Finite Group Databasebeta

← C3×#68927 #11784115

Subgroups — C3×#68927 #11784115

Order 1152.

629 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#907C211yesmaximal, characteristic
2C2×C3×#825C211yesmaximal, characteristic
3C3×#5122C211yesmaximal, characteristic, complemented
4C3×#5113C211yesmaximal, characteristic
5C2×C3×#815C211yesmaximal, characteristic, complemented
6C32×#341C211yesmaximal, characteristic, complemented
7C3×#4555C211yesmaximal, characteristic, complemented
8C3×#1765—31nomaximal
9#68927C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C22×C32×C8C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×C4×Dic3C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C3×#907—22no—
18C3×#907C414yescomplemented
19C2×C3×ΓL(1,𝔽25)—21no—
20C2×C3×D3×C8—21no—
21C3×#825C414yescomplemented
22C3×#821C412yescomplemented
23C3×#816C412yescomplemented
24C3×#815—22no—
25C3×#815—21no—
26C3×C8×Dic3—21no—
27C2×C3×#131—31no—
28C2×#907C2×C311yescharacteristic
29C2×C3×#104—31no—
30C3×#416—31no—
31C3×#405—31no—
32C2×C3×#97—31no—
33C2×#825C2×C311yescharacteristic
34#5122C2×C311yescharacteristic, complemented
35#5113C2×C311yescharacteristic
36C2×#815C2×C311yescharacteristic, complemented
37C3×#341—21no—
38C3×#341D311yescharacteristic, complemented
39C3×#341C2×C311yescharacteristic, complemented
40C3×#339—31no—
41#4555C2×C311yescharacteristic, complemented
42C22×C32×C4C2311yescharacteristic
43C2×C3×#67—21no—
44C22×C3×Dic3—21no—
45C3×#235—21no—
46C3×#235—21no—
47C3×#235C2×C412yes—
48C3×#235C2×C412yes—
49C2×C3×D12C2×C411yescharacteristic
50C2×C3×C4×D3—21no—
51C2×C3×Dic6C2×C411yescharacteristic
52C32×M16—21no—
53C32×M16—21no—
54C32×M16C2×C412yes—
55C32×M16C2×C412yes—
56C2×C32×C8—21no—
57C2×C32×C8—21no—
58C2×C32×C8—21no—
59C2×C32×C8—21no—
60C2×C32×C8C2×C412yes—
61C3×#217—21no—
62C3×C4⋊Dic3C2×C412yes—
63C3×C4×Dic3—21no—
64C3×C4×Dic3—21no—
65C3×C4×Dic3C2×C412yes—
66C2×C3×C3⋊C8C2×C412yes—
67C2×C3×C3⋊C8D411yescharacteristic
68C2×C3×C3⋊C8D411yescharacteristic
69C2×C3×C3⋊C8D412yes—
70C3×ΓL(1,𝔽25)—24no—
71C3×D3×C8—24no—
72#1765—31noSylow
73C2×C3×#41—31no—
74C2×#235C22×C311yescharacteristic
75C3×#131—34no—
76C3×#131—62no—
77C2×C3×M16—21no—
78C2×C3×M16—21no—
79C2×C3×M16—61no—
80C2×C3×M16C2×D311yescharacteristic
81C2×C3×M16C2×D311yescharacteristic
82C2×C3×M16C22×C311yescharacteristic
83C2×C3×M16C22×C311yescharacteristic
84C22×C3×C8—21no—
85C22×C3×C8—31no—
86C22×C3×C8—61no—
87C22×C3×C8C2×D311yescharacteristic
88C22×C3×C8C22×C311yescharacteristic
89C3×#117—31no—
90C2×C3×C42—31no—
91#926C22×C311yescharacteristic
92C2×C4×Dic3C22×C311yescharacteristic
93C22×C3⋊C8C22×C311yescharacteristic
94#907—22no—
95#907C3×C414yescomplemented
96C2×ΓL(1,𝔽25)—21no—
97C2×D3×C8—21no—
98C3×#104—34no—
99C3×#103—32no—
100C3×#102—32no—
101C3×#97—62no—
102C3×#97—61no—
103C3×C4×C8—61no—
104#825C3×C414yescomplemented
105#821C3×C412yescomplemented
106#816C3×C412yescomplemented
107#815—21no—
108#815—22no—
109C8×Dic3—21no—
110C23×C32#4111yescharacteristic
111C22×C3×D3—21no—
112C2×C32×C4C22×C411yescharacteristic
113C2×C32×C4C22×C411yescharacteristic
114C2×C32×C4C22×C411yescharacteristic
115C2×C32×C4C2×D411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4#4111yescharacteristic
118C3×#67—22no—
119C3×#67—22no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—42no—
124C3×D12—21no—
125C3×D12C4212yes—
126C3×C4×D3—22no—
127C3×C4×D3—22no—
128C3×Dic6—21no—
129C3×Dic6C4212yes—
130C32×C8—22no—
131C32×C8—22no—
132C32×C8—22no—
133C3×C3⋊C8C22⋊C412yes—
134C3×C3⋊C8C22⋊C412yes—
