Finite Group Databasebeta

← C3×#70013 #11792657

Subgroups — C3×#70013 #11792657

Order 1152.

779 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5822C212yesmaximal, complemented
2C3×#5748C211yesmaximal, characteristic, complemented
3C32×#408C211yesmaximal, characteristic, complemented
4C3×#5110C212yesmaximal, complemented
5C3×C3⋊#338C211yesmaximal, characteristic, complemented
6C3×#2017—31nomaximal
7#70013C311yesmaximal, characteristic, direct product factor
8C2×C32×#41C2211yescharacteristic
9C2×C3×#235C2212yes—
10C2×C32×M16C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C3×#951—22no—
13C3×#950—24no—
14C3×#949—22no—
15C3×#942—22no—
16C3×#932—22no—
17C2×C3×#208C2212yes—
18C3×#909—22no—
19C3×#908—22no—
20C3×#904—22no—
21C32×C22⋊C8—22no—
22C3×(C2×D3)⋊C8—24no—
23C3×#575—32no—
24C3×#575—31no—
25#5822C2×C312yescomplemented
26#5748C2×C311yescharacteristic, complemented
27C3×#408—21no—
28C3×#408—32no—
29C3×#408D311yescharacteristic, complemented
30C3×#408C2×C311yescharacteristic, complemented
31#5110C2×C312yescomplemented
32C3×#338—31no—
33C3⋊#338C2×C311yescharacteristic, complemented
34C32×#41—22no—
35C32×#41—22no—
36C2×C32×Q8D411yescharacteristic
37C2×C32×D4—21no—
38C2×C32×D4D411yescharacteristic
39C22×C32×C4—21no—
40C22×C32×C4C2311yescharacteristic
41C2×C3×#67—22no—
42C3×#235—22no—
43C3×#235—22no—
44C3×#235—22no—
45C3×#235—22no—
46C2×C3×D12D412yes—
47C2×C3×C4×D3—22no—
48C2×C3×Dic6D412yes—
49C32×M16—22no—
50C2×C32×C8—21no—
51C2×C32×C8—22no—
52C2×C32×C8D412yes—
53C3×#216—24no—
54C3×#215—24no—
55C3×#214—24no—
56C3×#213—24no—
57C3×#208—22no—
58C3×#208—22no—
59C2×C3×C3⋊C8D414yes—
60C3×Dic12—22no—
61C3×D24—22no—
62C3×#204—24no—
63#2017—31noSylow
64C2×C3×#41—21no—
65C2×C3×#41—32no—
66C2×C3×#41C2×D311yescharacteristic
67C2×C3×#41C22×C311yescharacteristic
68C2×#235C22×C312yes—
69C3×#137—62no—
70C3×#137—62no—
71C3×#137—62no—
72C3×AGL(1,ℤ/8)—62no—
73C3×AGL(1,ℤ/8)—62no—
74C3×AGL(1,ℤ/8)—62no—
75C3×#135—64no—
76C3×#135—62no—
77C2×C3×M16—21no—
78C2×C3×M16—32no—
79C2×C3×M16C2×D311yescharacteristic
80C2×C3×M16C22×C311yescharacteristic
81C22×C3×C8—21no—
82C22×C3×C8C2×D311yescharacteristic
83C22×C3×C8C22×C311yescharacteristic
84#951—22no—
85#950—24no—
86#949—22no—
87#942—22no—
88#932—22no—
89C2×#208C22×C312yes—
90#909—22no—
91#908—22no—
92#904—22no—
93C3×C22⋊C8—24no—
94C3×C22⋊C8—22no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—64no—
97(C2×D3)⋊C8—24no—
98C23×C32—21no—
99C23×C32#4111yescharacteristic
100C22×C3×D3—22no—
101C32×Q8—22no—
102C32×D4—22no—
103C32×D4—42no—
104C2×C32×C4—21no—
105C2×C32×C4—41no—
106C2×C32×C4—41no—
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4C2×D412yes—
109C2×C32×C4C2×D412yes—
110C2×C32×C4C2×D411yescharacteristic
111C3×#67—42no—
112C3×#67—42no—
113C2×C3×Dic3—22no—
114C3×D12—22no—
115C3×D12—22no—
116C3×C4×D3—42no—
117C3×C4×D3—42no—
118C3×Dic6—22no—
119C3×Dic6—22no—
120C32×C8—22no—
121C32×C8—41no—
122C3×C3⋊C8—24no—
123#575—32no—
124#575—31no—
125#408—32no—
126#408C3×D311yescharacteristic, complemented
127#338—31no—
128C3×#41—22no—
129C3×#41—22no—
