Finite Group Databasebeta

← C3×#70124 #11788901

Subgroups — C3×#70124 #11788901

Order 1152.

689 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5816C211yesmaximal, characteristic
2C3×#5745C211yesmaximal, characteristic
3C2×C32×#101C211yesmaximal, characteristic, complemented
4C2×C3×#836C211yesmaximal, characteristic, complemented
5C3×#5132C211yesmaximal, characteristic
6C3×#4559C211yesmaximal, characteristic, complemented
7C3×#4556C211yesmaximal, characteristic, complemented
8C3×#1915—31nomaximal
9#70124C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C3×#947—21no—
15C3×#946—22no—
16C3×#944—21no—
17C3×#929C414yes—
18C3×#926C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C2×C3×#208C2211yescharacteristic
21C3×#907—22no—
22C3×D3×M16—22no—
23C3×#901—22no—
24C32×#101—22no—
25C3×#836—22no—
26C3×#824—21no—
27C3×#823—21no—
28C3×#568—31no—
29C3×#536—31no—
30#5816C2×C311yescharacteristic
31#5745C2×C311yescharacteristic
32C3×#413—31no—
33C2×C3×#101—21no—
34C2×C3×#101—31no—
35C2×C3×#101D311yescharacteristic, complemented
36C2×C3×#101C2×C311yescharacteristic, complemented
37C2×#836C2×C311yescharacteristic, complemented
38#5132C2×C311yescharacteristic
39C3×#344—31no—
40C3×#344—31no—
41#4559C2×C311yescharacteristic, complemented
42#4556C2×C311yescharacteristic, complemented
43C2×C32×Q8—21no—
44C2×C32×Q8—21no—
45C2×C32×Q8—21no—
46C2×C32×Q8—21no—
47C2×C32×Q8—21no—
48C2×C32×Q8—21no—
49C22×C32×C4C2×C412yes—
50C22×C32×C4C2311yescharacteristic
51C2×C3×#67C2×C412yes—
52C3×#235—21no—
53C3×#235—21no—
54C3×#235—21no—
55C3×#235—21no—
56C2×C3×D12D411yescharacteristic
57C2×C3×C4×D3D412yes—
58C2×C3×Dic6D411yescharacteristic
59C32×M16—21no—
60C32×M16—21no—
61C32×M16—21no—
62C32×M16D412yes—
63C2×C32×C8—21no—
64C2×C32×C8—21no—
65C3×#217C2×C412yes—
66C3×(C2×D3)⋊C4—24no—
67C3×C4⋊Dic3—21no—
68C3×Dic3⋊C4—24no—
69C3×C4×Dic3—21no—
70C3×#208—21no—
71C3×#208—21no—
72C3×#208—21no—
73C3×#208D412yes—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76C3×ΓL(1,𝔽25)—24no—
77C3×D3×C8—24no—
78#1915—31noSylow
79C2×C3×#41—31no—
80C22×C3×Q8—21no—
81C22×C3×Q8C2×D311yescharacteristic
82C22×C3×Q8C22×C311yescharacteristic
83C2×#235C22×C311yescharacteristic
84C3×#131—62no—
85C3×#131—62no—
86C2×C3×M16—21no—
87C2×C3×M16—21no—
88C2×C3×M16—31no—
89C2×C3×M16—31no—
90C2×C3×M16—62no—
91C2×C3×M16C2×D311yescharacteristic
92C2×C3×M16C2×D311yescharacteristic
93C2×C3×M16C22×C311yescharacteristic
94C2×C3×M16C22×C311yescharacteristic
95C3×C4⋊Q8—61no—
96C3×#124—61no—
97C3×#123—34no—
98C3×C22⋊Q8—62no—
99C3×#117—31no—
100#947—21no—
101#946—22no—
102#944—21no—
103#929C3×C414yes—
104#926C22×C311yescharacteristic
105C2×#208C22×C311yescharacteristic
106C2×#208C22×C311yescharacteristic
107#907—22no—
108D3×M16—22no—
109#901—22no—
110C3×#101—24no—
111C3×#101—22no—
112C3×#101—22no—
113C3×#101—61no—
114C3×#101—62no—
115C3×#100—61no—
116#836—22no—
117#824—21no—
118#823—21no—
119C23×C32C22×C411yescharacteristic
120C22×C3×D3C22⋊C412yes—
121C32×Q8—41no—
122C32×Q8—41no—
123C32×Q8—41no—
124C32×Q8—41no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4#4111yescharacteristic
133C2×C32×C4#4111yescharacteristic
134C3×#67—22no—
135C3×#67—22no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3C22⋊C412yes—
138C3×D12—21no—
139C3×D12—21no—
140C3×C4×D3—22no—
141C3×C4×D3—22no—
142C3×Dic6—21no—
143C3×Dic6—21no—
144C32×C8—22no—
145C32×C8—41no—
146C3×C3⋊C8—22no—
147C3×C3⋊C8—41no—
148#568—31no—
149#536—31no—
150#413—31no—
151C2×#101—31no—
