Finite Group Databasebeta

← C3×#69712 #11778546

Subgroups — C3×#69712 #11778546

Order 1152.

583 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5510C211yesmaximal, characteristic
2C32×#398C211yesmaximal, characteristic, complemented
3C3×#4967C212yesmaximal, complemented
4C3×#4528C211yesmaximal, characteristic, complemented
5C3×#4451C212yesmaximal, complemented
6C3×#1610—31nomaximal
7#69712C311yesmaximal, characteristic, direct product factor
8C2×C3×#235—21no—
9C2×C32×C4⋊C4—21no—
10C2×C32×C4⋊C4C2211yescharacteristic
11C2×C32×C4⋊C4C2211yescharacteristic
12C3×#930—21no—
13C3×#930C2212yes—
14C3×#897—22no—
15C3×#895—22no—
16C32×#95—21no—
17C32×#95C2211yescharacteristic
18C3×#830C2212yes—
19C3×#809—22no—
20C3×#807—22no—
21C3×#537—31no—
22#5510C2×C311yescharacteristic
23C3×#480—32no—
24C3×#398—21no—
25C3×#398D311yescharacteristic, complemented
26C3×#398C2×C311yescharacteristic, complemented
27#4967C2×C312yescomplemented
28C3×#327—32no—
29C3×#324—31no—
30#4528C2×C311yescharacteristic, complemented
31#4451C2×C312yescomplemented
32C22×C32×C4—21no—
33C22×C32×C4—21no—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4D411yescharacteristic
36C22×C32×C4C2311yescharacteristic
37C2×C3×#67—22no—
38C3×#235—42no—
39C3×#235—42no—
40C2×C3×D12—21no—
41C2×C3×D12—21no—
42C2×C3×D12D412yes—
43C2×C3×C4×D3—22no—
44C2×C3×Dic6—21no—
45C32×C4⋊C4—22no—
46C32×C4⋊C4—22no—
47C32×C4⋊C4—42no—
48C3×(C2×D3)⋊C4—22no—
49C3×(C2×D3)⋊C4—22no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3D412yes—
52C2×C3×C3⋊C8—22no—
53#1610—31noSylow
54C2×C3×#41—61no—
55C2×#235—21no—
56C3×C22⋊Q8—62no—
57C3×#121—32no—
58C3×#121—62no—
59C3×#121—61no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4C2×D311yescharacteristic
66C2×C3×C4⋊C4C2×D311yescharacteristic
67C2×C3×C4⋊C4C22×C311yescharacteristic
68C2×C3×C4⋊C4C22×C311yescharacteristic
69#930—21no—
70#930C22×C312yes—
71#897—22no—
72#895—22no—
73C3×#107—62no—
74C3×#102—62no—
75C3×C22⋊C8—32no—
76C3×#95—22no—
77C3×#95—21no—
78C3×#95—21no—
79C3×#95—21no—
80C3×#95C2×D311yescharacteristic
81C3×#95C22×C311yescharacteristic
82#830C22×C312yes—
83#809—22no—
84#807—22no—
85C23×C32C2×D411yescharacteristic
86C22×C3×D3—22no—
87C2×C32×C4—22no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—41no—
92C2×C32×C4—41no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4—41no—
96C2×C32×C4—41no—
97C2×C32×C4—41no—
98C2×C32×C4D812yes—
99C2×C32×C4C2×D412yes—
100C3×#67—42no—
101C3×#67—42no—
102C2×C3×Dic3—22no—
103C3×D12—42no—
104C3×D12—42no—
105C3×D12—81no—
106C3×C4×D3—44no—
107C3×Dic6—42no—
108C3×C3⋊C8—42no—
109#537—31no—
110#480—32no—
111#398C3×D311yescharacteristic, complemented
112#327—32no—
113#324—31no—
114C3×#41—122no—
115C3×#41—122no—
116C2×C3×Q8—61no—
117C2×C3×D4—32no—
118C2×C3×D4—61no—
119C2×C3×D4—61no—
120C2×C3×D4—62no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—62no—
131C22×C3×C4#6711yescharacteristic
132C22×C3×C4#6711yescharacteristic
133C22×C3×C4C3×D411yescharacteristic
134C22×C3×C4C3×D411yescharacteristic
135C22×C3×C4C22×D311yescharacteristic
136C22×C3×C4C23×C311yescharacteristic
137C2×#67—22no—
138#235—42no—
139#235—42no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12C3×D412yes—
143C2×C4×D3—22no—
144C2×Dic6—21no—
145C2×C3×C8—62no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—32no—
151C3×C4⋊C4—42no—
152C3×C4⋊C4—42no—
153C3×C4⋊C4—44no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4—61no—
157C3×C22⋊C4—62no—
158C3×C22⋊C4—62no—
159(C2×D3)⋊C4—22no—
160(C2×D3)⋊C4—22no—
161C4⋊Dic3—21no—
162C4⋊Dic3C3×D412yes—
163C2×C3⋊C8—22no—
164C22×C32—21no—
165C22×C32—21no—
166C22×C32#12011yescharacteristic
167C22×C32C2×D811yescharacteristic
168C22×C32AGL(1,ℤ/8)11yescharacteristic
169C2×C3×D3—42no—
170C2×C3×D3—42no—
171C2×C3×D3—42no—
172C32×C4—22no—
173C32×C4—22no—
174C32×C4—42no—
175C32×C4—41no—
176C32×C4—81no—
177C3×Dic3—42no—
178C2×#41—61no—
179C22⋊Q8—62no—
180#121—32no—
181#121—62no—