130C3×#41—22no—
131C3×#41—22no—
132C3×#41—42no—
133C3×#41—42no—
134C3×#41—62no—
135C3×#41—62no—
136C3×#41—62no—
137C3×#41—62no—
138C2×C3×Q8—21no—
139C2×C3×Q8—32no—
140C2×C3×Q8D1211yescharacteristic
141C2×C3×Q8C3×D411yescharacteristic
142C2×C3×D4—22no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—32no—
147C2×C3×D4—62no—
148C2×C3×D4D1211yescharacteristic
149C2×C3×D4C3×D411yescharacteristic
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—62no—
155C22×C3×C4C22×D311yescharacteristic
156C22×C3×C4C23×C311yescharacteristic
157C2×#67—22no—
158#235—22no—
159#235—22no—
160#235—22no—
161#235—22no—
162C2×D12C3×D412yes—
163C2×C4×D3—22no—
164C2×Dic6C3×D412yes—
165C3×Q16—62no—
166C3×Q16—64no—
167C3×SD16—64no—
168C3×SD16—64no—
169C3×SD16—64no—
170C3×D8—62no—
171C3×D8—64no—
172C3×M16—22no—
173C3×M16—22no—
174C3×M16—42no—
175C3×M16—62no—
176C3×M16—62no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—21no—
180C2×C3×C8—22no—
181C2×C3×C8—21no—
182C2×C3×C8—22no—
183C2×C3×C8—34no—
184C2×C3×C8—42no—
185C2×C3×C8#6712yes—
186C2×C3×C8C3×D412yes—
187#216—24no—
188#215—24no—
189#214—24no—
190#213—24no—
191#208—22no—
192#208—22no—
193C2×C3⋊C8C3×D414yes—
194Dic12—22no—
195D24—22no—
196#204—24no—
197C22×C32—41no—
198C22×C32—42no—
199C22×C32—41no—
200C22×C32#12112yes—
201C22×C32#12111yescharacteristic
202C2×C3×D3—44no—
203C2×C3×D3—42no—
204C32×C4—41no—
205C32×C4#12012yes—
206C32×C4#12011yescharacteristic
207C32×C4#12711yescharacteristic
208C3×Dic3—42no—
209C2×#41—32no—
210C2×#41C2×C3×D311yescharacteristic
211#137—62no—
212#137—62no—
213#137—62no—
214AGL(1,ℤ/8)—62no—
215AGL(1,ℤ/8)—62no—
216AGL(1,ℤ/8)—62no—
217#135—64no—
218#135—62no—
219C2×M16—32no—
220C2×M16C2×C3×D311yescharacteristic
221C22×C8C2×C3×D311yescharacteristic
222C22⋊C8—22no—
223C22⋊C8—64no—
224C23×C3—22no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—62no—
229C23×C3#23511yescharacteristic
230C23×C3C3×#4111yescharacteristic
231C22×D3—22no—
232C3×Q8—22no—
233C3×Q8—22no—
234C3×Q8—42no—
235C3×Q8—62no—
236C3×Q8—62no—
237C3×D4—22no—
238C3×D4—22no—
239C3×D4—42no—
240C3×D4—44no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—62no—
244C3×D4—62no—
245C3×D4—122no—
246C3×D4—122no—
247C2×C3×C4—22no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—21no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—62no—
261C2×C3×C4—122no—
262C2×C3×C4—122no—
263C2×C3×C4C2×D1211yescharacteristic
264C2×C3×C4D3×D412yes—
265C2×C3×C4D3×D412yes—
266C2×C3×C4C2×#6711yescharacteristic
267C2×C3×C4C2×C3×D411yescharacteristic
268C2×C3×C4C2×C3×D412yes—
269C2×C3×C4C2×C3×D412yes—
270C2×C3×C4C2×C3×D411yescharacteristic
271#67—42no—
272#67—42no—
273C2×Dic3—22no—
274D12—22no—
275D12—22no—
276C4×D3—42no—
277C4×D3—42no—
278Dic6—22no—
279Dic6—22no—
280C3×C8—22no—
281C3×C8—22no—
282C3×C8—42no—
283C3×C8—42no—
284C3×C8—41no—
285C3×C8—41no—
286C3×C8—64no—
287C3⋊C8—24no—
288C2×C32—21no—
289C2×C32—22no—
290C2×C32—81no—
291C2×C32#39211yescharacteristic
292C3×D3—82no—
293#41—22no—
294#41—22no—
295#41—62no—
296#41—62no—
297#41—62no—
298#41—62no—
299C2×Q8—32no—
