Finite Group Databasebeta

← C3×#69032 #11785540

Subgroups — C3×#69032 #11785540

Order 1152.

557 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#907C211yesmaximal, characteristic
2C3×#5139C211yesmaximal, characteristic
3C2×C3×#822C211yesmaximal, characteristic, complemented
4C3×#5005C211yesmaximal, characteristic
5C3×#4998C211yesmaximal, characteristic, complemented
6C32×#344C211yesmaximal, characteristic, complemented
7C3×#4559C211yesmaximal, characteristic, complemented
8C3×#1814—31nomaximal
9#69032C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C3×#907—21no—
18C3×#907—21no—
19C3×#907C412yescomplemented
20C3×#907C412yescomplemented
21C2×C3×ΓL(1,𝔽25)—21no—
22C2×C3×D3×C8—21no—
23C3×#825—22no—
24C3×#822—21no—
25C3×#822—21no—
26C3×#808—21no—
27C3×#807—21no—
28C3×#805—22no—
29C2×C3×#131—31no—
30C2×#907C2×C311yescharacteristic
31C2×C3×#108—31no—
32C3×#427—31no—
33C3×#421—31no—
34C3×#421—31no—
35#5139C2×C311yescharacteristic
36C2×#822C2×C311yescharacteristic, complemented
37#5005C2×C311yescharacteristic
38#4998C2×C311yescharacteristic, complemented
39C3×#344—21no—
40C3×#344—31no—
41C3×#344D311yescharacteristic, complemented
42C3×#344C2×C311yescharacteristic, complemented
43#4559C2×C311yescharacteristic, complemented
44C22×C32×C4C2311yescharacteristic
45C2×C3×#67—21no—
46C3×#235—21no—
47C3×#235—21no—
48C3×#235C2×C412yes—
49C3×#235D412yes—
50C2×C3×D12Q811yescharacteristic
51C2×C3×C4×D3—21no—
52C2×C3×Dic6Q811yescharacteristic
53C32×M16—21no—
54C32×M16—21no—
55C32×M16—21no—
56C32×M16C2×C412yes—
57C2×C32×C8—21no—
58C2×C32×C8—21no—
59C32×C4⋊C4—21no—
60C32×C22⋊C4—21no—
61C32×C42—21no—
62C3×#217—21no—
63C3×C4⋊Dic3—21no—
64C3×C4×Dic3—21no—
65C3×#208—21no—
66C3×#208—21no—
67C2×C3×C3⋊C8—21no—
68C2×C3×C3⋊C8C2×C411yescharacteristic
69C2×C3×C3⋊C8C2×C411yescharacteristic
70C2×C3×C3⋊C8D411yescharacteristic
71C2×C3×C3⋊C8D411yescharacteristic
72C2×C3×C3⋊C8D411yescharacteristic
73C2×C3×C3⋊C8D411yescharacteristic
74C3×ΓL(1,𝔽25)—22no—
75C3×ΓL(1,𝔽25)—22no—
76C3×D3×C8—22no—
77C3×D3×C8—22no—
78#1814—31noSylow
79C2×C3×#41—31no—
80C2×#235C22×C311yescharacteristic
81C3×#131—32no—
82C3×#131—32no—
83C3×#131—61no—
84C3×#131—61no—
85C2×C3×M16—21no—
86C2×C3×M16—21no—
87C2×C3×M16—31no—
88C2×C3×M16—61no—
89C2×C3×M16C2×D311yescharacteristic
90C2×C3×M16C2×D311yescharacteristic
91C2×C3×M16C22×C311yescharacteristic
92C2×C3×M16C22×C311yescharacteristic
93C22×C3×C8—31no—
94C22×C3×C8—61no—
95C3×#117—21no—
96C3×#117—31no—
97C3×#117C2×D311yescharacteristic
98C3×#117C22×C311yescharacteristic
99#926C22×C311yescharacteristic
100C2×#208C22×C311yescharacteristic
101C22×C3⋊C8C22×C311yescharacteristic
102#907—21no—
103#907—21no—
104#907C3×C412yescomplemented
