Finite Group Databasebeta

← C3×#72078 #11796677

Subgroups — C3×#72078 #11796677

Order 1152.

597 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#135C211yesmaximal, characteristic
2C2×C3×#818C211yesmaximal, characteristic, complemented
3C2×C3×#812C211yesmaximal, characteristic, complemented
4C3×#4564C212yesmaximal, complemented
5C3×#4562C212yesmaximal, complemented
6C3×#2139—31nomaximal
7#72078C311yesmaximal, characteristic, direct product factor
8C2×C32×#41—21no—
9C32×#135—22no—
10C32×#135C414yescomplemented
11C2×C32×Q16C2211yescharacteristic
12C2×C32×SD16—21no—
13C2×C32×D8C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C2×C3×C4⋊Dic3—21no—
16C3×#818—21no—
17C3×#818C2212yes—
18C3×#812—22no—
19C2×C3×C3⋊C16C2212yes—
20C2×C3×#135—21no—
21C2×C3×#135D311yescharacteristic
22C2×C3×#135C2×C311yescharacteristic
23C2×C3×M32—31no—
24C2×C3×#107—31no—
25C2×#818C2×C311yescharacteristic, complemented
26C2×#812C2×C311yescharacteristic, complemented
27C3×#358—32no—
28C3×#357—32no—
29#4564C2×C312yescomplemented
30#4562C2×C312yescomplemented
31C32×#41—24no—
32C32×#41—42no—
33C2×C32×Q8—21no—
34C2×C32×D4—21no—
35C2×C32×D4—41no—
36C22×C32×C4—41no—
37C22×C32×C4D411yescharacteristic
38C22×C3×Dic3—41no—
39C32×Q16—21no—
40C32×Q16C2×C412yescomplemented
41C32×SD16—24no—
42C32×D8—21no—
43C32×D8C2×C412yescomplemented
44C2×C32×C8C2×C412yes—
45C2×C32×C8D411yescharacteristic
46C2×C32×C8D412yes—
47C2×C32×C8C2311yescharacteristic
48C3×C4⋊Dic3—22no—
49C3×C4⋊Dic3—41no—
50C3×C3⋊C16—22no—
51#2139—31noSylow
52C2×C3×#41—22no—
53C2×C3×#41—21no—
54C2×C3×#41—21no—
55C3×#135—24no—
56C3×#135—22no—
57C3×#135—22no—
58C3×#135—24no—
59C3×#135Dic314yescomplemented
60C3×#135C3×C414yescomplemented
61C2×C3×Q16—21no—
62C2×C3×Q16C2×D311yescharacteristic
63C2×C3×Q16C22×C311yescharacteristic
64C2×C3×SD16—22no—
65C2×C3×SD16—21no—
66C2×C3×SD16—21no—
67C2×C3×D8—21no—
68C2×C3×D8C2×D311yescharacteristic
69C2×C3×D8C22×C311yescharacteristic
70C22×C3×C8—21no—
71C22×C3×C8C2×D311yescharacteristic
72C22×C3×C8C22×C311yescharacteristic
73C2×C3×C4⋊C4—61no—
74C2×C4⋊Dic3—21no—
75C3×M32—62no—
76C2×C3×C16—32no—
77C3×#107—32no—
78C3×#107—61no—
79#818—21no—
80#818C22×C312yes—
81#812—22no—
82C2×C3⋊C16C22×C312yes—
83C23×C32—41no—
84C23×C32D811yescharacteristic
85C32×Q8—22no—
86C32×Q8—41no—
87C32×D4—22no—
88C32×D4—44no—
89C32×D4—41no—
90C2×C32×C4—41no—
91C2×C32×C4—44no—
92C2×C32×C4C22⋊C412yes—
93C2×C32×C4D811yescharacteristic
94C2×C32×C4SD1612yes—
95C2×C32×C4C2×D411yescharacteristic
96C2×C3×Dic3—42no—
97C2×C3×Dic3—82no—
98C32×C8C22⋊C412yes—
99C32×C8C22×C411yescharacteristic
100C32×C8C2×D411yescharacteristic
101C2×#135C3×D311yescharacteristic
102C2×M32—31no—
103C2×#107—31no—
104#358—32no—
105#357—32no—
106C3×#41—28no—
107C3×#41—24no—
108C3×#41—24no—
109C3×#41—44no—
110C3×#41—42no—
111C3×#41—42no—
112C2×C3×Q8—22no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×D4—22no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—42no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C22×C3×C4—21no—
