Finite Group Databasebeta

← C3×#72081 #11796940

Subgroups — C3×#72081 #11796940

Order 1152.

573 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#135C211yesmaximal, characteristic
2C2×C3×#819C211yesmaximal, characteristic, complemented
3C2×C3×#812C211yesmaximal, characteristic, complemented
4C3×#4565C212yesmaximal, complemented
5C3×#4563C212yesmaximal, complemented
6C3×#2148—31nomaximal
7#72081C311yesmaximal, characteristic, direct product factor
8C2×C32×#41—21no—
9C32×#135—21no—
10C32×#135—21no—
11C32×#135C412yes—
12C32×#135C412yes—
13C2×C32×Q16C2211yescharacteristic
14C2×C32×SD16—21no—
15C2×C32×D8C2211yescharacteristic
16C22×C32×C8C2211yescharacteristic
17C2×C3×#208—21no—
18C3×#819—21no—
19C3×#819C2212yes—
20C3×#812—21no—
21C3×#812C2212yes—
22C2×C3×C3⋊C16—21no—
23C2×C3×#135—21no—
24C2×C3×#135D311yescharacteristic
25C2×C3×#135C2×C311yescharacteristic
26C2×C3×M32—31no—
27C2×C3×#108—31no—
28C2×#819C2×C311yescharacteristic, complemented
29C2×#812C2×C311yescharacteristic, complemented
30C3×#362—32no—
31C3×#361—32no—
32#4565C2×C312yescomplemented
33#4563C2×C312yescomplemented
34C32×#41—22no—
35C32×#41—22no—
36C32×#41—41no—
37C32×#41—41no—
38C2×C32×Q8—21no—
39C2×C32×D4—21no—
40C2×C32×D4—41no—
41C22×C32×C4—41no—
42C22×C32×C4D411yescharacteristic
43C32×Q16—21no—
44C32×Q16C2×C412yes—
45C32×SD16—22no—
46C32×SD16—22no—
47C32×D8—21no—
48C32×D8C2×C412yes—
49C2×C32×C8C2×C411yescharacteristic
50C2×C32×C8C2×C411yescharacteristic
51C2×C32×C8D411yescharacteristic
52C2×C32×C8D411yescharacteristic
53C2×C32×C8D411yescharacteristic
54C2×C32×C8C2311yescharacteristic
55C3×#208—22no—
56C3×#208—41no—
57C2×C3×C3⋊C8—41no—
58C3×C3⋊C16—22no—
59#2148—31noSylow
60C2×C3×#41—22no—
61C2×C3×#41—21no—
62C2×C3×#41—21no—
63C3×#135—22no—
64C3×#135—22no—
65C3×#135—21no—
66C3×#135—21no—
67C3×#135—21no—
68C3×#135—21no—
69C3×#135—22no—
70C3×#135—22no—
71C3×#135Dic312yes—
72C3×#135Dic312yes—
73C3×#135C3×C412yes—
74C3×#135C3×C412yes—
75C2×C3×Q16—21no—
76C2×C3×Q16C2×D311yescharacteristic
77C2×C3×Q16C22×C311yescharacteristic
78C2×C3×SD16—22no—
79C2×C3×SD16—21no—
80C2×C3×SD16—21no—
81C2×C3×D8—21no—
82C2×C3×D8C2×D311yescharacteristic
83C2×C3×D8C22×C311yescharacteristic
84C2×C3×M16—61no—
85C22×C3×C8—21no—
86C22×C3×C8C2×D311yescharacteristic
87C22×C3×C8C22×C311yescharacteristic
88C2×#208—21no—
89C3×M32—32no—
90C3×M32—61no—
91C2×C3×C16—61no—
92C3×#108—32no—
93C3×#108—61no—
94#819—21no—
95#819C22×C312yes—
96#812—21no—
97#812C22×C312yes—
98C2×C3⋊C16—21no—
99C23×C32—41no—
100C23×C32SD1611yescharacteristic
101C32×Q8—22no—
102C32×Q8—41no—
103C32×D4—22no—
104C32×D4—42no—
105C32×D4—42no—
106C32×D4—41no—
107C2×C32×C4—41no—
108C2×C32×C4—42no—
109C2×C32×C4—42no—
110C2×C32×C4C22⋊C411yescharacteristic
111C2×C32×C4C22⋊C411yescharacteristic
112C2×C32×C4D811yescharacteristic
113C2×C32×C4D811yescharacteristic
114C2×C32×C4SD1611yescharacteristic
115C2×C32×C4C2×D411yescharacteristic
116C32×C8C22⋊C411yescharacteristic
117C32×C8C22⋊C411yescharacteristic
118C32×C8C22×C411yescharacteristic
119C32×C8C2×D411yescharacteristic
120C3×C3⋊C8—42no—
121C2×#135C3×D311yescharacteristic
122C2×M32—31no—
123C2×#108—31no—
124#362—32no—
125#361—32no—
126C3×#41—24no—
127C3×#41—24no—
128C3×#41—22no—
129C3×#41—22no—
130C3×#41—22no—
131C3×#41—22no—
132C3×#41—42no—
133C3×#41—42no—
134C3×#41—41no—