182#121—61no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C2×C3×D311yescharacteristic
185C2×C4⋊C4C2×C3×D311yescharacteristic
186#107—62no—
187#102—62no—
188C22⋊C8—32no—
189#95—21no—
190#95C2×C3×D311yescharacteristic
191C23×C3—21no—
192C23×C3—62no—
193C23×C3C2×#6711yescharacteristic
194C23×C3C2×C3×D411yescharacteristic
195C22×D3—22no—
196C3×Q8—122no—
197C3×D4—122no—
198C3×D4—122no—
199C3×D4—122no—
200C3×D4—122no—
201C3×D4—241no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—24no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—62no—
238C2×C3×C4—124no—
239C2×C3×C4#21312yes—
240C2×C3×C4C3×D812yes—
241C2×C3×C4D3×D412yes—
242C2×C3×C4C2×C3×D412yes—
243#67—42no—
244#67—42no—
245C2×Dic3—22no—
246D12—42no—
247D12—42no—
248D12—81no—
249C4×D3—44no—
250Dic6—42no—
251C3×C8—122no—
252C3⋊C8—42no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32#44711yescharacteristic
256C2×C32#45611yescharacteristic
257C2×C32#45811yescharacteristic
258C3×D3—82no—
259#41—122no—
260#41—122no—
261C2×Q8—61no—
262C2×D4—32no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—62no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—62no—
269C22×C4C3×#6711yescharacteristic
270C22×C4C3×#6711yescharacteristic
271C22×C4C22×C3×D311yescharacteristic
272C2×C8—62no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—32no—
276C4⋊C4—42no—
277C4⋊C4—61no—
278C22⋊C4—62no—
279C22⋊C4—62no—
280C22×C3—22no—
281C22×C3—22no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—122no—
290C22×C3—122no—
291C22×C3—122no—
292C22×C3C2×#21311yescharacteristic
293C22×C3#93711yescharacteristic
294C22×C3#94911yescharacteristic
295C22×C3C3×#12011yescharacteristic
296C22×C3C2×C3×D811yescharacteristic
297C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
298C2×D3—42no—
299C2×D3—42no—
300C2×D3—42no—
301C3×C4—22no—
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—22no—
305C3×C4—42no—
306C3×C4—42no—
307C3×C4—44no—
308C3×C4—42no—
309C3×C4—42no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—81no—
313C3×C4—82no—
314C3×C4—81no—
315C3×C4—81no—
316C3×C4—122no—
317Dic3—42no—
318C32#161011yescharacteristic, complemented, Sylow
319C23—62no—
320C23C2×C3×#6711yescharacteristic
321Q8—122no—
322D4—122no—
323D4—122no—
324D4—122no—
325D4—122no—
326D4—241no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—62no—
339C2×C4—124no—
340C2×C4C3×#21312yes—
341C2×C4C3×D3×D412yes—
342C8—122no—
343C2×C3—22no—
344C2×C3—22no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—242no—
353C2×C3#503611yescharacteristic
354C2×C3#505311yescharacteristic
355C2×C3#509511yescharacteristic
356C2×C3C3×#44711yescharacteristic
357C2×C3C3×#45611yescharacteristic
358C2×C3C3×#45811yescharacteristic
359D3—82no—
360C22—21no—
361C22—21no—
362C22—122no—
363C22—122no—
364C22—122no—
365C22C2×C3×#21311yescharacteristic
366C22C3×#93711yescharacteristic
367C22C3×#94911yescharacteristic
368C4—22no—
369C4—22no—
370C4—42no—
371C4—41no—
372C4—81no—
373C4—122no—
374C3—21no—
375C3#6971211yescharacteristic, direct product factor
376C3C3×#161011yescharacteristic, complemented
377C2—21no—
378C2—21no—
379C2—242no—
380C2C3×#503611yescharacteristic
381C2C3×#505311yescharacteristic
382C2C3×#509511yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#398C2
socleC22×C32#120
prime radicalC3×#5510C2
hypercenterC3×#398D3
hypocenterC3C3×#1610
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#398

Complements

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

NH ≅ G/Ntype
2. C32×#398C2semidirect
3. C3×#4967C2semidirect
4. C3×#4528C2semidirect
5. C3×#4451C2semidirect
7. #69712C3direct (N × H)
25. C3×#398D3semidirect
26. C3×#398C2×C3semidirect
27. #4967C2×C3semidirect
30. #4528C2×C3semidirect
31. #4451C2×C3semidirect
111. #398C3×D3semidirect
318. C32#1610semidirect
375. C3#69712direct (N × H)
376. C3C3×#1610semidirect

Maximal subgroups

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

subgroupindex
C3×#55102
C32×#3982
C3×#49672
C3×#45282
C3×#44512
C3×#16103
#697123