122C22×C3×C4—42no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—121no—
126C22×C3×C4#6711yescharacteristic
127C22×C3×C4C3×D411yescharacteristic
128C22×Dic3—41no—
129C3×Q16—22no—
130C3×Q16—21no—
131C3×Q16—21no—
132C3×Q16—22no—
133C3×Q16C2×Dic312yescomplemented
134C3×Q16C2×C3×C412yescomplemented
135C3×SD16—28no—
136C3×SD16—24no—
137C3×SD16—24no—
138C3×D8—22no—
139C3×D8—21no—
140C3×D8—21no—
141C3×D8—22no—
142C3×D8C2×Dic312yescomplemented
143C3×D8C2×C3×C412yescomplemented
144C2×C3×C8—22no—
145C2×C3×C8—21no—
146C2×C3×C8—22no—
147C2×C3×C8—21no—
148C2×C3×C8C2×Dic312yes—
149C2×C3×C8#6711yescharacteristic
150C2×C3×C8#6712yes—
151C2×C3×C8C2×C3×C412yes—
152C2×C3×C8C3×D411yescharacteristic
153C2×C3×C8C3×D412yes—
154C2×C3×C8C22×D311yescharacteristic
155C2×C3×C8C23×C311yescharacteristic
156C3×C4⋊C4—62no—
157C3×C4⋊C4—121no—
158C4⋊Dic3—22no—
159C4⋊Dic3—41no—
160C3×C16—62no—
161C3⋊C16—22no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—42no—
165C22×C32—81no—
166C22×C32—81no—
167C22×C32#10212yes—
168C22×C32C2×D811yescharacteristic
169C32×C4—42no—
170C32×C4#10212yes—
171C32×C4C2×C22⋊C411yescharacteristic
172C32×C4C2×SD1611yescharacteristic
173C3×Dic3—82no—
174C2×#41—21no—
175#135—22no—
176#135C3×Dic314yescomplemented
177C2×Q16C2×C3×D311yescharacteristic
178C2×SD16—21no—
179C2×D8C2×C3×D311yescharacteristic
180C22×C8C2×C3×D311yescharacteristic
181C2×C4⋊C4—61no—
182M32—62no—
183C2×C16—32no—
184#107—32no—
185#107—61no—
186C23×C3—21no—
187C23×C3—42no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3#21311yescharacteristic
191C23×C3C3×D811yescharacteristic
192C3×Q8—24no—
193C3×Q8—22no—
194C3×Q8—22no—
195C3×Q8—42no—
196C3×Q8—41no—
197C3×Q8—41no—
198C3×D4—24no—
199C3×D4—22no—
200C3×D4—22no—
201C3×D4—42no—
202C3×D4—48no—
203C3×D4—44no—
204C3×D4—44no—
205C3×D4—41no—
206C3×D4—41no—
207C2×C3×C4—21no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—21no—
211C2×C3×C4—48no—
212C2×C3×C4—42no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—44no—
216C2×C3×C4—44no—
217C2×C3×C4—122no—
218C2×C3×C4—242no—
219C2×C3×C4#21311yescharacteristic
220C2×C3×C4#21412yes—
221C2×C3×C4#21712yes—
222C2×C3×C4C3×C22⋊C412yes—
223C2×C3×C4C3×D811yescharacteristic
224C2×C3×C4C3×SD1612yes—
225C2×C3×C4C2×#6711yescharacteristic
226C2×C3×C4C2×C3×D411yescharacteristic
227C2×Dic3—42no—
228C2×Dic3—82no—
229C3×C8—22no—
230C3×C8—21no—
231C3×C8—21no—
232C3×C8#21712yes—
233C3×C8C3×C22⋊C412yes—
234C3×C8C22×Dic311yescharacteristic
235C3×C8C2×#6711yescharacteristic
236C3×C8C22×C3×C411yescharacteristic
237C3×C8C2×C3×D411yescharacteristic
238C2×C32—22no—
239C2×C32—82no—
240C2×C32C2×#10211yescharacteristic
241C2×C32#50911yescharacteristic
242C2×C32#51011yescharacteristic
243#41—24no—
244#41—42no—
245C2×Q8—21no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—41no—
249C22×C4—121no—
250C22×C4C3×#6711yescharacteristic
251Q16—21no—
252Q16C2×C3×Dic312yescomplemented
253SD16—24no—
254D8—21no—
255D8C2×C3×Dic312yescomplemented
256C2×C8C2×C3×Dic312yes—
257C2×C8C3×#6711yescharacteristic
258C2×C8C3×#6712yes—
259C2×C8C22×C3×D311yescharacteristic
