Finite Group Databasebeta

← C3×#70009 #11784835

Subgroups — C3×#70009 #11784835

Order 1152.

1175 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic
2C32×#408C211yesmaximal, characteristic, complemented
3C3×#5237C211yesmaximal, characteristic, complemented
4C3×#5132C212yesmaximal, complemented
5C3×#5110C212yesmaximal, complemented
6C3×#1791—31nomaximal
7#70009C311yesmaximal, characteristic, direct product factor
8C2×C32×#41C2211yescharacteristic
9C3×#1010—22no—
10C3×#1009—22no—
11C3×D3×#41—24no—
12C3×#1007—21no—
13C3×#1004—21no—
14C3×#1004C414yes—
15C2×C3×#235—22no—
16C2×C3×#235C2212yes—
17C2×C32×M16C2211yescharacteristic, complemented
18C22×C32×C8C2211yescharacteristic, complemented
19C2×C3×#208C2212yescomplemented
20C32×C22⋊C8—22no—
21C3×#836—21no—
22C3×#833—21no—
23C3×#824—22no—
24C3×#823—22no—
25C3×(C2×D3)⋊C8—24no—
26C3×#585—31no—
27#5964C2×C311yescharacteristic
28C3×#413—31no—
29C3×#413—32no—
30C3×#408—21no—
31C3×#408—32no—
32C3×#408D311yescharacteristic, complemented
33C3×#408C2×C311yescharacteristic, complemented
34#5237C2×C311yescharacteristic, complemented
35#5132C2×C312yescomplemented
36#5110C2×C312yescomplemented
37C32×#41—22no—
38C32×#41—22no—
39C2×C32×Q8D411yescharacteristic
40C2×C32×D4C2×C412yes—
41C2×C32×D4D411yescharacteristic
42C22×C32×C4D412yes—
43C22×C32×C4C2311yescharacteristic
44C2×C3×#67—24no—
45C2×C3×#67C2×C414yes—
46C3×#239—24no—
47C3×D3×Q8—24no—
48C3×#237—24no—
49C3×#237—28no—
50C3×D3×D4—24no—
51C3×D3×D4—28no—
52C3×#235—24no—
53C3×#235—24no—
54C3×#235—24no—
55C3×#235—22no—
56C3×#235—22no—
57C3×#235—22no—
58C3×#235—22no—
59C2×C3×D12—22no—
60C2×C3×D12D412yes—
61C2×C3×C4×D3—24no—
62C2×C3×C4×D3D414yes—
63C2×C3×Dic6—22no—
64C2×C3×Dic6D412yes—
65C32×M16—22no—
66C2×C32×C8—21no—
67C2×C32×C8—21no—
68C2×C32×C8—22no—
69C3×#208—22no—
70C3×#208—22no—
71C2×C3×C3⋊C8—22no—
72#1791—31noSylow
73C3×2−1+4—62no—
74C3×2−1+4—61no—
75C3×H(5,𝔽2)—34no—
76C3×H(5,𝔽2)—62no—
77C3×H(5,𝔽2)—61no—
78C2×C3×#41—21no—
79C2×C3×#41—32no—
80C2×C3×#41—64no—
81C2×C3×#41—62no—
82C2×C3×#41C2×D311yescharacteristic
83C2×C3×#41C22×C311yescharacteristic
84#1010—22no—
85#1009—22no—
86D3×#41—24no—
87#1007—21no—
88#1004—21no—
89#1004C3×C414yes—
90C2×#235—22no—
91C2×#235C22×C312yes—
92C2×C3×M16—21no—
93C2×C3×M16—32no—
94C2×C3×M16C2×D311yescharacteristic, complemented
95C2×C3×M16C22×C311yescharacteristic, complemented
96C22×C3×C8—21no—
97C22×C3×C8C2×D311yescharacteristic, complemented
98C22×C3×C8C22×C311yescharacteristic, complemented
99C2×#208C22×C312yescomplemented
100C3×#101—62no—
101C3×#101—61no—
102C3×#100—62no—
103C3×#100—61no—
104C3×C22⋊C8—24no—
105C3×C22⋊C8—22no—
106C3×C22⋊C8—22no—
107C3×C22⋊C8—64no—
108#836—21no—
109#833—21no—
110#824—22no—
