Finite Group Databasebeta

← C2×#11722850 #13170606

Subgroups — C2×#11722850 #13170606

Order 1728.

1298 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1298 conjugacy classes of subgroups — too many to draw. Only its 212 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
1C2×C3×#3795C212yesmaximal, complemented
2C2×#88812C211yesmaximal, characteristic, complemented
3C2×#88792C212yesmaximal, complemented
4C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C2×C3×#3713C211yesmaximal, characteristic, complemented
6#11722850C218yesmaximal, direct product factor
7C2×#63933—32nomaximal
8C2×#63822—31nomaximal
9C2×C3×#634C2212yes—
10C2×C32×D12C2212yes—
11C2×C32×C4×D3C2211yescharacteristic
12C2×C3×#625—22no—
13C2×C3×D3×Dic3—22no—
14C3×#3795—24no—
15C3×#3795C2218yescomplemented
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18#88812C2214yescomplemented
19#88792C2218yescomplemented
20D3×C3⋊(C3⋊C8)C2214yescomplemented
21C3×#3713C2214yescomplemented
22C2×C3×#237—32no—
23C2×C3×#235—32no—
24C2×#3795D312yescomplemented
25C2×#3754—32no—
26C22×C3⋊(C3⋊C8)—31no—
27C2×C3×#213—32no—
28#63933—316no—
29C2×#3719—32no—
30C2×#3718—31no—
31C2×#3715—32no—
32C2×#3715—32no—
33C2×#3715—62no—
34#63822—38no—
35C2×#3713D311yescharacteristic, complemented
36C2×D3×C3⋊C8—32no—
37C2×D3×C3⋊C8—61no—
38C22×C32×D3—22no—
39C22×C32×D3D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3—22no—
42C3×#634—22no—
43C3×#634C2314yes—
44C2×C32×Dic3D411yescharacteristic
45C32×D12—22no—
46C32×D12C2314yes—
47C32×C4×D3D412yes—
48C32×C4×D3C2312yes—
49C3×#625—28no—
50C3×D3×Dic3—28no—
51C3⋊(C3⋊(C3⋊C8))C2312yes—
52C3×C3⋊(C3⋊C8)C2312yes—
53C2×#950—92no—
54C2×#912—91no—
55C2×C32×D4—32no—
56C22×C32×C4—31no—
57C2×#634C2×D312yes—
58C2×C3×#67—62no—
59C2×C3×#67—62no—
60C22×C3×Dic3—62no—
61C3×#237—38no—
62C3×#237—64no—
63C3×#235—38no—
64C3×#235—64no—
65C2×C3×D12—22no—
66C2×C3×D12C2×D312yes—
67C2×C3×D12C2×D312yes—
68C2×C3×C4×D3—21no—
69C2×C3×C4×D3—62no—
70C2×C3×C4×D3C2×D312yes—
71C2×C3×Dic6—32no—
72C2×C3×Dic6—32no—
73C2×C3×Dic6—62no—
74C2×#625—22no—
75C2×D3×Dic3—22no—
76#3795—24no—
77#3795C2×D318yescomplemented
78#3754—38no—
79C2×C3⋊(C3⋊C8)—32no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—62no—
84C2×C3⋊(C3⋊C8)—121no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×#213—38no—
87C2×C3×C3⋊C8—32no—
88C2×C3×C3⋊C8—61no—
89#3719—38no—
90#3718—34no—
91#3715—38no—
92#3715—38no—
93#3715—68no—
94#3713C2×D314yescomplemented
95D3×C3⋊C8—38no—
96D3×C3⋊C8—64no—
97C22×C33C2×D411yescharacteristic
98C2×C32×D3—22no—
99C2×C32×D3—24no—
100C2×C32×D3—44no—
101C2×C32×D3C2×D412yes—
102C33×C4C2×D411yescharacteristic
103C33×C4C2411yescharacteristic
104C3×C3⋊Dic3—24no—
105C32×Dic3C2×D412yes—
106C2×C3×#41—92no—
107C2×#237—32no—
108C2×#235—32no—
109C2×C3×D8—91no—
110#950—916no—
111C2×#216—92no—
112C2×#215—92no—
113C2×#214—92no—
114C2×#214—92no—
115C2×#214—182no—
116C2×#213—32no—
117C22×C3⋊C8—92no—
118C22×C3⋊C8—181no—
119#912—98no—
120C2×Dic12—91no—
121C2×D3×C8—91no—
122C23×C32—31no—
123C23×C32—62no—
124C22×C3×D3—22no—
125C22×C3×D3—22no—
126C22×C3×D3—21no—
127C22×C3×D3—42no—
128C22×C3×D3#6712yes—
129C32×D4—34no—
