24×C3#6713yescomplemented
128C24×C3C3×D413yescomplemented
129C24×C3C22×D311yescharacteristic
130C24×C3C23×C311yescharacteristic
131C23×D3C23×C311yescharacteristic, complemented
132C3×#41—224no—
133C3×#41—212no—
134C3×#41—212no—
135C3×#41—224no—
136C3×#41—212no—
137C3×#41—212no—
138C2×C3×Q8—22no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×D4—26no—
142C2×C3×D4—23no—
143C2×C3×D4—23no—
144C2×C3×D4—212no—
145C2×C3×D4—224no—
146C2×C3×D4—212no—
147C2×C3×D4—212no—
148C2×C3×D4—224no—
149C2×C3×D4—212no—
150C2×C3×D4—26no—
151C2×C3×D4—212no—
152C2×C3×D4—26no—
153C2×C3×D4—26no—
154C2×C3×D4—212no—
155C2×C3×D4—26no—
156C2×C3×D4—26no—
157C2×C3×D4—26no—
158C2×C3×D4—63no—
159C2×C3×D4—63no—
160C2×C3×D4—66no—
161C2×C3×D4#6716yescomplemented
162C2×C3×D4C3×D416yescomplemented
163C2×C3×D4C22×D316yes—
164C2×C3×D4C23×C316yes—
165C22×C3×C4—26no—
166C22×C3×C4—23no—
167C22×C3×C4—23no—
168C22×C3×C4—23no—
169C22×C3×C4—31no—
170C22×C3×C4—63no—
171C22×C3×C4#6713yescomplemented
172C22×C3×C4C3×D413yescomplemented
173C2×#67—23no—
174C2×#67—23no—
175C2×#67—26no—
176C22×Dic3—23no—
177C22×Dic3C23×C311yescharacteristic, complemented
178C3×C4⋊C4—66no—
179C3×C22⋊C4—36no—
180C3×C22⋊C4—63no—
181C3×C22⋊C4—66no—
182C3×C22⋊C4—66no—
183#217—23no—
184#217—26no—
185#217C23×C316yescomplemented
186(C2×D3)⋊C4—26no—
187Dic3⋊C4—26no—
188C22×C32—26no—
189C22×C32—43no—
190C22×C32—46no—
191C22×C32—43no—
192C22×C32—41no—
193C22×C32—46no—
194C22×C32—83no—
195C22×C32—84no—
196C22×C32#12011yescharacteristic
197C22×C32#12013yes—
198C22×C32C22×D413yes—
199C2×C3×D3—46no—
200C2×C3×D3—41no—
201C32×C4—26no—
202C3×Dic3—46no—
203C3×Dic3—41no—
204H(5,𝔽2)—26no—
205H(5,𝔽2)C2×C3×D314yes—
206C2×#41—23no—
207C22×D4—23no—
208C22×D4—31no—
209C22×D4C2×C3×D313yes—
210#123—36no—
211#123—63no—
212#121—66no—
213#120—36no—
214#120—63no—
215C2×C22⋊C4—33no—
216#99—312no—
217C23×C3—26no—
218C23×C3—28no—
219C23×C3—24no—
220C23×C3—24no—
221C23×C3—26no—
222C23×C3—23no—
223C23×C3—23no—
224C23×C3—23no—
225C23×C3—23no—
226C23×C3—21no—
227C23×C3—23no—
228C23×C3—26no—
229C23×C3—23no—
230C23×C3—63no—
231C23×C3—81no—
232C23×C3—81no—
233C23×C3—83no—
234C23×C3—83no—
235C23×C3—82no—
236C23×C3—86no—
237C23×C3—84no—
238C23×C3—82no—
239C23×C3—82no—
240C23×C3—241no—
241C23×C3C2×#6716yes—
242C23×C3C2×#6713yes—
243C23×C3C2×#6713yes—
244C23×C3C2×C3×D416yes—
245C23×C3C2×C3×D413yes—
246C23×C3C2×C3×D413yes—
247C23×C3C23×D311yescharacteristic
248C23×C3C24×C311yescharacteristic