111#823—22no—
112(C2×D3)⋊C8—24no—
113C23×C32C22⋊C412yes—
114C23×C32C22×C411yescharacteristic
115C22×C3×D3—24no—
116C22×C3×D3C22⋊C414yes—
117C32×Q8—22no—
118C32×D4—24no—
119C32×D4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4C22⋊C412yes—
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D412yes—
125C2×C32×C4C2×D412yes—
126C2×C32×C4C2×D411yescharacteristic
127C3×#67—28no—
128C3×#67—24no—
129C3×#67—24no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3C22⋊C414yes—
132C3×D12—24no—
133C3×D12—22no—
134C3×D12—22no—
135C3×C4×D3—28no—
136C3×C4×D3—24no—
137C3×C4×D3—24no—
138C3×Dic6—24no—
139C3×Dic6—22no—
140C3×Dic6—22no—
141C32×C8—41no—
142C32×C8—41no—
143C3×C3⋊C8—42no—
144#585—31no—
145#413—31no—
146#413—32no—
147#408—32no—
148#408C3×D311yescharacteristic, complemented
149C3×#41—22no—
150C3×#41—22no—
151C3×#41—22no—
152C3×#41—22no—
153C3×#41—42no—
154C3×#41—42no—
155C3×#41—64no—
156C3×#41—64no—
157C3×#41—68no—
158C3×#41—64no—
159C3×#41—64no—
160C3×#41—64no—
161C3×#41—62no—
162C3×#41—62no—
163C3×#41—62no—
164C3×#41—62no—
165C2×C3×Q8—21no—
166C2×C3×Q8—32no—
167C2×C3×Q8—62no—
168C2×C3×Q8—64no—
169C2×C3×Q8D1211yescharacteristic
170C2×C3×Q8C3×D411yescharacteristic
171C2×C3×D4—22no—
172C2×C3×D4—21no—
173C2×C3×D4—34no—
174C2×C3×D4—32no—
175C2×C3×D4—62no—
176C2×C3×D4—68no—
177C2×C3×D4—64no—
178C2×C3×D4—64no—
179C2×C3×D4C4×D312yes—
180C2×C3×D4D1211yescharacteristic
181C2×C3×D4C2×C3×C412yes—
182C2×C3×D4C3×D411yescharacteristic
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—34no—
186C22×C3×C4—64no—
187C22×C3×C4#6712yes—
188C22×C3×C4C3×D412yes—
189C22×C3×C4C22×D311yescharacteristic
190C22×C3×C4C23×C311yescharacteristic
191C2×#67—24no—
192C2×#67C2×C3×C414yes—
193#239—24no—
194D3×Q8—24no—
195#237—24no—
196#237—28no—
197D3×D4—24no—
198D3×D4—28no—
199#235—24no—
200#235—24no—
201#235—24no—
202#235—22no—
203#235—22no—
204#235—22no—
205#235—22no—
206C2×D12—22no—
207C2×D12C3×D412yes—
208C2×C4×D3—24no—
209C2×C4×D3C3×D414yes—
210C2×Dic6—22no—
211C2×Dic6C3×D412yes—
212C3×M16—22no—
213C3×M16—22no—
214C3×M16—42no—
215C3×M16—62no—
216C3×M16—62no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—22no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—42no—
226C2×C3×C8—62no—
227#208—22no—
228#208—22no—
229C2×C3⋊C8—22no—
230C22×C32—22no—
231C22×C32—41no—
232C22×C32—42no—
233C22×C32C2×C22⋊C412yes—
234C22×C32C2×C22⋊C411yescharacteristic
235C2×C3×D3—24no—
236C2×C3×D3—44no—
237C2×C3×D3—84no—
238C32×C4—22no—
239C32×C4#12012yes—
240C32×C4#12011yescharacteristic
241C32×C4#12011yescharacteristic
242C3×Dic3—24no—
2432−1+4—62no—
2442−1+4—61no—
245H(5,𝔽2)—34no—
246H(5,𝔽2)—62no—
247H(5,𝔽2)—61no—
248C2×#41—32no—
249C2×#41—64no—
250C2×#41—62no—