260C4⋊C4—62no—
261C4⋊C4—121no—
262C16—62no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—22no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—44no—
272C22×C3—42no—
273C22×C3—42no—
274C22×C3—82no—
275C22×C3—82no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3—81no—
279C22×C3—81no—
280C22×C3#83212yes—
281C22×C3C3×#10212yes—
282C22×C3C2×#21311yescharacteristic
283C22×C3C2×C3×D811yescharacteristic
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—22no—
287C3×C4—44no—
288C3×C4—42no—
289C3×C4—42no—
290C3×C4—242no—
291C3×C4#83212yes—
292C3×C4C3×#10212yes—
293C3×C4C2×#21411yescharacteristic
294C3×C4C2×#21711yescharacteristic
295C3×C4C2×C3×C22⋊C411yescharacteristic
296C3×C4C2×C3×SD1611yescharacteristic
297Dic3—82no—
298C32#213911yescharacteristic, complemented, Sylow
299C23—41no—
300C23C3×#21311yescharacteristic
301Q8—22no—
302Q8—41no—
303D4—22no—
304D4—44no—
305D4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—122no—
309C2×C4—242no—
310C2×C4C3×#21311yescharacteristic
311C2×C4C3×#21412yes—
312C2×C4C3×#21712yes—
313C2×C4C2×C3×#6711yescharacteristic
314C8C3×#21712yes—
315C8C22×C3×Dic311yescharacteristic
316C8C2×C3×#6711yescharacteristic
317C2×C3—22no—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—42no—
323C2×C3—84no—
324C2×C3—82no—
325C2×C3—82no—
326C2×C3#519311yescharacteristic
327C2×C3#519511yescharacteristic
328C2×C3C2×#83211yescharacteristic
329C2×C3C2×C3×#10211yescharacteristic
330C2×C3C3×#50911yescharacteristic
331C2×C3C3×#51011yescharacteristic
332C22—21no—
333C22—21no—
334C22—42no—
335C22—81no—
336C22—81no—
337C22C3×#83212yes—
338C22C2×C3×#21311yescharacteristic
339C4—42no—
340C4—242no—
341C4C3×#83212yes—
342C4C2×C3×#21411yescharacteristic
343C4C2×C3×#21711yescharacteristic
344C3—21no—
345C3#7207811yescharacteristic, direct product factor
346C3C3×#213911yescharacteristic, complemented
347C2—22no—
348C2—82no—
349C2C3×#519311yescharacteristic
350C2C3×#519511yescharacteristic
351C2C2×C3×#83211yescharacteristic
subgroupG/H
centerC22×C3C2×#213
commutatorC3×C8C22×C3×C4
FrattiniC2×C8C22×C3×D3
Fitting, prime radicalC2×C32×#135C2
socleC22×C32C2×D8
cosocleC2×C3×C8C23×C3
hypercenterC2×C3×#135D3
hypocenterC3C3×#2139
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ 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×#818C2semidirect
3. C2×C3×#812C2semidirect
4. C3×#4564C2semidirect
5. C3×#4562C2semidirect
7. #72078C3direct (N × H)
10. C32×#135C4semidirect
25. C2×#818C2×C3semidirect
26. C2×#812C2×C3semidirect
29. #4564C2×C3semidirect
30. #4562C2×C3semidirect
40. C32×Q16C2×C4semidirect
43. C32×D8C2×C4semidirect
59. C3×#135Dic3semidirect
60. C3×#135C3×C4semidirect
133. C3×Q16C2×Dic3semidirect
134. C3×Q16C2×C3×C4semidirect
142. C3×D8C2×Dic3semidirect
143. C3×D8C2×C3×C4semidirect
176. #135C3×Dic3semidirect
252. Q16C2×C3×Dic3semidirect
255. D8C2×C3×Dic3semidirect
298. C32#2139semidirect
345. C3#72078direct (N × H)
346. C3C3×#2139semidirect

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×#8182
C2×C3×#8122
C3×#45642
C3×#45622
C3×#21393
#720783