Finite Group Databasebeta

← C3×#10707070 #72222603

Subgroups — C3×#10707070 #72222603

Order 2304.

588 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#70885C211yesmaximal, characteristic
2C3×#68077C211yesmaximal, characteristic, complemented
3C3×#67464C211yesmaximal, characteristic, complemented
4C3×#67762C211yesmaximal, characteristic, complemented
5C3×#67465C211yesmaximal, characteristic, complemented
6C3×#67442C211yesmaximal, characteristic, complemented
7C32×#1580C211yesmaximal, characteristic, complemented
8#10707070C311yesmaximal, characteristic, direct product factor
9C3×#12067—31nomaximal
10C32×C4×D8C2211yescharacteristic
11C32×C8×D4—21no—
12C2×C3×#826C411yescharacteristic
13C3×C4×D24C2211yescharacteristic
14C3×C8×D12—21no—
15C32×#334C2211yescharacteristic
16C32×D4⋊C8—21no—
17C32×#322C2211yescharacteristic
18C3×#4562C411yescharacteristic
19C3×#4491C411yescharacteristic
20C3×#4490C411yescharacteristic
21C3×#4461C2211yescharacteristic
22C3×#4461C2211yescharacteristic
23C3×#4459—21no—
24C3×#4457C2211yescharacteristic
25C3×#2177—31no—
26C3×#1580—21no—
27C3×#1580—31no—
28C3×#1580D311yescharacteristic, complemented
29C3×#1580C2×C311yescharacteristic, complemented
30#70885C2×C311yescharacteristic
31#68077C2×C311yescharacteristic, complemented
32C3×#1368—31no—
33C3×#1333—31no—
34C3×#1333—31no—
35C3×#1313—31no—
36#67762C2×C311yescharacteristic, complemented
37#67465C2×C311yescharacteristic, complemented
38#67464C2×C311yescharacteristic, complemented
39#67442C2×C311yescharacteristic, complemented
40C2×C32×D8C2×C411yescharacteristic
41C22×C32×C8—41no—
42C32×C4×D4—21no—
43C2×C3×D24C2×C411yescharacteristic
44C2×C3×D3×C8—41no—
45C3×C4×D12—21no—
46C32×#107C2×C411yescharacteristic
47C32×C4⋊C8—21no—
48C32×#102—21no—
49C32×C22⋊C8—41no—
50C32×C4×C8D411yescharacteristic
51C32×C4×C8D411yescharacteristic
52C32×C4×C8C2311yescharacteristic
53C3×#826C814yescomplemented
54C3×#821—21no—
55C3×(C2×D3)⋊C8—41no—
56C3×#818C2×C411yescharacteristic
57C2×C3×C3⋊C16—21no—
58C2×C3×C3⋊C16C2×C411yescharacteristic
59C2×C3×C3⋊C16C2×C411yescharacteristic
60C3×C4⋊(C3⋊C8)—21no—
61#12067—31noSylow
62C2×C3×D16—31no—
63C3×C4×D8—21no—
64C3×C4×D8—31no—
65C3×C4×D8C2×D311yescharacteristic
66C3×C4×D8C22×C311yescharacteristic
67C3×C8×D4—22no—
68C3×C8×D4—21no—
69C3×C8×D4—21no—
70C3×C8×D4—61no—
71C2×#826C3×C411yescharacteristic
72C4×D24C22×C311yescharacteristic
73C8×D12—21no—
74C3×#367—31no—
75C3×#357—31no—
76C3×#357—31no—
77C3×#346—31no—
78C3×#346—31no—
79C3×#334—21no—
80C3×#334—31no—
81C3×#334C2×D311yescharacteristic
82C3×#334C22×C311yescharacteristic
83C3×D4⋊C8—22no—
84C3×D4⋊C8—21no—
85C3×D4⋊C8—21no—
86C3×D4⋊C8—61no—
87C3×#322—21no—
88C3×#322C2×D311yescharacteristic
89C3×#322C22×C311yescharacteristic
90#4562C3×C411yescharacteristic
91#4491C3×C411yescharacteristic
92#4490C3×C411yescharacteristic
93#4461C22×C311yescharacteristic
94#4461C22×C311yescharacteristic
95#4459—21no—
96#4457C22×C311yescharacteristic
97C2×C32×D4—21no—
98C22×C32×C4—41no—
99C2×C3×D12—21no—
100C2×C3×C4×D3—41no—
101C32×D8—21no—
102C32×D8C2×C812yescomplemented
103C2×C32×C8—21no—
104C2×C32×C8—21no—
105C2×C32×C8—41no—
106C2×C32×C8—82no—
107C2×C32×C8C22×C411yescharacteristic
108C2×C32×C8#4111yescharacteristic
109C32×C4⋊C4—21no—
110C32×C22⋊C4—41no—
111C32×C42C2×D411yescharacteristic
112C3×(C2×D3)⋊C4—41no—
113C3×C4⋊Dic3—21no—
114C2×C3×C3⋊C8—41no—
115C3×D24—21no—
116C3×D24C2×C812yescomplemented
117C3×D3×C8—82no—
118C3×C3⋊C16—21no—
119C3×C3⋊C16—41no—
120C3×C3⋊C16C2×C812yescomplemented
121#2177—31no—
122#1580—31no—
123#1580C3×D311yescharacteristic, complemented
124#1368—31no—
125#1333—31no—
126#1333—31no—
127#1313—31no—
128C2×C3×D8—21no—
129C2×C3×D8—31no—
130C2×C3×D8C4×D311yescharacteristic
131C2×C3×D8C2×C3×C411yescharacteristic
132C22×C3×C8—42no—
133C22×C3×C8—41no—