251C2×#41C2×C3×D311yescharacteristic
252C2×M16—32no—
253C2×M16C2×C3×D311yescharacteristic, complemented
254C22×C8C2×C3×D311yescharacteristic, complemented
255#101—62no—
256#101—61no—
257#100—62no—
258#100—61no—
259C22⋊C8—22no—
260C22⋊C8—64no—
261C23×C3—22no—
262C23×C3—21no—
263C23×C3—34no—
264C23×C3—64no—
265C23×C3(C2×D3)⋊C412yes—
266C23×C3C3×C22⋊C412yes—
267C23×C3C2×C4×D311yescharacteristic
268C23×C3C22×C3×C411yescharacteristic
269C22×D3—24no—
270C22×D3C3×C22⋊C414yes—
271C3×Q8—22no—
272C3×Q8—22no—
273C3×Q8—42no—
274C3×Q8—64no—
275C3×Q8—62no—
276C3×Q8—62no—
277C3×D4—24no—
278C3×D4—24no—
279C3×D4—22no—
280C3×D4—22no—
281C3×D4—42no—
282C3×D4—44no—
283C3×D4—64no—
284C3×D4—68no—
285C3×D4—64no—
286C3×D4—64no—
287C3×D4—62no—
288C3×D4—62no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—21no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—21no—
298C2×C3×C4—34no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—68no—
302C2×C3×C4—64no—
303C2×C3×C4—64no—
304C2×C3×C4—64no—
305C2×C3×C4(C2×D3)⋊C412yes—
306C2×C3×C4C3×C22⋊C412yes—
307C2×C3×C4C2×D1211yescharacteristic
308C2×C3×C4D3×D412yes—
309C2×C3×C4D3×D412yes—
310C2×C3×C4C2×#6711yescharacteristic
311C2×C3×C4C2×C3×D411yescharacteristic
312C2×C3×C4C2×C3×D412yes—
313C2×C3×C4C2×C3×D412yes—
314C2×C3×C4C2×C3×D411yescharacteristic
315#67—28no—
316#67—24no—
317#67—24no—
318C2×Dic3—24no—
319C2×Dic3C3×C22⋊C414yes—
320D12—24no—
321D12—22no—
322D12—22no—
323C4×D3—28no—
324C4×D3—24no—
325C4×D3—24no—
326Dic6—24no—
327Dic6—22no—
328Dic6—22no—
329C3×C8—42no—
330C3×C8—42no—
331C3×C8—41no—
332C3×C8—41no—
333C3×C8—41no—
334C3×C8—41no—
335C3×C8—122no—
336C3⋊C8—42no—
337C2×C32—21no—
338C2×C32—22no—
339C2×C32—42no—
340C2×C32#37911yescharacteristic
341C3×D3—44no—
342#41—22no—
343#41—22no—
344#41—64no—
345#41—64no—
346#41—68no—
347#41—64no—
348#41—64no—
349#41—64no—
350#41—62no—
351#41—62no—
352#41—62no—
353#41—62no—
354C2×Q8—32no—
355C2×Q8—62no—
356C2×Q8—64no—
357C2×Q8C3×D1211yescharacteristic
358C2×D4—34no—
359C2×D4—32no—
360C2×D4—62no—
361C2×D4—68no—
362C2×D4—64no—
363C2×D4—64no—
364C2×D4C3×C4×D312yes—
365C2×D4C3×D1211yescharacteristic
366C22×C4—34no—
367C22×C4—64no—
368C22×C4C3×#6712yes—
369C22×C4C22×C3×D311yescharacteristic
370M16—22no—
371M16—62no—
372M16—62no—
373C2×C8—22no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—62no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—21no—
381C22×C3—42no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—44no—
388C22×C3—64no—
389C22×C3—124no—
390C22×C3—244no—
391C22×C3D3×C22⋊C412yes—
392C22×C3C2×(C2×D3)⋊C411yescharacteristic
393C22×C3C2×C3×C22⋊C412yes—
394C22×C3C2×C3×C22⋊C411yescharacteristic
395C2×D3—24no—
396C2×D3—44no—
397C2×D3—84no—
398C3×C4—22no—