135C3×#41—41no—
136C3×#41—41no—
137C3×#41—41no—
138C2×C3×Q8—22no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—42no—
145C2×C3×D4—41no—
146C2×C3×D4—41no—
147C22×C3×C4—21no—
148C22×C3×C4—42no—
149C22×C3×C4—41no—
150C22×C3×C4—41no—
151C22×C3×C4#6711yescharacteristic
152C22×C3×C4C3×D411yescharacteristic
153C3×Q16—22no—
154C3×Q16—21no—
155C3×Q16—21no—
156C3×Q16—22no—
157C3×Q16C2×Dic312yes—
158C3×Q16C2×C3×C412yes—
159C3×SD16—24no—
160C3×SD16—24no—
161C3×SD16—22no—
162C3×SD16—22no—
163C3×SD16—22no—
164C3×SD16—22no—
165C3×D8—22no—
166C3×D8—21no—
167C3×D8—21no—
168C3×D8—22no—
169C3×D8C2×Dic312yes—
170C3×D8C2×C3×C412yes—
171C3×M16—62no—
172C3×M16—121no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—121no—
180C2×C3×C8C2×Dic311yescharacteristic
181C2×C3×C8C2×Dic311yescharacteristic
182C2×C3×C8#6711yescharacteristic
183C2×C3×C8#6711yescharacteristic
184C2×C3×C8#6711yescharacteristic
185C2×C3×C8C2×C3×C411yescharacteristic
186C2×C3×C8C2×C3×C411yescharacteristic
187C2×C3×C8C3×D411yescharacteristic
188C2×C3×C8C3×D411yescharacteristic
189C2×C3×C8C3×D411yescharacteristic
190C2×C3×C8C22×D311yescharacteristic
191C2×C3×C8C23×C311yescharacteristic
192#208—22no—
193#208—41no—
194C2×C3⋊C8—41no—
195C3×C16—62no—
196C3⋊C16—22no—
197C22×C32—41no—
198C22×C32—42no—
199C22×C32—82no—
200C22×C32#10211yescharacteristic
201C22×C32#10211yescharacteristic
202C22×C32C2×SD1611yescharacteristic
203C32×C4—42no—
204C32×C4#10211yescharacteristic
205C32×C4#10211yescharacteristic
206C32×C4C2×C22⋊C411yescharacteristic
207C32×C4C2×D811yescharacteristic
208C2×#41—21no—
209#135—21no—
210#135—21no—
211#135C3×Dic312yes—
212#135C3×Dic312yes—
213C2×Q16C2×C3×D311yescharacteristic
214C2×SD16—21no—
215C2×D8C2×C3×D311yescharacteristic
216C2×M16—61no—
217C22×C8C2×C3×D311yescharacteristic
218M32—32no—
219M32—61no—
220C2×C16—61no—
221#108—32no—
222#108—61no—
223C23×C3—21no—
224C23×C3—42no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3#21411yescharacteristic
228C23×C3C3×SD1611yescharacteristic
229C3×Q8—24no—
230C3×Q8—22no—
231C3×Q8—22no—
232C3×Q8—42no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×D4—24no—
236C3×D4—22no—
237C3×D4—22no—
238C3×D4—42no—
239C3×D4—44no—
240C3×D4—44no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—42no—
244C3×D4—42no—
245C3×D4—41no—
246C3×D4—41no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—44no—
254C2×C3×C4—44no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4#21311yescharacteristic
263C2×C3×C4#21311yescharacteristic
264C2×C3×C4#21411yescharacteristic
265C2×C3×C4#21711yescharacteristic
266C2×C3×C4#21711yescharacteristic
267C2×C3×C4C3×C22⋊C411yescharacteristic
268C2×C3×C4C3×C22⋊C411yescharacteristic
269C2×C3×C4C3×D811yescharacteristic
270C2×C3×C4C3×D811yescharacteristic
271C2×C3×C4C3×SD1611yescharacteristic
272C2×C3×C4C2×#6711yescharacteristic
273C2×C3×C4C2×C3×D411yescharacteristic
274C3×C8—21no—
275C3×C8—21no—
276C3×C8—21no—
277C3×C8—21no—
278C3×C8—122no—
279C3×C8#21711yescharacteristic
280C3×C8#21711yescharacteristic
281C3×C8C3×C22⋊C411yescharacteristic
282C3×C8C3×C22⋊C411yescharacteristic
283C3×C8C22×Dic311yescharacteristic
284C3×C8C2×#6711yescharacteristic
285C3×C8C22×C3×C411yescharacteristic
286C3×C8C2×C3×D411yescharacteristic
287C3⋊C8—42no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—82no—
