Finite Group Databasebeta

← C3×#68110 #11778442

Subgroups — C3×#68110 #11778442

Order 1152.

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

normal subgroups only

This group has 609 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×#5493C211yesmaximal, characteristic
2C3×#4953C211yesmaximal, characteristic, complemented
3C3×#4736C212yesmaximal, complemented
4C32×#323C211yesmaximal, characteristic, complemented
5C3×#4470C212yesmaximal, complemented
6C3×#1607—31nomaximal
7#68110C311yesmaximal, characteristic, direct product factor
8C22×C3×#67—21no—
9C2×C32×C22⋊C4C2211yescharacteristic
10C2×C3×#217—21no—
11C3×#930—21no—
12C3×#930C2212yes—
13C2×C3×C4⋊Dic3C2211yescharacteristic, complemented
14C3×#886—22no—
15C3×#882—22no—
16C32×C22⋊C8C2212yescomplemented
17C3×C3⋊#95—21no—
18C3×C3⋊#95C2211yescharacteristic, complemented
19C3×#821—22no—
20C3×#818—22no—
21C3×#534—31no—
22#5493C2×C311yescharacteristic
23C3×#480—32no—
24C3×#397—31no—
25#4953C2×C311yescharacteristic, complemented
26#4736C2×C312yescomplemented
27C3×#327—32no—
28C3×#323—21no—
29C3×#323D311yescharacteristic, complemented
30C3×#323C2×C311yescharacteristic, complemented
31#4470C2×C312yescomplemented
32C24×C32D411yescharacteristic
33C23×C3×D3—21no—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4C2311yescharacteristic
36C2×C3×#67—22no—
37C2×C3×#67—22no—
38C2×C3×#67—42no—
39C2×C3×#67—42no—
40C22×C3×Dic3—21no—
41C22×C3×Dic3—21no—
42C22×C3×Dic3—21no—
43C2×C3×D12—21no—
44C2×C3×D12D412yes—
45C2×C32×C8—22no—
46C32×C22⋊C4—22no—
47C3×#217—42no—
48C3×(C2×D3)⋊C4—22no—
49C3×(C2×D3)⋊C4—22no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—22no—
52C3×C4⋊Dic3D412yes—
53#1607—31noSylow
54C22×C3×D4—61no—
55C22×#67—21no—
56C3×#123—62no—
57C3×#121—32no—
58C3×#121—61no—
59C3×#120—62no—
60C2×C3×C4⋊C4—31no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—61no—
63C2×C3×C22⋊C4C2×D311yescharacteristic
64C2×C3×C22⋊C4C22×C311yescharacteristic
65C2×#217—21no—
66#930—21no—
67#930C22×C312yes—
68C2×C4⋊Dic3C22×C311yescharacteristic, complemented
69#886—22no—
70#882—22no—
71C3×#107—62no—
72C3×#102—62no—
73C3×C22⋊C8—22no—
74C3×C22⋊C8C2×D312yescomplemented
75C3×C22⋊C8C22×C312yescomplemented
76C3×#95—31no—
77C3×#95—61no—
78C3⋊#95—21no—
79C3⋊#95C22×C311yescharacteristic, complemented
80#821—22no—
81#818—22no—
82C23×C32—21no—
83C23×C32—21no—
84C23×C32—41no—
85C23×C32—41no—
86C23×C32D812yes—
87C23×C32C2×D411yescharacteristic
88C22×C3×D3—22no—
89C22×C3×D3—42no—
90C22×C3×D3—81no—
91C22×C3×D3—81no—
92C2×C32×C4—21no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4C2×D412yes—
96C3×#67—44no—
97C3×#67—42no—
98C3×#67—42no—
99C2×C3×Dic3—22no—
100C2×C3×Dic3—42no—
101C2×C3×Dic3—41no—
102C2×C3×Dic3—41no—
103C2×C3×Dic3—41no—
104C2×C3×Dic3—41no—
105C2×C3×Dic3—41no—
106C2×C3×Dic3—41no—
107C3×D12—42no—
108C3×D12—81no—
109C32×C8—42no—
110#534—31no—
111#480—32no—
112#397—31no—
113#327—32no—
114#323C3×D311yescharacteristic, complemented
115C24×C3—21no—
116C24×C3—61no—
117C24×C3D1211yescharacteristic
118C24×C3C3×D411yescharacteristic
119C23×D3—21no—