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—42no—
404C3×C4—64no—
405C3×C4#88212yes—
406C3×C4#93711yescharacteristic
407C3×C4#93711yescharacteristic
408C3×C4C3×#12012yes—
409C3×C4C3×#12011yescharacteristic
410C3×C4C3×#12011yescharacteristic
411Dic3—24no—
412C32#179111yescharacteristic, complemented, Sylow
413C23—34no—
414C23—64no—
415C23C3×(C2×D3)⋊C412yes—
416C23C2×C3×C4×D311yescharacteristic
417Q8—22no—
418Q8—64no—
419Q8—62no—
420Q8—62no—
421D4—24no—
422D4—22no—
423D4—64no—
424D4—68no—
425D4—64no—
426D4—64no—
427D4—62no—
428D4—62no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—34no—
432C2×C4—68no—
433C2×C4—64no—
434C2×C4—64no—
435C2×C4—64no—
436C2×C4C3×(C2×D3)⋊C412yes—
437C2×C4C2×C3×D1211yescharacteristic
438C2×C4C3×D3×D412yes—
439C2×C4C3×D3×D412yes—
440C2×C4C2×C3×#6711yescharacteristic
441C8—41no—
442C8—41no—
443C8—122no—
444C2×C3—22no—
445C2×C3—22no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—22no—
449C2×C3—21no—
450C2×C3—42no—
451C2×C3—42no—
452C2×C3—42no—
453C2×C3—82no—
454C2×C3—124no—
455C2×C3#495511yescharacteristic
456C2×C3C3×#37911yescharacteristic
457D3—44no—
458C22—22no—
459C22—41no—
460C22—42no—
461C22—64no—
462C22—124no—
463C22—244no—
464C22C3×D3×C22⋊C412yes—
465C22C2×C3×(C2×D3)⋊C411yescharacteristic
466C4—22no—
467C4—64no—
468C4C3×#88212yes—
469C4C3×#93711yescharacteristic
470C4C3×#93711yescharacteristic
471C3—21no—
472C3#7000911yescharacteristic, direct product factor
473C3C3×#179111yescharacteristic, complemented
474C2—21no—
475C2—22no—
476C2—42no—
477C2—124no—
478C2C3×#495511yescharacteristic
subgroupG/H
centerC3×C4#937
commutatorC23×C3C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#408C2
socleC2×C32#379
prime radicalC3×#5964C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#408D3
hypocenterC3C3×#1791
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#408

Complements

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

NH ≅ G/Ntype
2. C32×#408C2semidirect
3. C3×#5237C2semidirect
4. C3×#5132C2semidirect
5. C3×#5110C2semidirect
7. #70009C3direct (N × H)
17. C2×C32×M16C22semidirect
18. C22×C32×C8C22semidirect
19. C2×C3×#208C22semidirect
32. C3×#408D3semidirect
33. C3×#408C2×C3semidirect
34. #5237C2×C3semidirect
35. #5132C2×C3semidirect
36. #5110C2×C3semidirect
94. C2×C3×M16C2×D3semidirect
95. C2×C3×M16C22×C3semidirect
97. C22×C3×C8C2×D3semidirect
98. C22×C3×C8C22×C3semidirect
99. C2×#208C22×C3semidirect
148. #408C3×D3semidirect
253. C2×M16C2×C3×D3semidirect
254. C22×C8C2×C3×D3semidirect
412. C32#1791semidirect
472. C3#70009direct (N × H)
473. C3C3×#1791semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
C32×#4082
C3×#52372
C3×#51322
C3×#51102
C3×#17913
#700093