105#907C3×C412yescomplemented
106C2×ΓL(1,𝔽25)—21no—
107C2×D3×C8—21no—
108C3×#108—61no—
109C3×#108—61no—
110C3×#107—61no—
111C3×#106—61no—
112C3×#104—62no—
113C3×#104—62no—
114#825—22no—
115#822—21no—
116#822—21no—
117#808—21no—
118#807—21no—
119#805—22no—
120C23×C32#4111yescharacteristic
121C22×C3×D3—21no—
122C2×C32×C4—41no—
123C2×C32×C4C22×C411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2×Q811yescharacteristic
128C2×C32×C4#4111yescharacteristic
129C3×#67—22no—
130C3×#67—22no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—41no—
133C3×D12—21no—
134C3×D12C4⋊C412yes—
135C3×C4×D3—22no—
136C3×C4×D3—22no—
137C3×Dic6—21no—
138C3×Dic6C4⋊C412yes—
139C32×C8—22no—
140C32×C8—41no—
141C3×C3⋊C8—41no—
142C3×C3⋊C8C22⋊C411yescharacteristic
143C3×C3⋊C8C22⋊C411yescharacteristic
144C3×C3⋊C8C22⋊C411yescharacteristic
145C3×C3⋊C8C22⋊C411yescharacteristic
146C2×#131—31no—
147C2×#108—31no—
148#427—31no—
149#421—31no—
150#421—31no—
151#344—31no—
152#344C3×D311yescharacteristic, complemented
153C3×#41—32no—
154C3×#41—32no—
155C3×#41—61no—
156C3×#41—61no—
157C2×C3×Q8—31no—
158C2×C3×D4—31no—
159C2×C3×D4—61no—
160C22×C3×C4—21no—
161C22×C3×C4—61no—
162C22×C3×C4C22×D311yescharacteristic
163C22×C3×C4C23×C311yescharacteristic
164C2×#67—21no—
165#235—21no—
166#235—21no—
167#235C2×C3×C412yes—
168#235C3×D412yes—
169C2×D12C3×Q811yescharacteristic
170C2×C4×D3—21no—
171C2×Dic6C3×Q811yescharacteristic
172C3×M16—21no—
173C3×M16—21no—
174C3×M16—21no—
175C3×M16—21no—
176C3×M16—21no—
177C3×M16—21no—
178C3×M16—22no—
179C3×M16—41no—
180C3×M16—41no—
181C3×M16—41no—
182C3×M16—62no—
183C3×M16—62no—
184C3×M16—61no—
185C3×M16—61no—
186C3×M16C4×D312yes—
187C3×M16C2×C3×C412yes—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—22no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—31no—
194C2×C3×C8—31no—
195C2×C3×C8—31no—
196C2×C3×C8—31no—
197C2×C3×C8—31no—
198C2×C3×C8—31no—
199C2×C3×C8—41no—
200C2×C3×C8—62no—
201C2×C3×C8—62no—
202C2×C3×C8—61no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—61no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—21no—
209C3×C22⋊C4—21no—
210C3×C22⋊C4—61no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—41no—
214C3×C42—61no—
215#217—21no—
216C4⋊Dic3—21no—
217C4×Dic3—21no—
218#208—21no—
219#208—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8C2×C3×C411yescharacteristic
222C2×C3⋊C8C2×C3×C411yescharacteristic
223C2×C3⋊C8C3×D411yescharacteristic
224C2×C3⋊C8C3×D411yescharacteristic
225C2×C3⋊C8C3×D411yescharacteristic
226C2×C3⋊C8C3×D411yescharacteristic
227ΓL(1,𝔽25)—22no—
228ΓL(1,𝔽25)—22no—
229D3×C8—22no—
230D3×C8—22no—
231C22×C32—41no—
232C22×C32#11711yescharacteristic
233C22×C32#12111yescharacteristic
234C22×C32C22⋊Q811yescharacteristic