130C32×D4—62no—
131C2×C32×C4—21no—
132C2×C32×C4—22no—
133C2×C32×C4—31no—
134C2×C32×C4—32no—
135C2×C32×C4—32no—
136C2×C32×C4—41no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D312yes—
139C2×C3⋊Dic3—22no—
140#634—22no—
141#634C22×D314yes—
142C3×#67—68no—
143C3×#67—68no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—68no—
148C2×C3×Dic3—122no—
149C2×C3×Dic3#6712yes—
150C3×D12—22no—
151C3×D12—22no—
152C3×D12—24no—
153C3×D12—42no—
154C3×D12C22×D314yes—
155C3×D12C22×D314yes—
156C3×C4×D3—22no—
157C3×C4×D3—22no—
158C3×C4×D3—68no—
159C3×C4×D3#6714yes—
160C3×C4×D3C22×D314yes—
161C3×Dic6—34no—
162C3×Dic6—34no—
163C3×Dic6—64no—
164C3×Dic6—62no—
165C3×Dic6—62no—
166C3×Dic6—122no—
167#625—28no—
168D3×Dic3—28no—
169C3⋊(C3⋊C8)—34no—
170C3⋊(C3⋊C8)—32no—
171C3⋊(C3⋊C8)—62no—
172C3⋊(C3⋊C8)—64no—
173C3⋊(C3⋊C8)—122no—
174C3⋊(C3⋊C8)C22×D312yes—
175C3×C3⋊C8—34no—
176C3×C3⋊C8—62no—
177C2×#135—271noSylow
178C2×C33#13512yescomplemented
179C2×C33C22×D411yescharacteristic
180C32×D3—22no—
181C32×D3—44no—
182C3×#41—98no—
183C3×#41—184no—
184C2×C3×Q8—92no—
185C2×C3×D4—32no—
186C2×C3×D4—32no—
187C2×C3×D4—62no—
188C2×C3×D4—182no—
189C22×C3×C4—32no—
190C22×C3×C4—61no—
191C22×C3×C4—182no—
192C2×#67—62no—
193C2×#67—62no—
194C22×Dic3—62no—
195#237—38no—
196#237—64no—
197#235—38no—
198#235—64no—
199C2×D12D3212yes—
200C2×C4×D3—62no—
201C2×C4×D3D3211yescharacteristic
202C2×Dic6—32no—
203C2×Dic6—32no—
204C2×Dic6—62no—
205C3×D8—94no—
206C2×C3×C8—91no—
207#216—98no—
208#215—98no—
209#214—98no—
210#214—98no—
211#214—188no—
212#213—38no—
213C2×C3⋊C8—32no—
214C2×C3⋊C8—61no—
215C2×C3⋊C8—98no—
216C2×C3⋊C8—91no—
217C2×C3⋊C8—92no—
218C2×C3⋊C8—184no—
219C2×C3⋊C8—181no—
220C2×C3⋊C8—182no—
221C2×C3⋊C8—361no—
222Dic12—94no—
223D3×C8—94no—
224C22×C32—21no—
225C22×C32—22no—
226C22×C32—32no—
227C22×C32—41no—
228C22×C32—62no—
229C22×C32—64no—
230C22×C32—124no—
231C22×C32D3×D411yescharacteristic
232C22×C32C2×#6712yes—
233C2×C3×D3—24no—
234C2×C3×D3—24no—
235C2×C3×D3—24no—
236C2×C3×D3—24no—
237C2×C3×D3—22no—
238C2×C3×D3—44no—
239C2×C3×D3—44no—
240C2×C3×D3—44no—
241C2×C3×D3—84no—
242C2×C3×D3C2×#6714yes—
243C32×C4—21no—
244C32×C4—22no—
245C32×C4—21no—
246C32×C4—22no—
247C32×C4—32no—
248C32×C4—41no—
249C32×C4—41no—
250C32×C4D3×D411yescharacteristic
251C32×C4C2×#6712yes—
252C32×C4C23×D311yescharacteristic
253C32×C4C23×D312yes—
254C3⋊Dic3—24no—
255C3×Dic3—22no—
256C3×Dic3—64no—
257C3×Dic3—64no—
258C3×Dic3—124no—
259C3×Dic3C2×#6714yes—
260C2×#41—92no—
261#135—278no—
262C2×Q16—271no—
263C2×SD16—272no—
264C2×D8—91no—
265C22×C8—271no—
266C33C2×#13511yescharacteristic, complemented, Sylow
267C23×C3—32no—
268C23×C3—61no—
269C23×C3—62no—
270C23×C3—62no—
271C23×C3—122no—
272C22×D3—22no—
273C22×D3#62511yescharacteristic
274C3×Q8—94no—
275C3×Q8—182no—
276C3×D4—34no—
277C3×D4—34no—
278C3×D4—64no—
279C3×D4—62no—
280C3×D4—62no—
281C3×D4—122no—
282C3×D4—188no—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—32no—
286C2×C3×C4—34no—
287C2×C3×C4—34no—
288C2×C3×C4—41no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—61no—
292C2×C3×C4—188no—
293C2×C3×C4—182no—
294C2×C3×C4C2×D3211yescharacteristic
295C2×C3×C4C2×D3212yes—