249C22×D3—23no—
250C22×D3—81no—
251C3×Q8—28no—
252C3×Q8—24no—
253C3×Q8—24no—
254C3×D4—224no—
255C3×D4—212no—
256C3×D4—212no—
257C3×D4—224no—
258C3×D4—224no—
259C3×D4—212no—
260C3×D4—212no—
261C3×D4—212no—
262C3×D4—212no—
263C3×D4—121no—
264C3×D4—123no—
265C3×D4—246no—
266C2×C3×C4—26no—
267C2×C3×C4—224no—
268C2×C3×C4—212no—
269C2×C3×C4—212no—
270C2×C3×C4—26no—
271C2×C3×C4—23no—
272C2×C3×C4—23no—
273C2×C3×C4—23no—
274C2×C3×C4—23no—
275C2×C3×C4—26no—
276C2×C3×C4—66no—
277C2×C3×C4—63no—
278C2×C3×C4—243no—
279C2×C3×C4C2×#6716yes—
280C2×C3×C4C2×C3×D416yes—
281#67—41no—
282#67—43no—
283#67—86no—
284C2×Dic3—23no—
285C2×Dic3—26no—
286C2×Dic3—83no—
287C2×C32—21no—
288C2×C32—23no—
289C2×C32—23no—
290C2×C32—46no—
291C2×C32C2×#12011yescharacteristic
292C3×D3—81no—
293C24—21no—
294C24—31no—
295C24C3×#6713yescomplemented
296C24C22×C3×D311yescharacteristic
297#41—212no—
298#41—212no—
299C2×Q8—21no—
300C2×D4—23no—
301C2×D4—212no—
302C2×D4—26no—
303C2×D4—212no—
304C2×D4—26no—
305C2×D4—63no—
306C2×D4—63no—
307C2×D4—66no—
308C2×D4C3×#6716yescomplemented
309C2×D4C22×C3×D316yes—
310C22×C4—23no—
311C22×C4—31no—
312C22×C4—63no—
313C22×C4C3×#6713yescomplemented
314C4⋊C4—66no—
315C22⋊C4—36no—
316C22⋊C4—63no—
317C22⋊C4—66no—
318C22⋊C4—66no—
319C22×C3—212no—
320C22×C3—26no—
321C22×C3—26no—
322C22×C3—21no—
323C22×C3—23no—
324C22×C3—23no—
325C22×C3—43no—
326C22×C3—43no—
327C22×C3—41no—
328C22×C3—43no—
329C22×C3—43no—
330C22×C3—41no—
331C22×C3—46no—
332C22×C3—46no—
333C22×C3—46no—
334C22×C3—412no—
335C22×C3—46no—
336C22×C3—46no—
337C22×C3—83no—
338C22×C3—81no—
339C22×C3—86no—
340C22×C3—83no—
341C22×C3—83no—
342C22×C3—86no—
343C22×C3—88no—
344C22×C3—84no—
345C22×C3—84no—
346C22×C3—126no—
347C22×C3—121no—
348C22×C3#95311yescharacteristic
349C22×C3#95313yes—
350C22×C3C3×#12011yescharacteristic
351C22×C3C3×#12013yes—
352C22×C3C22×#6713yes—
353C22×C3C22×C3×D413yes—
354C2×D3—46no—
355C2×D3—41no—
356C3×C4—212no—
357C3×C4—26no—
358C3×C4—26no—
359C3×C4—126no—
360C3×C4—121no—
361Dic3—46no—
362Dic3—41no—
363C32#302611yescharacteristic, complemented, Sylow
364C23—24no—
365C23—23no—
366C23—23no—
367C23—63no—
368C23—81no—
369C23—83no—
370C23—82no—
371C23—241no—
372C23C2×C3×#6716yes—
373C23C2×C3×#6713yes—
374C23C2×C3×#6713yes—
375C23C23×C3×D311yescharacteristic
376Q8—24no—
377D4—212no—
378D4—212no—
379D4—212no—
380D4—121no—
381D4—123no—
382D4—246no—
383C2×C4—212no—
384C2×C4—23no—
385C2×C4—23no—
386C2×C4—66no—
387C2×C4—63no—
