Finite Group Databasebeta

← C3×#68273 #11780191

Subgroups — C3×#68273 #11780191

Order 1152.

503 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×D12C211yesmaximal, characteristic
2C3×#5031C211yesmaximal, characteristic, complemented
3C3×#4846C211yesmaximal, characteristic
4C3×#4682C211yesmaximal, characteristic, complemented
5C32×Q8⋊C8C211yesmaximal, characteristic, complemented
6C3×#4486C211yesmaximal, characteristic, complemented
7C3×#4459C211yesmaximal, characteristic
8C3×#1649—31nomaximal
9#68273C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×Q8—21no—
11C32×C4×Q8C2211yescharacteristic
12C2×C3×#216—21no—
13C3×#897—21no—
14C3×#897—22no—
15C3×C4×D12—21no—
16C3×C4×D12C2211yescharacteristic
17C3×#869—21no—
18C3×#869C2211yescharacteristic
19C3×#869C2211yescharacteristic
20C32×C4⋊C8C2211yescharacteristic
21C32×C4×C8C2211yescharacteristic
22C3×#821—21no—
23C3×#818—22no—
24C3×#818—21no—
25C3×#810—21no—
26C3×C4⋊(C3⋊C8)C2211yescharacteristic
27C3×D4×Q8—31no—
28Q8×D12C2×C311yescharacteristic
29C3×#500—31no—
30C3×#474—31no—
31C3×#462—31no—
32#5031C2×C311yescharacteristic, complemented
33#4846C2×C311yescharacteristic
34#4682C2×C311yescharacteristic, complemented
35C3×#332—31no—
36C3×Q8⋊C8—21no—
37C3×Q8⋊C8D311yescharacteristic, complemented
38C3×Q8⋊C8C2×C311yescharacteristic, complemented
39C3×D4⋊C8—31no—
40#4486C2×C311yescharacteristic, complemented
41#4459C2×C311yescharacteristic
42C2×C32×Q8D411yescharacteristic
43C3×D3×Q8—42no—
44C3×D3×Q8—42no—
45C2×C3×D12D411yescharacteristic
46C2×C3×C4×D3—22no—
47C2×C3×C4×D3—21no—
48C2×C3×Dic6—22no—
49C2×C3×Dic6—21no—
50C2×C3×Dic6—21no—
51C2×C32×C8—21no—
52C2×C32×C8—21no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4D411yescharacteristic
55C32×C42—21no—
56C32×C42C2311yescharacteristic
57C3×#216—42no—
58C3×(C2×D3)⋊C4—22no—
59C3×(C2×D3)⋊C4—21no—
60C3×C4⋊Dic3—22no—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3D412yes—
65C3×C4⋊Dic3D411yescharacteristic
66C2×C3×C3⋊C8—21no—
67#1649—31noSylow
68C22×C3×Q8—61no—
69C2×D3×Q8—21no—
70C2×C3×Q16—61no—
71C3×C4⋊Q8—31no—
72C3×C4⋊Q8—31no—
73C3×C4⋊Q8—61no—
74C3×C22⋊Q8—62no—
75C3×C22⋊Q8—61no—
76C3×C4×Q8—21no—
77C3×C4×Q8C2×D311yescharacteristic
78C3×C4×Q8C22×C311yescharacteristic
79C3×C4×D4—31no—
80C3×C4×D4—61no—
81C2×#216—21no—
82#897—21no—
83#897—22no—
84C4×D12—21no—
85C4×D12C22×C311yescharacteristic
86#869—21no—
87#869C22×C311yescharacteristic
88#869C22×C311yescharacteristic
89C3×#107—61no—
90C3×#107—62no—
91C3×C4⋊C8—21no—
92C3×C4⋊C8—31no—
93C3×C4⋊C8C2×D311yescharacteristic
94C3×C4⋊C8C22×C311yescharacteristic
95C3×#103—61no—
96C3×#102—61no—
97C3×C4×C8—21no—
98C3×C4×C8C2×D311yescharacteristic
99C3×C4×C8C22×C311yescharacteristic
100#821—21no—
101#818—22no—
102#818—21no—
103#810—21no—
104C4⋊(C3⋊C8)C22×C311yescharacteristic
105C22×C3×D3—21no—
106C32×Q8—21no—
107C32×Q8D812yes—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C3×Dic3—21no—
114C2×C3×Dic3—22no—
115C2×C3×Dic3—41no—
116C3×D12—21no—
117C3×D12Q1612yescomplemented
118C3×C4×D3—44no—
119C3×C4×D3—41no—
120C3×C4×D3—41no—
121C3×Dic6—44no—
122C3×Dic6—41no—
123C3×Dic6—41no—
124C3×Dic6—41no—