152C2×#101C3×D311yescharacteristic, complemented
153#344—31no—
154#344—31no—
155C3×#41—61no—
156C3×#41—61no—
157C3×#41—61no—
158C3×#41—61no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—31no—
174C2×C3×Q8—41no—
175C2×C3×Q8—41no—
176C2×C3×Q8—41no—
177C2×C3×Q8—41no—
178C2×C3×D4—32no—
179C2×C3×D4—31no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—32no—
183C22×C3×C4C4×D312yes—
184C22×C3×C4C2×C3×C412yes—
185C22×C3×C4C22×D311yescharacteristic
186C22×C3×C4C23×C311yescharacteristic
187C2×#67C2×C3×C412yes—
188#235—21no—
189#235—21no—
190#235—21no—
191#235—21no—
192C2×D12C3×D411yescharacteristic
193C2×C4×D3C3×D412yes—
194C2×Dic6C3×D411yescharacteristic
195C3×M16—21no—
196C3×M16—21no—
197C3×M16—22no—
198C3×M16—21no—
199C3×M16—21no—
200C3×M16—21no—
201C3×M16—21no—
202C3×M16—22no—
203C3×M16—32no—
204C3×M16—41no—
205C3×M16—41no—
206C3×M16—64no—
207C3×M16—61no—
208C3×M16—61no—
209C3×M16—61no—
210C3×M16#6712yes—
211C3×M16C3×D412yes—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—41no—
217C2×C3×C8—41no—
218C2×C3×C8—64no—
219C2×C3×C8—61no—
220C2×C3×C8—61no—
221C3×C4⋊C4—64no—
222C3×C4⋊C4—61no—
223C3×C22⋊C4—32no—
224C3×C22⋊C4—64no—
225C3×C42—61no—
226#217C2×C3×C412yes—
227(C2×D3)⋊C4—24no—
228C4⋊Dic3—21no—
229Dic3⋊C4—24no—
230C4×Dic3—21no—
231#208—21no—
232#208—21no—
233#208—21no—
234#208C3×D412yes—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—21no—
237ΓL(1,𝔽25)—24no—
238D3×C8—24no—
239C22×C32—41no—
240C22×C32C2×C22⋊C411yescharacteristic
241C22×C32C4×D411yescharacteristic
242C22×C32C4×D411yescharacteristic
243C2×C3×D3—22no—
244C2×C3×D3—42no—
245C32×C4—42no—
246C32×C4#12011yescharacteristic
247C32×C4#12111yescharacteristic
248C32×C4#12111yescharacteristic
249C32×C4#12311yescharacteristic
250C3×Dic3—22no—
251C3×Dic3—42no—
252C2×#41—31no—
253C22×Q8C2×C3×D311yescharacteristic
254#131—62no—
255#131—62no—
256C2×M16—31no—
257C2×M16—31no—
258C2×M16—62no—
259C2×M16C2×C3×D311yescharacteristic
260C2×M16C2×C3×D311yescharacteristic
261C4⋊Q8—61no—
262#124—61no—
263#123—34no—
264C22⋊Q8—62no—
265#117—31no—
266#101—22no—
267#101—61no—
268#101—62no—
269#100—61no—
270C23×C3—21no—
271C23×C3—32no—
272C23×C3C2×C4×D311yescharacteristic
273C23×C3C22×C3×C411yescharacteristic
274C22×D3C3×C22⋊C412yes—
275C3×Q8—42no—
276C3×Q8—42no—
277C3×Q8—42no—
278C3×Q8—42no—
279C3×Q8—41no—
280C3×Q8—41no—
281C3×Q8—41no—
282C3×Q8—41no—
283C3×Q8—41no—
284C3×Q8—41no—
285C3×Q8—41no—
286C3×Q8—41no—
287C3×Q8—61no—
288C3×Q8—61no—
289C3×D4—62no—
290C3×D4—62no—
291C3×D4—61no—
292C3×D4—61no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—32no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4#23511yescharacteristic
313C2×C3×C4D3×D411yescharacteristic
314C2×C3×C4D3×D411yescharacteristic
315C2×C3×C4D3×D411yescharacteristic
316C2×C3×C4#23711yescharacteristic
317C2×C3×C4C2×#6711yescharacteristic
318C2×C3×C4C2×C3×D411yescharacteristic
319C2×C3×C4C2×C3×D411yescharacteristic
320C2×C3×C4C2×C3×D411yescharacteristic
321C2×C3×C4C2×C3×D411yescharacteristic
322C2×C3×C4C3×#4111yescharacteristic
323C2×C3×C4C3×#4111yescharacteristic
324#67—22no—
325#67—22no—
326C2×Dic3—22no—
327C2×Dic3C3×C22⋊C412yes—
328D12—21no—
329D12—21no—
330C4×D3—22no—
331C4×D3—22no—
332Dic6—21no—
333Dic6—21no—
334C3×C8—22no—
335C3×C8—22no—
336C3×C8—42no—
337C3×C8—42no—
338C3×C8—41no—
339C3×C8—41no—
340C3×C8—62no—
341C3×C8—121no—
342C3⋊C8—22no—
343C3⋊C8—41no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32#38611yescharacteristic
348C3×D3—42no—
349#41—61no—