120C2×C3×D4—32no—
121C2×C3×D4—61no—
122C2×C3×D4—62no—
123C2×C3×D4—62no—
124C2×C3×D4—122no—
125C2×C3×D4—122no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—61no—
129C22×C3×C4—61no—
130C22×C3×C4—61no—
131C22×C3×C4D1211yescharacteristic
132C22×C3×C4C3×D411yescharacteristic
133C22×C3×C4C22×D311yescharacteristic
134C22×C3×C4C23×C311yescharacteristic
135C2×#67—22no—
136C2×#67—22no—
137C2×#67—42no—
138C2×#67—42no—
139C22×Dic3—21no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C2×D12—21no—
143C2×D12C3×D412yes—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8—42no—
147C3×C4⋊C4—32no—
148C3×C4⋊C4—61no—
149C3×C4⋊C4—62no—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—22no—
152C3×C22⋊C4—42no—
153C3×C22⋊C4—62no—
154C3×C22⋊C4—62no—
155C3×C22⋊C4—122no—
156#217—42no—
157(C2×D3)⋊C4—22no—
158(C2×D3)⋊C4—22no—
159C4⋊Dic3—21no—
160C4⋊Dic3—22no—
161C4⋊Dic3C3×D412yes—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—22no—
165C22×C32—42no—
166C22×C32—41no—
167C22×C32—41no—
168C22×C32—81no—
169C22×C32#12011yescharacteristic
170C22×C32C2×D811yescharacteristic
171C22×C32AGL(1,ℤ/8)11yescharacteristic
172C2×C3×D3—42no—
173C2×C3×D3—42no—
174C2×C3×D3—42no—
175C2×C3×D3—84no—
176C32×C4—22no—
177C32×C4—41no—
178C3×Dic3—42no—
179C3×Dic3—81no—
180C3×Dic3—81no—
181C22×D4—61no—
182#123—62no—
183#121—32no—
184#121—61no—
185#120—62no—
186C2×C4⋊C4—31no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4C2×C3×D311yescharacteristic
189#107—62no—
190#102—62no—
191C22⋊C8C2×C3×D312yescomplemented
192#95—31no—
193#95—61no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—22no—
199C23×C3—21no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—62no—
207C23×C3—81no—
208C23×C3—81no—
209C23×C3—122no—
210C23×C3—241no—
211C23×C3—241no—
212C23×C3D2412yes—
213C23×C3C3×D812yes—
214C23×C3C2×D1211yescharacteristic
215C23×C3C2×C3×D411yescharacteristic
216C22×D3—22no—
217C22×D3—42no—
218C22×D3—81no—
219C22×D3—81no—
220C3×D4—122no—
221C3×D4—124no—
222C3×D4—122no—
223C3×D4—122no—
224C3×D4—241no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—22no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—62no—
234C2×C3×C4—81no—
235C2×C3×C4—81no—
236C2×C3×C4—122no—
237C2×C3×C4—121no—
238C2×C3×C4—121no—
239C2×C3×C4—121no—
240C2×C3×C4—121no—
241C2×C3×C4—121no—
242C2×C3×C4—121no—
243C2×C3×C4D3×D412yes—
244C2×C3×C4C2×C3×D412yes—
245#67—44no—
246#67—42no—
247#67—42no—
248C2×Dic3—22no—
249C2×Dic3—42no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256D12—42no—
257D12—81no—
258C3×C8—42no—
259C3×C8—42no—
260C3×C8—82no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—42no—
264C2×C32#44711yescharacteristic
265C2×C32#45511yescharacteristic
266C2×C32#45811yescharacteristic
267C3×D3—82no—
268C24—61no—
269C24C3×D1211yescharacteristic
270C2×D4—32no—
271C2×D4—61no—
272C2×D4—62no—
273C2×D4—62no—
274C2×D4—122no—
275C2×D4—122no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4C3×D1211yescharacteristic
280C22×C4C22×C3×D311yescharacteristic
281C2×C8—22no—
282C4⋊C4—32no—
283C4⋊C4—61no—
284C4⋊C4—62no—
285C22⋊C4—22no—
286C22⋊C4—62no—