292C2×C32C2×#10211yescharacteristic
293#41—22no—
294#41—22no—
295#41—41no—
296#41—41no—
297C2×Q8—21no—
298C2×D4—21no—
299C2×D4—41no—
300C22×C4—41no—
301C22×C4C3×#6711yescharacteristic
302Q16—21no—
303Q16C2×C3×Dic312yes—
304SD16—22no—
305SD16—22no—
306D8—21no—
307D8C2×C3×Dic312yes—
308M16—62no—
309M16—121no—
310C2×C8—121no—
311C2×C8C2×C3×Dic311yescharacteristic
312C2×C8C2×C3×Dic311yescharacteristic
313C2×C8C3×#6711yescharacteristic
314C2×C8C3×#6711yescharacteristic
315C2×C8C3×#6711yescharacteristic
316C2×C8C22×C3×D311yescharacteristic
317C16—62no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—42no—
324C22×C3—44no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—84no—
328C22×C3—82no—
329C22×C3—82no—
330C22×C3#83211yescharacteristic
331C22×C3#83211yescharacteristic
332C22×C3C3×#10211yescharacteristic
333C22×C3C3×#10211yescharacteristic
334C22×C3C2×#21411yescharacteristic
335C22×C3C2×C3×SD1611yescharacteristic
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—44no—
341C3×C4—42no—
342C3×C4—42no—
343C3×C4#83211yescharacteristic
344C3×C4#83211yescharacteristic
345C3×C4C3×#10211yescharacteristic
346C3×C4C3×#10211yescharacteristic
347C3×C4C2×#21311yescharacteristic
348C3×C4C2×#21711yescharacteristic
349C3×C4C2×C3×C22⋊C411yescharacteristic
350C3×C4C2×C3×D811yescharacteristic
351C32#214811yescharacteristic, complemented, Sylow
352C23—41no—
353C23C3×#21411yescharacteristic
354Q8—22no—
355Q8—41no—
356D4—22no—
357D4—42no—
358D4—42no—
359D4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4C3×#21311yescharacteristic
364C2×C4C3×#21311yescharacteristic
365C2×C4C3×#21411yescharacteristic
366C2×C4C3×#21711yescharacteristic
367C2×C4C3×#21711yescharacteristic
368C2×C4C2×C3×#6711yescharacteristic
369C8—122no—
370C8C3×#21711yescharacteristic
371C8C3×#21711yescharacteristic
372C8C22×C3×Dic311yescharacteristic
373C8C2×C3×#6711yescharacteristic
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—84no—
385C2×C3—82no—
386C2×C3—82no—
387C2×C3C2×#83211yescharacteristic
388C2×C3C2×C3×#10211yescharacteristic
389C22—41no—
390C22—42no—
391C22—82no—
392C22C3×#83211yescharacteristic
393C22C3×#83211yescharacteristic
394C22C2×C3×#21411yescharacteristic
395C4—42no—
396C4C3×#83211yescharacteristic
397C4C3×#83211yescharacteristic
398C4C2×C3×#21311yescharacteristic
399C4C2×C3×#21711yescharacteristic
400C3—21no—
401C3#7208111yescharacteristic, direct product factor
402C3C3×#214811yescharacteristic, complemented
403C2—21no—
404C2—21no—
405C2—21no—
406C2—82no—
407C2C2×C3×#83211yescharacteristic
subgroupG/H
centerC3×C4#832
commutatorC3×C8C22×C3×C4
FrattiniC2×C8C22×C3×D3
Fitting, prime radicalC2×C32×#135C2
socleC2×C32C2×#102
cosocleC2×C3×C8C23×C3
hypercenterC2×C3×#135D3
hypocenterC3C3×#2148
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ C2×C3×#135

Complements

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

NH ≅ G/Ntype
2. C2×C3×#819C2semidirect
3. C2×C3×#812C2semidirect
4. C3×#4565C2semidirect
5. C3×#4563C2semidirect
7. #72081C3direct (N × H)
28. C2×#819C2×C3semidirect
29. C2×#812C2×C3semidirect
32. #4565C2×C3semidirect
33. #4563C2×C3semidirect
351. C32#2148semidirect
401. C3#72081direct (N × H)
402. C3C3×#2148semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1352
C2×C3×#8192
C2×C3×#8122
C3×#45652
C3×#45632
C3×#21483
#720813