134C22×C3×C8—41no—
135C22×C3×C8—121no—
136C3×C4×D4—22no—
137C3×C4×D4—21no—
138C3×C4×D4—21no—
139C3×C4×D4—61no—
140C2×D24C2×C3×C411yescharacteristic
141C2×D3×C8—41no—
142C4×D12—21no—
143C3×D16—34no—
144C2×C3×C16—31no—
145C2×C3×C16—31no—
146C2×C3×C16—61no—
147C3×#107—21no—
148C3×#107—31no—
149C3×#107C4×D311yescharacteristic
150C3×#107C2×C3×C411yescharacteristic
151C3×C4⋊C8—22no—
152C3×C4⋊C8—21no—
153C3×C4⋊C8—21no—
154C3×C4⋊C8—61no—
155C3×#102—22no—
156C3×#102—21no—
157C3×#102—21no—
158C3×#102—61no—
159C3×C22⋊C8—42no—
160C3×C22⋊C8—41no—
161C3×C22⋊C8—41no—
162C3×C22⋊C8—121no—
163C3×C4×C8—21no—
164C3×C4×C8—21no—
165C3×C4×C8—21no—
166C3×C4×C8#6711yescharacteristic
167C3×C4×C8#6711yescharacteristic
168C3×C4×C8C3×D411yescharacteristic
169C3×C4×C8C3×D411yescharacteristic
170C3×C4×C8C22×D311yescharacteristic
171C3×C4×C8C23×C311yescharacteristic
172#826C3×C814yescomplemented
173#821—21no—
174(C2×D3)⋊C8—41no—
175#818C2×C3×C411yescharacteristic
176C2×C3⋊C16—21no—
177C2×C3⋊C16C2×C3×C411yescharacteristic
178C2×C3⋊C16C2×C3×C411yescharacteristic
179C4⋊(C3⋊C8)—21no—
180C23×C32—41no—
181C22×C3×D3—41no—
182C32×D4—22no—
183C32×D4—41no—
184C2×C32×C4—41no—
185C2×C32×C4—81no—
186C2×C32×C4—81no—
187C2×C32×C4C4×D411yescharacteristic
188C2×C32×C4AGL(1,ℤ/8)11yescharacteristic
189C2×C32×C4AGL(1,ℤ/8)11yescharacteristic
190C2×C3×Dic3—41no—
191C3×D12—22no—
192C3×D12—41no—
193C3×C4×D3—81no—
194C3×C4×D3—81no—
195C32×C8—21no—
196C32×C8—41no—
197C32×C8—41no—
198C32×C8—81no—
199C32×C8C22×C811yescharacteristic
200C32×C8#13111yescharacteristic
201C3×C3⋊C8—81no—
202C2×D16—31no—
203C4×D8—31no—
204C4×D8C2×C3×D311yescharacteristic
205C8×D4—21no—
206C8×D4—61no—
207#367—31no—
208#357—31no—
209#357—31no—
210#346—31no—
211#346—31no—
212#334—31no—
213#334C2×C3×D311yescharacteristic
214D4⋊C8—21no—
215D4⋊C8—61no—
216#322C2×C3×D311yescharacteristic
217C2×C3×D4—22no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—61no—
221C22×C3×C4—42no—
222C22×C3×C4—41no—
223C22×C3×C4—41no—
224C22×C3×C4—121no—
225C2×D12—21no—
226C2×C4×D3—41no—
227C3×D8—22no—
228C3×D8—21no—
229C3×D8—21no—
230C3×D8—22no—
231C3×D8—32no—
232C3×D8—61no—
233C3×D8D3×C812yescomplemented
234C3×D8C2×C3×C812yescomplemented
235C2×C3×C8—22no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—41no—
243C2×C3×C8—42no—
244C2×C3×C8—41no—
245C2×C3×C8—41no—
246C2×C3×C8—84no—
247C2×C3×C8—82no—
248C2×C3×C8—82no—
249C2×C3×C8—121no—
250C2×C3×C8—242no—
251C2×C3×C8C2×C4×D311yescharacteristic
252C2×C3×C8#23511yescharacteristic
253C2×C3×C8C22×C3×C411yescharacteristic
254C2×C3×C8C3×#4111yescharacteristic
255C3×C4⋊C4—22no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—61no—
259C3×C22⋊C4—42no—
260C3×C22⋊C4—41no—
261C3×C22⋊C4—41no—
262C3×C22⋊C4—121no—
263C3×C42—21no—
264C3×C42C2×#6711yescharacteristic
265C3×C42C2×C3×D411yescharacteristic
266(C2×D3)⋊C4—41no—
267C4⋊Dic3—21no—
268C2×C3⋊C8—41no—
269D24—21no—
270D24C2×C3×C812yescomplemented
271D3×C8—82no—
272C3×C16—32no—
273C3×C16—61no—
274C3×C16—121no—
275C3⋊C16—21no—
276C3⋊C16—41no—
277C3⋊C16C2×C3×C812yescomplemented
278C22×C32—42no—
279C22×C32—81no—
280C22×C32—81no—
281C22×C32#44211yescharacteristic
282C2×C3×D3—42no—
283C2×C3×D3—81no—
284C2×C3×D3—81no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—81no—
288C32×C4C8×D411yescharacteristic
289C32×C4#44411yescharacteristic
290C3×Dic3—81no—
291C2×D8—31no—
292C2×D8C3×C4×D311yescharacteristic
293C22×C8—41no—
294C22×C8—121no—
295C4×D4—21no—
296C4×D4—61no—
297D16—34no—
298C2×C16—31no—
299C2×C16—31no—
300C2×C16—61no—
301#107—31no—
302#107C3×C4×D311yescharacteristic
303C4⋊C8—21no—
304C4⋊C8—61no—
305#102—21no—
306#102—61no—
307C22⋊C8—41no—
308C22⋊C8—121no—
309C4×C8C3×#6711yescharacteristic
310C4×C8C3×#6711yescharacteristic
311C4×C8C22×C3×D311yescharacteristic
312C23×C3—42no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—121no—