125C3×Dic6—41no—
126C32×C8—22no—
127C32×C8—41no—
128C3×C3⋊C8—41no—
129D4×Q8—31no—
130#500—31no—
131#474—31no—
132#462—31no—
133#332—31no—
134Q8⋊C8C3×D311yescharacteristic, complemented
135D4⋊C8—31no—
136C2×C3×Q8—21no—
137C2×C3×Q8—62no—
138C2×C3×Q8—61no—
139C2×C3×Q8—61no—
140C2×C3×Q8—122no—
141C2×C3×Q8—122no—
142C2×C3×Q8D1211yescharacteristic
143C2×C3×Q8C3×D411yescharacteristic
144C2×C3×D4—31no—
145C22×C3×C4—62no—
146C22×C3×C4—61no—
147D3×Q8—42no—
148D3×Q8—42no—
149C2×D12C3×D411yescharacteristic
150C2×C4×D3—22no—
151C2×C4×D3—21no—
152C2×Dic6—22no—
153C2×Dic6—21no—
154C2×Dic6—21no—
155C3×Q16—122no—
156C2×C3×C8—21no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—41no—
161C2×C3×C8—41no—
162C2×C3×C8—61no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—32no—
168C3×C4⋊C4—41no—
169C3×C4⋊C4—62no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—61no—
172C3×C4⋊C4—61no—
173C3×C4⋊C4D1211yescharacteristic
174C3×C4⋊C4C3×D411yescharacteristic
175C3×C22⋊C4—62no—
176C3×C22⋊C4—61no—
177C3×C42—21no—
178C3×C42—21no—
179C3×C42—21no—
180C3×C42—41no—
181C3×C42C22×D311yescharacteristic
182C3×C42C23×C311yescharacteristic
183#216—42no—
184(C2×D3)⋊C4—22no—
185(C2×D3)⋊C4—21no—
186C4⋊Dic3—22no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3C3×D411yescharacteristic
191C4⋊Dic3C3×D412yes—
192C2×C3⋊C8—21no—
193C22×C32#12011yescharacteristic
194C2×C3×D3—22no—
195C2×C3×D3—41no—
196C2×C3×D3—41no—
197C32×C4—22no—
198C32×C4—21no—
199C32×C4—41no—
200C32×C4C2×D811yescharacteristic
201C32×C4C2×Q1611yescharacteristic
202C32×C4AGL(1,ℤ/8)11yescharacteristic
203C32×C4#13711yescharacteristic
204C3×Dic3—41no—
205C3×Dic3—42no—
206C3×Dic3—81no—
207C22×Q8—61no—
208C2×Q16—61no—
209C4⋊Q8—31no—
210C4⋊Q8—31no—
211C4⋊Q8—61no—
212C22⋊Q8—62no—
213C22⋊Q8—61no—
214C4×Q8C2×C3×D311yescharacteristic
215C4×D4—31no—
216C4×D4—61no—
217#107—61no—
218#107—62no—
219C4⋊C8—31no—
220C4⋊C8C2×C3×D311yescharacteristic
221#103—61no—
222#102—61no—
223C4×C8C2×C3×D311yescharacteristic
224C23×C3—61no—
225C22×D3—21no—
226C3×Q8—21no—
227C3×Q8—21no—
228C3×Q8—22no—
229C3×Q8—41no—
230C3×Q8—124no—
231C3×Q8—121no—
232C3×Q8—121no—
233C3×Q8—121no—
234C3×Q8—121no—
235C3×Q8D2412yes—
236C3×Q8C3×D812yes—
237C3×D4—32no—
238C3×D4—61no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—62no—
249C2×C3×C4—61no—
250C2×C3×C4—124no—
251C2×C3×C4—121no—
252C2×C3×C4—121no—
253C2×C3×C4—121no—
254C2×C3×C4C2×D1211yescharacteristic
255C2×C3×C4D3×D411yescharacteristic
256C2×C3×C4D3×D411yescharacteristic
257C2×C3×C4C2×C3×D411yescharacteristic
258C2×C3×C4C2×C3×D411yescharacteristic
259C2×C3×C4C2×C3×D411yescharacteristic
260C2×Dic3—21no—
261C2×Dic3—22no—
262C2×Dic3—41no—
263D12—21no—
264D12C3×Q1612yescomplemented
265C4×D3—44no—
266C4×D3—41no—
267C4×D3—41no—
268Dic6—44no—
269Dic6—41no—
270Dic6—41no—
271Dic6—41no—
272Dic6—41no—
273C3×C8—22no—
274C3×C8—22no—
275C3×C8—42no—
276C3×C8—41no—
277C3×C8—41no—
278C3×C8—81no—
279C3×C8—121no—
280C3⋊C8—41no—
281C2×C32#44711yescharacteristic
282C2×C32#45111yescharacteristic
283C2×C32#45611yescharacteristic
284C3×D3—42no—
285C2×Q8—62no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—122no—