235C2×C3×D3—22no—
236C2×C3×D3—42no—
237C32×C4—81no—
238C32×C4C2×C22⋊C411yescharacteristic
239C32×C4C2×C4⋊C411yescharacteristic
240C32×C4#12111yescharacteristic
241C32×C4C22⋊Q811yescharacteristic
242C3×Dic3—22no—
243C3×Dic3—81no—
244C2×#41—31no—
245#131—32no—
246#131—32no—
247#131—61no—
248#131—61no—
249C2×M16—31no—
250C2×M16—61no—
251C2×M16C2×C3×D311yescharacteristic
252C2×M16C2×C3×D311yescharacteristic
253C22×C8—31no—
254C22×C8—61no—
255#117—31no—
256#117C2×C3×D311yescharacteristic
257#108—61no—
258#108—61no—
259#107—61no—
260#106—61no—
261#104—62no—
262#104—62no—
263C23×C3—21no—
264C23×C3—61no—
265C23×C3#23511yescharacteristic
266C23×C3C3×#4111yescharacteristic
267C22×D3—21no—
268C3×Q8—32no—
269C3×Q8—61no—
270C3×D4—32no—
271C3×D4—62no—
272C3×D4—62no—
273C3×D4—61no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—61no—
284C2×C3×C4—62no—
285C2×C3×C4—62no—
286C2×C3×C4—121no—
287C2×C3×C4C2×C4×D311yescharacteristic
288C2×C3×C4#23511yescharacteristic
289C2×C3×C4D3×D411yescharacteristic
290C2×C3×C4D3×D411yescharacteristic
291C2×C3×C4D3×D411yescharacteristic
292C2×C3×C4D3×Q811yescharacteristic
293C2×C3×C4C22×C3×C411yescharacteristic
294C2×C3×C4C2×C3×D411yescharacteristic
295C2×C3×C4C2×C3×D411yescharacteristic
296C2×C3×C4C2×C3×D411yescharacteristic
297C2×C3×C4C2×C3×Q811yescharacteristic
298C2×C3×C4C3×#4111yescharacteristic
299#67—22no—
300#67—22no—
301C2×Dic3—21no—
302C2×Dic3—41no—
303D12—21no—
304D12C3×C4⋊C412yes—
305C4×D3—22no—
306C4×D3—22no—
307Dic6—21no—
308Dic6C3×C4⋊C412yes—
309C3×C8—22no—
310C3×C8—22no—
311C3×C8—31no—
312C3×C8—31no—
313C3×C8—31no—
314C3×C8—31no—
315C3×C8—42no—
316C3×C8—42no—
317C3×C8—41no—
318C3×C8—41no—
319C3×C8—121no—
320C3⋊C8—41no—
321C3⋊C8C3×C22⋊C411yescharacteristic
322C3⋊C8C3×C22⋊C411yescharacteristic
323C3⋊C8C3×C22⋊C411yescharacteristic
324C3⋊C8C3×C22⋊C411yescharacteristic
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32#38011yescharacteristic
329C3×D3—42no—
330#41—32no—
331#41—32no—
332#41—61no—
333#41—61no—
334C2×Q8—31no—
335C2×D4—31no—
336C2×D4—61no—
337C22×C4—61no—
338C22×C4C22×C3×D311yescharacteristic
339M16—21no—
340M16—21no—
341M16—21no—
342M16—62no—
343M16—62no—
344M16—61no—
345M16—61no—
346M16C3×C4×D312yes—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—31no—
350C2×C8—31no—
351C2×C8—31no—
352C2×C8—31no—
353C2×C8—31no—
354C2×C8—31no—
355C2×C8—62no—
356C2×C8—62no—
357C2×C8—61no—
358C4⋊C4—21no—
359C4⋊C4—61no—
360C22⋊C4—21no—
361C22⋊C4—61no—
362C42—21no—
363C42—61no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—62no—
370C22×C3—81no—
371C22×C3—122no—
372C22×C3#87211yescharacteristic
373C22×C3#88411yescharacteristic
374C22×C3#89611yescharacteristic
375C22×C3C3×#11711yescharacteristic