300C2×Q8C3×D1211yescharacteristic
301C2×D4—21no—
302C2×D4—32no—
303C2×D4—62no—
304C2×D4C3×D1211yescharacteristic
305C22×C4—21no—
306C22×C4—62no—
307C22×C4C22×C3×D311yescharacteristic
308Q16—62no—
309Q16—64no—
310SD16—64no—
311SD16—64no—
312SD16—64no—
313D8—62no—
314D8—64no—
315M16—22no—
316M16—62no—
317M16—62no—
318C2×C8—22no—
319C2×C8—21no—
320C2×C8—34no—
321C2×C8C3×#6712yes—
322C22×C3—22no—
323C22×C3—21no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—44no—
329C22×C3—42no—
330C22×C3—42no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—124no—
334C22×C3—122no—
335C22×C3#88412yes—
336C22×C3#93011yescharacteristic
337C22×C3C3×#12112yes—
338C22×C3C3×#12111yescharacteristic
339C2×D3—44no—
340C2×D3—42no—
341C3×C4—22no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—42no—
345C3×C4—41no—
346C3×C4—41no—
347C3×C4—122no—
348C3×C4#88212yes—
349C3×C4#93711yescharacteristic
350C3×C4#94011yescharacteristic
351C3×C4C3×#12012yes—
352C3×C4C3×#12011yescharacteristic
353C3×C4C3×#12711yescharacteristic
354Dic3—42no—
355C32#201711yescharacteristic, complemented, Sylow
356C23—21no—
357C23—62no—
358C23C3×#23511yescharacteristic
359Q8—22no—
360Q8—62no—
361Q8—62no—
362D4—22no—
363D4—42no—
364D4—62no—
365D4—62no—
366D4—122no—
367D4—122no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—62no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4C2×C3×D1211yescharacteristic
375C2×C4C3×D3×D412yes—
376C2×C4C3×D3×D412yes—
377C2×C4C2×C3×#6711yescharacteristic
378C8—22no—
379C8—41no—
380C8—64no—
381C2×C3—22no—
382C2×C3—22no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—22no—
386C2×C3—21no—
387C2×C3—42no—
388C2×C3—82no—
389C2×C3—81no—
390C2×C3—81no—
391C2×C3—242no—
392C2×C3#496011yescharacteristic
393C2×C3C3×#39211yescharacteristic
394D3—82no—
395C22—41no—
396C22—42no—
397C22—41no—
398C22—124no—
399C22—122no—
400C22C3×#88412yes—
401C22C3×#93011yescharacteristic
402C4—41no—
403C4—122no—
404C4C3×#88212yes—
405C4C3×#93711yescharacteristic
406C4C3×#94011yescharacteristic
407C3—21no—
408C3#7001311yescharacteristic, direct product factor
409C3C3×#201711yescharacteristic, complemented
410C2—21no—
411C2—22no—
412C2—81no—
413C2—242no—
414C2C3×#496011yescharacteristic
subgroupG/H
centerC3×C4#940
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#408C2
socleC2×C32#392
hypercenterC3×#408D3
hypocenterC3C3×#2017
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#408

Complements

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

NH ≅ G/Ntype
1. C3×#5822C2semidirect
2. C3×#5748C2semidirect
3. C32×#408C2semidirect
4. C3×#5110C2semidirect
5. C3×C3⋊#338C2semidirect
7. #70013C3direct (N × H)
25. #5822C2×C3semidirect
26. #5748C2×C3semidirect
29. C3×#408D3semidirect
30. C3×#408C2×C3semidirect
31. #5110C2×C3semidirect
33. C3⋊#338C2×C3semidirect
126. #408C3×D3semidirect
355. C32#2017semidirect
408. C3#70013direct (N × H)
409. C3C3×#2017semidirect

Maximal subgroups

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

subgroupindex
C3×#58222
C3×#57482
C32×#4082
C3×#51102
C3×C3⋊#3382
C3×#20173
#700133