289C2×Q8—122no—
290C2×Q8C3×D1211yescharacteristic
291C2×D4—31no—
292C22×C4—62no—
293C22×C4—61no—
294Q16—122no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—61no—
298C4⋊C4—21no—
299C4⋊C4—31no—
300C4⋊C4—32no—
301C4⋊C4—62no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4C3×D1211yescharacteristic
306C22⋊C4—62no—
307C22⋊C4—61no—
308C42—21no—
309C42C22×C3×D311yescharacteristic
310C22×C3—21no—
311C22×C3—62no—
312C22×C3—121no—
313C22×C3—121no—
314C22×C3#88211yescharacteristic
315C22×C3C3×#12011yescharacteristic
316C2×D3—22no—
317C2×D3—41no—
318C2×D3—41no—
319C3×C4—22no—
320C3×C4—22no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—41no—
328C3×C4—42no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—81no—
332C3×C4—122no—
333C3×C4—121no—
334C3×C4—241no—
335C3×C4C2×D2411yescharacteristic
336C3×C4#90811yescharacteristic
337C3×C4#91511yescharacteristic
338C3×C4D3×Q1611yescharacteristic
339C3×C4C2×C3×D811yescharacteristic
340C3×C4C2×C3×Q1611yescharacteristic
341C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
342C3×C4C3×#13711yescharacteristic
343Dic3—41no—
344Dic3—42no—
345Dic3—81no—
346C32#164911yescharacteristic, complemented, Sylow
347C23—61no—
348Q8—21no—
349Q8—124no—
350Q8—121no—
351Q8—121no—
352Q8—121no—
353Q8—121no—
354Q8C3×D2412yes—
355D4—32no—
356D4—61no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—62no—
360C2×C4—61no—
361C2×C4—124no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4C2×C3×D1211yescharacteristic
366C2×C4C3×D3×D411yescharacteristic
367C2×C4C3×D3×D411yescharacteristic
368C8—22no—
369C8—41no—
370C8—121no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—122no—
375C2×C3#473211yescharacteristic
376C2×C3#480911yescharacteristic
377C2×C3#482611yescharacteristic
378C2×C3C3×#44711yescharacteristic
379C2×C3C3×#45111yescharacteristic
380C2×C3C3×#45611yescharacteristic
381D3—42no—
382C22—62no—
383C22—121no—
384C22—121no—
385C22C3×#88211yescharacteristic
386C4—22no—
387C4—21no—
388C4—41no—
389C4—122no—
390C4—121no—
391C4—241no—
392C4C2×C3×D2411yescharacteristic
393C4C3×#90811yescharacteristic
394C4C3×#91511yescharacteristic
395C4C3×D3×Q1611yescharacteristic
396C3—21no—
397C3#6827311yescharacteristic, direct product factor
398C3C3×#164911yescharacteristic, complemented
399C2—122no—
400C2C3×#473211yescharacteristic
401C2C3×#480911yescharacteristic
402C2C3×#482611yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×Q8⋊C8C2
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC3×Q8⋊C8D3
hypocenterC3C3×#1649
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C3×#5031C2semidirect
4. C3×#4682C2semidirect
5. C32×Q8⋊C8C2semidirect
6. C3×#4486C2semidirect
9. #68273C3direct (N × H)
32. #5031C2×C3semidirect
34. #4682C2×C3semidirect
37. C3×Q8⋊C8D3semidirect
38. C3×Q8⋊C8C2×C3semidirect
40. #4486C2×C3semidirect
117. C3×D12Q16semidirect
134. Q8⋊C8C3×D3semidirect
264. D12C3×Q16semidirect
346. C32#1649semidirect
397. C3#68273direct (N × H)
398. C3C3×#1649semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×D122
C3×#50312
C3×#48462
C3×#46822
C32×Q8⋊C82
C3×#44862
C3×#44592
C3×#16493
#682733