316C22×D3—41no—
317C3×D4—24no—
318C3×D4—22no—
319C3×D4—22no—
320C3×D4—42no—
321C3×D4—41no—
322C3×D4—41no—
323C3×D4—62no—
324C3×D4—121no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—82no—
332C2×C3×C4—81no—
333C2×C3×C4—81no—
334C2×C3×C4—82no—
335C2×C3×C4—81no—
336C2×C3×C4—81no—
337C2×C3×C4—121no—
338C2×C3×C4—241no—
339C2×C3×C4—241no—
340C2×C3×C4C4×#6711yescharacteristic
341C2×C3×C4#93211yescharacteristic
342C2×C3×C4#94911yescharacteristic
343C2×C3×C4C3×C4×D411yescharacteristic
344C2×C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
345C2×C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
346C2×Dic3—41no—
347D12—22no—
348D12—41no—
349C4×D3—81no—
350C4×D3—81no—
351C3×C8—21no—
352C3×C8—21no—
353C3×C8—21no—
354C3×C8—21no—
355C3×C8—41no—
356C3×C8—41no—
357C3×C8—42no—
358C3×C8—41no—
359C3×C8—41no—
360C3×C8—41no—
361C3×C8—81no—
362C3×C8—82no—
363C3×C8—81no—
364C3×C8—81no—
365C3×C8—241no—
366C3×C8C2×D3×C811yescharacteristic
367C3×C8#90111yescharacteristic
368C3×C8C22×C3×C811yescharacteristic
369C3×C8C3×#13111yescharacteristic
370C3⋊C8—81no—
371C2×C32—82no—
372C2×C32#158011yescharacteristic
373C2×C32AGL(1,ℤ/16)11yescharacteristic
374C2×C32AGL(1,ℤ/16)11yescharacteristic
375C3×D3—82no—
376C2×D4—21no—
377C2×D4—61no—
378C22×C4—41no—
379C22×C4—121no—
380D8—21no—
381D8—32no—
382D8—61no—
383D8C3×D3×C812yescomplemented
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—41no—
387C2×C8—82no—
388C2×C8—121no—
389C2×C8—242no—
390C2×C8C2×C3×C4×D311yescharacteristic
391C2×C8C3×#23511yescharacteristic
392C4⋊C4—21no—
393C4⋊C4—61no—
394C22⋊C4—41no—
395C22⋊C4—121no—
396C42C2×C3×#6711yescharacteristic
397C16—32no—
398C16—61no—
399C16—121no—
400C22×C3—21no—
401C22×C3—44no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—122no—
411C22×C3—241no—
412C22×C3—241no—
413C22×C3#501411yescharacteristic
414C22×C3C3×#44211yescharacteristic
415C2×D3—42no—
416C2×D3—81no—
417C2×D3—81no—
418C3×C4—22no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—41no—
426C3×C4—82no—
427C3×C4—81no—
428C3×C4—81no—
429C3×C4—241no—
430C3×C4C8×#6711yescharacteristic
431C3×C4#514511yescharacteristic
432C3×C4C3×C8×D411yescharacteristic
433C3×C4C3×#44411yescharacteristic
434Dic3—81no—
435C32#1206711yescharacteristic, complemented, Sylow
436C23—41no—
437C23—121no—
438D4—22no—
439D4—41no—
440D4—62no—
441D4—121no—
442C2×C4—41no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—121no—
446C2×C4—241no—
447C2×C4—241no—
448C2×C4C3×C4×#6711yescharacteristic
449C2×C4C3×#93211yescharacteristic
450C2×C4C3×#94911yescharacteristic
451C8—21no—
452C8—41no—
453C8—41no—
454C8—81no—
455C8—241no—
456C8C2×C3×D3×C811yescharacteristic
457C8C3×#90111yescharacteristic
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—84no—
462C2×C3—82no—
463C2×C3—82no—
464C2×C3—242no—
465C2×C3#7084011yescharacteristic
466C2×C3#7091811yescharacteristic
467C2×C3#7092911yescharacteristic
468C2×C3C3×#158011yescharacteristic
469C2×C3C3×AGL(1,ℤ/16)11yescharacteristic
470C2×C3C3×AGL(1,ℤ/16)11yescharacteristic
471D3—82no—
472C22—42no—
473C22—81no—
474C22—81no—
475C22—122no—
476C22—241no—
477C22—241no—
478C22C3×#501411yescharacteristic
479C4—21no—
480C4—21no—
481C4—81no—
482C4—241no—
483C4C3×C8×#6711yescharacteristic
484C4C3×#514511yescharacteristic
485C3—21no—
486C3C3×#1206711yescharacteristic, complemented
487C3#1070707011yescharacteristic, direct product factor
488C2—82no—
489C2—242no—
490C2C3×#7084011yescharacteristic
491C2C3×#7091811yescharacteristic
492C2C3×#7092911yescharacteristic
subgroupG/H
centerC22×C3#5014
commutatorC3×C8C22×C3×C8
FrattiniC4×C8C22×C3×D3
FittingC32×#1580C2
socleC22×C32#442
prime radicalC2×C3×#826C4
cosocleC3×C4×C8C23×C3
hypercenterC3×#1580D3
hypocenterC3C3×#12067
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#322 ◁ C3×#1580