135C2×#131—31no—
136C2×#104—31no—
137#416—31no—
138#405—31no—
139C2×#97—31no—
140#341C3×D311yescharacteristic, complemented
141#339—31no—
142C3×#41—32no—
143C3×#41—32no—
144C3×#41—61no—
145C3×#41—61no—
146C2×C3×Q8—31no—
147C2×C3×D4—31no—
148C2×C3×D4—61no—
149C22×C3×C4—21no—
150C22×C3×C4—61no—
151C22×C3×C4—61no—
152C22×C3×C4C22×D311yescharacteristic
153C22×C3×C4C23×C311yescharacteristic
154C2×#67—21no—
155C22×Dic3—21no—
156#235—21no—
157#235—21no—
158#235C2×C3×C412yes—
159#235C2×C3×C412yes—
160C2×D12C2×C3×C411yescharacteristic
161C2×C4×D3—21no—
162C2×Dic6C2×C3×C411yescharacteristic
163C3×M16—21no—
164C3×M16—21no—
165C3×M16—21no—
166C3×M16—21no—
167C3×M16—22no—
168C3×M16—22no—
169C3×M16—41no—
170C3×M16—41no—
171C3×M16—64no—
172C3×M16C4×D312yes—
173C3×M16C4×D312yes—
174C3×M16C2×C3×C412yes—
175C3×M16C2×C3×C412yes—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—22no—
185C2×C3×C8—32no—
186C2×C3×C8—32no—
187C2×C3×C8—31no—
188C2×C3×C8—31no—
189C2×C3×C8—41no—
190C2×C3×C8—41no—
191C2×C3×C8—41no—
192C2×C3×C8—41no—
193C2×C3×C8—64no—
194C2×C3×C8C4×D312yes—
195C2×C3×C8C2×C3×C412yes—
196C3×C4⋊C4—32no—
197C3×C22⋊C4—61no—
198C3×C42—32no—
199C3×C42—61no—
200C3×C42—61no—
201#217—21no—
202C4⋊Dic3C2×C3×C412yes—
203C4×Dic3—21no—
204C4×Dic3—21no—
205C4×Dic3C2×C3×C412yes—
206C2×C3⋊C8C2×C3×C412yes—
207C2×C3⋊C8C3×D411yescharacteristic
208C2×C3⋊C8C3×D411yescharacteristic
209C2×C3⋊C8C3×D412yes—
210ΓL(1,𝔽25)—24no—
211D3×C8—24no—
212C22×C32—22no—
213C22×C32#11711yescharacteristic
214C22×C32C4×D411yescharacteristic
215C22×C32C4×D411yescharacteristic
216C2×C3×D3—22no—
217C2×C3×D3—42no—
218C32×C4C2×C4211yescharacteristic
219C32×C4C2×C22⋊C411yescharacteristic
220C32×C4C4×D411yescharacteristic
221C32×C4C4×D411yescharacteristic
222C3×Dic3—22no—
223C3×Dic3—42no—
224C3×Dic3—42no—
225C2×#41—31no—
226#131—34no—
227#131—62no—
228C2×M16—61no—
229C2×M16C2×C3×D311yescharacteristic
230C2×M16C2×C3×D311yescharacteristic
231C22×C8—31no—
232C22×C8—61no—
233C22×C8C2×C3×D311yescharacteristic
234#117—31no—
235C2×C42—31no—
236#104—34no—
237#103—32no—
238#102—32no—
239#97—62no—
240#97—61no—
241C4×C8—61no—
242C23×C3—21no—
243C23×C3—61no—
244C23×C3#23911yescharacteristic
245C23×C3C3×#4111yescharacteristic
246C22×D3—21no—
247C3×Q8—32no—
248C3×Q8—61no—
249C3×D4—32no—
250C3×D4—62no—
251C3×D4—62no—
252C3×D4—61no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—61no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—122no—
265C2×C3×C4C2×C4×D311yescharacteristic
266C2×C3×C4C2×C4×D311yescharacteristic
267C2×C3×C4C2×C4×D311yescharacteristic
268C2×C3×C4D3×D411yescharacteristic
269C2×C3×C4D3×D411yescharacteristic
270C2×C3×C4#23711yescharacteristic
271C2×C3×C4C22×C3×C411yescharacteristic
272C2×C3×C4C22×C3×C411yescharacteristic
273C2×C3×C4C22×C3×C411yescharacteristic
274C2×C3×C4C2×C3×D411yescharacteristic
275C2×C3×C4C2×C3×D411yescharacteristic
276C2×C3×C4C3×#4111yescharacteristic
277#67—22no—
278#67—22no—
279C2×Dic3—21no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—42no—
283D12—21no—
284D12C3×C4212yes—
285C4×D3—22no—
286C4×D3—22no—
287Dic6—21no—
288Dic6C3×C4212yes—
289C3×C8—22no—
290C3×C8—22no—
291C3×C8—22no—
292C3×C8—22no—
293C3×C8—22no—
294C3×C8—22no—
295C3×C8—32no—
296C3×C8—32no—
297C3×C8—42no—
298C3×C8—42no—
299C3×C8—42no—
300C3⋊C8C3×C22⋊C412yes—
301C3⋊C8C3×C22⋊C412yes—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32C4×C22⋊C411yescharacteristic
305C2×C32#44412yes—
306C3×D3—42no—
307#41—32no—
308#41—32no—
309#41—61no—
310#41—61no—