376C22×C3C3×#12111yescharacteristic
377C22×C3C3×C22⋊Q811yescharacteristic
378C2×D3—22no—
379C2×D3—42no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—62no—
385C3×C4—81no—
386C3×C4—81no—
387C3×C4—82no—
388C3×C4—241no—
389C3×C4D3×C22⋊C411yescharacteristic
390C3×C4#88411yescharacteristic
391C3×C4D3×C4⋊C411yescharacteristic
392C3×C4#89611yescharacteristic
393C3×C4C2×C3×C22⋊C411yescharacteristic
394C3×C4C2×C3×C4⋊C411yescharacteristic
395C3×C4C3×#12111yescharacteristic
396C3×C4C3×C22⋊Q811yescharacteristic
397Dic3—22no—
398Dic3—81no—
399C32#181411yescharacteristic, complemented, Sylow
400C23—61no—
401C23C3×#23511yescharacteristic
402Q8—32no—
403Q8—61no—
404D4—32no—
405D4—62no—
406D4—62no—
407D4—61no—
408C2×C4—41no—
409C2×C4—61no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—121no—
413C2×C4C2×C3×C4×D311yescharacteristic
414C2×C4C3×#23511yescharacteristic
415C2×C4C3×D3×D411yescharacteristic
416C2×C4C3×D3×D411yescharacteristic
417C2×C4C3×D3×D411yescharacteristic
418C2×C4C3×D3×Q811yescharacteristic
419C8—22no—
420C8—31no—
421C8—31no—
422C8—31no—
423C8—31no—
424C8—41no—
425C8—121no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3—41no—
436C2×C3—122no—
437C2×C3#466711yescharacteristic
438C2×C3C3×#38011yescharacteristic
439D3—42no—
440C22—41no—
441C22—62no—
442C22—122no—
443C22C3×#87211yescharacteristic
444C22C3×#88411yescharacteristic
445C22C3×#89611yescharacteristic
446C4—62no—
447C4—81no—
448C4—241no—
449C4C3×D3×C22⋊C411yescharacteristic
450C4C3×#88411yescharacteristic
451C4C3×D3×C4⋊C411yescharacteristic
452C4C3×#89611yescharacteristic
453C3—21no—
454C3#6903211yescharacteristic, direct product factor
455C3C3×#181411yescharacteristic, complemented
456C2—21no—
457C2—21no—
458C2—21no—
459C2—122no—
460C2C3×#466711yescharacteristic
subgroupG/H
centerC3×C4#884
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#344C2
socleC2×C32#380
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#344D3
hypocenterC3C3×#1814
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#344

Complements

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

NH ≅ G/Ntype
3. C2×C3×#822C2semidirect
5. C3×#4998C2semidirect
6. C32×#344C2semidirect
7. C3×#4559C2semidirect
9. #69032C3direct (N × H)
19. C3×#907C4semidirect
20. C3×#907C4semidirect
36. C2×#822C2×C3semidirect
38. #4998C2×C3semidirect
41. C3×#344D3semidirect
42. C3×#344C2×C3semidirect
43. #4559C2×C3semidirect
104. #907C3×C4semidirect
105. #907C3×C4semidirect
152. #344C3×D3semidirect
399. C32#1814semidirect
454. C3#69032direct (N × H)
455. C3C3×#1814semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9072
C3×#51392
C2×C3×#8222
C3×#50052
C3×#49982
C32×#3442
C3×#45592
C3×#18143
#690323