Complements

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

NH ≅ G/Ntype
2. C3×#68077C2semidirect
3. C3×#67464C2semidirect
4. C3×#67762C2semidirect
5. C3×#67465C2semidirect
6. C3×#67442C2semidirect
7. C32×#1580C2semidirect
8. #10707070C3direct (N × H)
28. C3×#1580D3semidirect
29. C3×#1580C2×C3semidirect
31. #68077C2×C3semidirect
36. #67762C2×C3semidirect
37. #67465C2×C3semidirect
38. #67464C2×C3semidirect
39. #67442C2×C3semidirect
53. C3×#826C8semidirect
102. C32×D8C2×C8semidirect
116. C3×D24C2×C8semidirect
120. C3×C3⋊C16C2×C8semidirect
123. #1580C3×D3semidirect
172. #826C3×C8semidirect
233. C3×D8D3×C8semidirect
234. C3×D8C2×C3×C8semidirect
270. D24C2×C3×C8semidirect
277. C3⋊C16C2×C3×C8semidirect
383. D8C3×D3×C8semidirect
435. C32#12067semidirect
486. C3C3×#12067semidirect
487. C3#10707070direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#708852
C3×#680772
C3×#674642
C3×#677622
C3×#674652
C3×#674422
C32×#15802
#107070703
C3×#120673