388C2×C4—243no—
389C2×C4C2×C3×#6716yes—
390C2×C3—22no—
391C2×C3—23no—
392C2×C3—21no—
393C2×C3—23no—
394C2×C3—23no—
395C2×C3—21no—
396C2×C3—23no—
397C2×C3—21no—
398C2×C3—43no—
399C2×C3—43no—
400C2×C3—412no—
401C2×C3—46no—
402C2×C3—46no—
403C2×C3—241no—
404C2×C3C2×#95311yescharacteristic
405C2×C3C2×C3×#12011yescharacteristic
406D3—81no—
407C22—26no—
408C22—43no—
409C22—43no—
410C22—41no—
411C22—46no—
412C22—46no—
413C22—83no—
414C22—84no—
415C22—126no—
416C22—121no—
417C22C3×#95311yescharacteristic
418C22C3×#95313yes—
419C22C22×C3×#6713yes—
420C4—26no—
421C4—126no—
422C4—121no—
423C3—21no—
424C3#8411011yescharacteristic, direct product factor
425C3C3×#302611yescharacteristic, complemented
426C2—23no—
427C2—21no—
428C2—23no—
429C2—46no—
430C2—241no—
431C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC2×C3C2×#953
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC2×C32×H(5,𝔽2)C2
socleC2×C32C2×#120
hypercenterC2×C3×H(5,𝔽2)D3
hypocenterC3C3×#3026
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C24×C3 ◁ C2×C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×C32×H(5,𝔽2)C2semidirect
2. C3×#5804C2semidirect
3. C3×#5244C2semidirect
4. C3×#5243C2semidirect
5. C2×C3×#834C2semidirect
7. #84110C3direct (N × H)
13. C22×C3×#67C22semidirect
15. C2×C3×#217C22semidirect
17. C3×#937C22semidirect
18. C3×#935C22semidirect
20. C3×#834C22semidirect
22. C2×C3×H(5,𝔽2)D3semidirect
23. C2×C3×H(5,𝔽2)C2×C3semidirect
25. #5804C2×C3semidirect
29. #5244C2×C3semidirect
30. #5243C2×C3semidirect
31. C2×#834C2×C3semidirect
33. C24×C32D4semidirect
35. C23×C3×D3C23semidirect
44. C2×C32×D4D4semidirect
47. C22×C32×C4D4semidirect
52. C22×C3×Dic3C23semidirect
55. C3×#217C23semidirect
75. C22×#67C22×C3semidirect
83. C2×#217C22×C3semidirect
85. #937C22×C3semidirect
86. #935C22×C3semidirect
89. #834C22×C3semidirect
116. C2×H(5,𝔽2)C3×D3semidirect
127. C24×C3#67semidirect
128. C24×C3C3×D4semidirect
131. C23×D3C23×C3semidirect
161. C2×C3×D4#67semidirect
162. C2×C3×D4C3×D4semidirect
171. C22×C3×C4#67semidirect
172. C22×C3×C4C3×D4semidirect
177. C22×Dic3C23×C3semidirect
185. #217C23×C3semidirect
295. C24C3×#67semidirect
308. C2×D4C3×#67semidirect
313. C22×C4C3×#67semidirect
363. C32#3026semidirect
424. C3#84110direct (N × H)
425. C3C3×#3026semidirect

Maximal subgroups

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

subgroupindex
C2×C32×H(5,𝔽2)2
C3×#58042
C3×#52442
C3×#52432
C2×C3×#8342
C3×#30263
#841103