350#41—61no—
351#41—61no—
352#41—61no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—31no—
360C2×D4—32no—
361C2×D4—31no—
362C22×C4—32no—
363C22×C4C3×C4×D312yes—
364C22×C4C22×C3×D311yescharacteristic
365M16—21no—
366M16—21no—
367M16—21no—
368M16—32no—
369M16—64no—
370M16—61no—
371M16—61no—
372M16—61no—
373M16C3×#6712yes—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—64no—
377C2×C8—61no—
378C2×C8—61no—
379C4⋊C4—64no—
380C4⋊C4—61no—
381C22⋊C4—32no—
382C22⋊C4—64no—
383C42—61no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—42no—
390C22×C3—62no—
391C22×C3—122no—
392C22×C3D3×C22⋊C411yescharacteristic
393C22×C3#67⋊C411yescharacteristic
394C22×C3C4×#6711yescharacteristic
395C22×C3C2×C3×C22⋊C411yescharacteristic
396C22×C3C3×C4×D411yescharacteristic
397C22×C3C3×C4×D411yescharacteristic
398C2×D3—22no—
399C2×D3—42no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—44no—
405C3×C4—42no—
406C3×C4—42no—
407C3×C4—62no—
408C3×C4—122no—
409C3×C4#88311yescharacteristic
410C3×C4#88411yescharacteristic
411C3×C4#93711yescharacteristic
412C3×C4#93811yescharacteristic
413C3×C4C3×#12011yescharacteristic
414C3×C4C3×#12111yescharacteristic
415C3×C4C3×#12111yescharacteristic
416C3×C4C3×#12311yescharacteristic
417Dic3—22no—
418Dic3—42no—
419C32#191511yescharacteristic, complemented, Sylow
420C23—32no—
421C23C2×C3×C4×D311yescharacteristic
422Q8—41no—
423Q8—41no—
424Q8—41no—
425Q8—41no—
426Q8—61no—
427Q8—61no—
428D4—62no—
429D4—62no—
430D4—61no—
431D4—61no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—32no—
436C2×C4—62no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4C3×#23511yescharacteristic
440C2×C4C3×D3×D411yescharacteristic
441C2×C4C3×D3×D411yescharacteristic
442C2×C4C3×D3×D411yescharacteristic
443C2×C4C3×#23711yescharacteristic
444C2×C4C2×C3×#6711yescharacteristic
445C8—22no—
446C8—41no—
447C8—62no—
448C8—121no—
449C2×C3—22no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—41no—
458C2×C3—41no—
459C2×C3—122no—
460C2×C3#495611yescharacteristic
461C2×C3C3×#38611yescharacteristic
462D3—42no—
463C22—41no—
464C22—62no—
465C22—122no—
466C22C3×D3×C22⋊C411yescharacteristic
467C22C3×#67⋊C411yescharacteristic
468C22C3×C4×#6711yescharacteristic
469C4—42no—
470C4—62no—
471C4—122no—
472C4C3×#88311yescharacteristic
473C4C3×#88411yescharacteristic
474C4C3×#93711yescharacteristic
475C4C3×#93811yescharacteristic
476C3—21no—
477C3#7012411yescharacteristic, direct product factor
478C3C3×#191511yescharacteristic, complemented
479C2—21no—
480C2—21no—
481C2—21no—
482C2—122no—
483C2C3×#495611yescharacteristic
subgroupG/H
centerC2×C3#4956
commutatorC23×C3C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#101C2
socleC2×C32#386
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#101D3
hypocenterC3C3×#1915
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#101

Complements

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

NH ≅ G/Ntype
3. C2×C32×#101C2semidirect
4. C2×C3×#836C2semidirect
6. C3×#4559C2semidirect
7. C3×#4556C2semidirect
9. #70124C3direct (N × H)
35. C2×C3×#101D3semidirect
36. C2×C3×#101C2×C3semidirect
37. C2×#836C2×C3semidirect
41. #4559C2×C3semidirect
42. #4556C2×C3semidirect
152. C2×#101C3×D3semidirect
419. C32#1915semidirect
477. C3#70124direct (N × H)
478. C3C3×#1915semidirect

Maximal subgroups

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

subgroupindex
C3×#58162
C3×#57452
C2×C32×#1012
C2×C3×#8362
C3×#51322
C3×#45592
C3×#45562
C3×#19153
#701243