287C22⋊C4—62no—
288C22⋊C4—122no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3—82no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—82no—
313C22×C3—122no—
314C22×C3—122no—
315C22×C3—122no—
316C22×C3—244no—
317C22×C3#88211yescharacteristic
318C22×C3C2×D2411yescharacteristic
319C22×C3#90811yescharacteristic
320C22×C3C3×#12011yescharacteristic
321C22×C3C2×C3×D811yescharacteristic
322C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
323C2×D3—42no—
324C2×D3—42no—
325C2×D3—42no—
326C2×D3—84no—
327C3×C4—22no—
328C3×C4—22no—
329C3×C4—42no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—81no—
333C3×C4—122no—
334C3×C4—241no—
335C3×C4—241no—
336Dic3—42no—
337Dic3—81no—
338Dic3—81no—
339C32#160711yescharacteristic, complemented, Sylow
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—62no—
345C23—122no—
346C23—241no—
347C23—241no—
348C23C3×D2412yes—
349C23C2×C3×D1211yescharacteristic
350D4—122no—
351D4—124no—
352D4—122no—
353D4—122no—
354D4—241no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—62no—
359C2×C4—122no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4C3×D3×D412yes—
367C8—42no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—42no—
378C2×C3—42no—
379C2×C3—82no—
380C2×C3—242no—
381C2×C3#473211yescharacteristic
382C2×C3#474211yescharacteristic
383C2×C3#474611yescharacteristic
384C2×C3C3×#44711yescharacteristic
385C2×C3C3×#45511yescharacteristic
386C2×C3C3×#45811yescharacteristic
387D3—82no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—42no—
392C22—41no—
393C22—41no—
394C22—81no—
395C22—122no—
396C22—122no—
397C22—122no—
398C22—244no—
399C22C3×#88211yescharacteristic
400C22C2×C3×D2411yescharacteristic
401C22C3×#90811yescharacteristic
402C4—22no—
403C4—41no—
404C4—122no—
405C4—241no—
406C4—241no—
407C3—21no—
408C3#6811011yescharacteristic, direct product factor
409C3C3×#160711yescharacteristic, complemented
410C2—21no—
411C2—21no—
412C2—42no—
413C2—242no—
414C2C3×#473211yescharacteristic
415C2C3×#474211yescharacteristic
416C2C3×#474611yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#323C2
socleC22×C32#120
prime radicalC3×#5493C2
hypercenterC3×#323D3
hypocenterC3C3×#1607
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#323

Complements

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

NH ≅ G/Ntype
2. C3×#4953C2semidirect
3. C3×#4736C2semidirect
4. C32×#323C2semidirect
5. C3×#4470C2semidirect
7. #68110C3direct (N × H)
13. C2×C3×C4⋊Dic3C22semidirect
16. C32×C22⋊C8C22semidirect
18. C3×C3⋊#95C22semidirect
25. #4953C2×C3semidirect
26. #4736C2×C3semidirect
29. C3×#323D3semidirect
30. C3×#323C2×C3semidirect
31. #4470C2×C3semidirect
68. C2×C4⋊Dic3C22×C3semidirect
74. C3×C22⋊C8C2×D3semidirect
75. C3×C22⋊C8C22×C3semidirect
79. C3⋊#95C22×C3semidirect
114. #323C3×D3semidirect
191. C22⋊C8C2×C3×D3semidirect
339. C32#1607semidirect
408. C3#68110direct (N × H)
409. C3C3×#1607semidirect

Maximal subgroups

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

subgroupindex
C3×#54932
C3×#49532
C3×#47362
C32×#3232
C3×#44702
C3×#16073
#681103