311C2×Q8—31no—
312C2×D4—31no—
313C2×D4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4C22×C3×D311yescharacteristic
317M16—21no—
318M16—21no—
319M16—64no—
320M16C3×C4×D312yes—
321M16C3×C4×D312yes—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—32no—
327C2×C8—31no—
328C2×C8—31no—
329C2×C8—32no—
330C2×C8—64no—
331C2×C8C3×C4×D312yes—
332C4⋊C4—32no—
333C22⋊C4—61no—
334C42—32no—
335C42—61no—
336C42—61no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—42no—
343C22×C3—62no—
344C22×C3—122no—
345C22×C3#89211yescharacteristic
346C22×C3#89311yescharacteristic
347C22×C3#89311yescharacteristic
348C22×C3C3×#11711yescharacteristic
349C22×C3C3×C4×D411yescharacteristic
350C22×C3C3×C4×D411yescharacteristic
351C2×D3—22no—
352C2×D3—42no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—62no—
358C3×C4—122no—
359C3×C4—122no—
360C3×C4C42×D311yescharacteristic
361C3×C4D3×C22⋊C411yescharacteristic
362C3×C4#67⋊C411yescharacteristic
363C3×C4#67⋊C411yescharacteristic
364C3×C4C2×C3×C4211yescharacteristic
365C3×C4C2×C3×C22⋊C411yescharacteristic
366C3×C4C3×C4×D411yescharacteristic
367C3×C4C3×C4×D411yescharacteristic
368Dic3—22no—
369Dic3—42no—
370Dic3—42no—
371C32#176511yescharacteristic, complemented, Sylow
372C23—61no—
373C23C3×#23911yescharacteristic
374Q8—32no—
375Q8—61no—
376D4—32no—
377D4—62no—
378D4—62no—
379D4—61no—
380C2×C4—61no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—122no—
386C2×C4C2×C3×C4×D311yescharacteristic
387C2×C4C2×C3×C4×D311yescharacteristic
388C2×C4C2×C3×C4×D311yescharacteristic
389C2×C4C3×D3×D411yescharacteristic
390C2×C4C3×D3×D411yescharacteristic
391C2×C4C3×#23711yescharacteristic
392C8—22no—
393C8—22no—
394C8—22no—
395C8—32no—
396C8—32no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—22no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3—122no—
406C2×C3#466611yescharacteristic
407C2×C3#489412yes—
408C2×C3C3×C4×C22⋊C411yescharacteristic
409C2×C3C3×#44412yes—
410D3—42no—
411C22—22no—
412C22—62no—
413C22—122no—
414C22C3×#89211yescharacteristic
415C22C3×#89311yescharacteristic
416C22C3×#89311yescharacteristic
417C4—62no—
418C4—122no—
419C4—122no—
420C4C3×C42×D311yescharacteristic
421C4C3×D3×C22⋊C411yescharacteristic
422C4C3×#67⋊C411yescharacteristic
423C4C3×#67⋊C411yescharacteristic
424C3—21no—
425C3#6892711yescharacteristic, direct product factor
426C3C3×#176511yescharacteristic, complemented
427C2—21no—
428C2—21no—
429C2—122no—
430C2C3×#466611yescharacteristic
431C2C3×#489412yes—
subgroupG/H
centerC2×C3×C4D3×D4
commutatorC3×C4C2×C3×C42
FrattiniC22×C4C22×C3×D3
FittingC32×#341C2
socleC22×C32C4×D4
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#341D3
hypocenterC3C3×#1765
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C3×#341

Complements

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

NH ≅ G/Ntype
3. C3×#5122C2semidirect
5. C2×C3×#815C2semidirect
6. C32×#341C2semidirect
7. C3×#4555C2semidirect
9. #68927C3direct (N × H)
18. C3×#907C4semidirect
21. C3×#825C4semidirect
22. C3×#821C4semidirect
23. C3×#816C4semidirect
34. #5122C2×C3semidirect
36. C2×#815C2×C3semidirect
38. C3×#341D3semidirect
39. C3×#341C2×C3semidirect
41. #4555C2×C3semidirect
95. #907C3×C4semidirect
104. #825C3×C4semidirect
105. #821C3×C4semidirect
106. #816C3×C4semidirect
140. #341C3×D3semidirect
371. C32#1765semidirect
425. C3#68927direct (N × H)
426. C3C3×#1765semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9072
C2×C3×#8252
C3×#51222
C3×#51132
C2×C3×#8152
C32×#3412
C3×#45552
C3×#17653
#689273