296#67—68no—
297#67—68no—
298C2×Dic3—62no—
299C2×Dic3—62no—
300C2×Dic3—68no—
301C2×Dic3—122no—
302C2×Dic3#62511yescharacteristic
303D12—22no—
304D12C2×D3214yes—
305C4×D3—68no—
306C4×D3#62512yes—
307C4×D3C2×D3212yes—
308Dic6—34no—
309Dic6—34no—
310Dic6—64no—
311Dic6—62no—
312Dic6—62no—
313Dic6—122no—
314C3×C8—92no—
315C3⋊C8—34no—
316C3⋊C8—62no—
317C3⋊C8—92no—
318C3⋊C8—94no—
319C3⋊C8—182no—
320C3⋊C8—184no—
321C3⋊C8—362no—
322C2×C32—21no—
323C2×C32—22no—
324C2×C32—22no—
325C2×C32—24no—
326C2×C32—42no—
327C2×C32—41no—
328C2×C32—62no—
329C2×C32—124no—
330C2×C32#91212yescomplemented
331C2×C32#95014yescomplemented
332C2×C32C2×D3×D411yescharacteristic
333C2×C32C22×#6712yes—
334C3×D3—24no—
335C3×D3—24no—
336C3×D3—44no—
337C3×D3—44no—
338C3×D3—84no—
339#41—98no—
340#41—184no—
341C2×Q8—92no—
342C2×D4—32no—
343C2×D4—182no—
344C22×C4—31no—
345C22×C4—182no—
346Q16—274no—
347SD16—278no—
348D8—94no—
349C2×C8—91no—
350C2×C8—274no—
351C2×C8—271no—
352C22×C3—22no—
353C22×C3—21no—
354C22×C3—34no—
355C22×C3—41no—
356C22×C3—62no—
357C22×C3—64no—
358C22×C3—64no—
359C22×C3—64no—
360C22×C3—64no—
361C22×C3—124no—
362C22×C3—124no—
363C22×C3—124no—
364C22×C3—244no—
365C22×C3C2×#62511yescharacteristic
366C22×C3D3×#6712yes—
367C2×D3—22no—
368C2×D3—24no—
369C2×D3—44no—
370C2×D3C2×#62512yes—
371C3×C4—21no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—21no—
375C3×C4—34no—
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—62no—
379C3×C4—184no—
380C3×C4C2×#62511yescharacteristic
381C3×C4D3×#6712yes—
382C3×C4C22×D3211yescharacteristic
383C3×C4C22×D3212yes—
384Dic3—64no—
385Dic3—64no—
386Dic3—124no—
387Dic3C2×#62512yes—
388C32—22no—
389C32—21no—
390C32—41no—
391C32C2×#91211yescharacteristic, complemented
392C32C2×#95012yescomplemented
393C23—31no—
394C23—62no—
395Q8—94no—
396Q8—182no—
397D4—34no—
398D4—62no—
399D4—188no—
400C2×C4—31no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—188no—
404C2×C4—182no—
405C2×C4D3311yescharacteristic
406C8—92no—
407C8—272no—
408C2×C3—22no—
409C2×C3—24no—
410C2×C3—22no—
411C2×C3—21no—
412C2×C3—42no—
413C2×C3—41no—
414C2×C3—64no—
415C2×C3—64no—
416C2×C3—124no—
417C2×C3—124no—
418C2×C3—244no—
419C2×C3#6382212yescomplemented
420C2×C3#6393314yescomplemented
421C2×C3C22×#62511yescharacteristic
422C2×C3C2×D3×#6712yes—
423D3—22no—
424D3—44no—
425C22—32no—
426C22—62no—
427C22—64no—
428C22—124no—
429C22D3×#62511yescharacteristic
430C4—32no—
431C4—184no—
432C4D3×#62511yescharacteristic
433C4C2×D3311yescharacteristic
434C3—21no—
435C3—22no—
436C3—41no—
437C3C2×#6382211yescharacteristic, complemented
438C3C2×#6393312yescomplemented
439C2—62no—
440C2—124no—
441C2#1172285012yesdirect product factor
442C2C2×D3×#62511yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
hypercenterC2×C4D33
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3795C2semidirect
2. C2×#88812C2semidirect
3. C2×#88792C2semidirect
4. C2×D3×C3⋊(C3⋊C8)C2semidirect
5. C2×C3×#3713C2semidirect
6. #11722850C2direct (N × H)
15. C3×#3795C22semidirect
18. #88812C22semidirect
19. #88792C22semidirect
20. D3×C3⋊(C3⋊C8)C22semidirect
21. C3×#3713C22semidirect
24. C2×#3795D3semidirect
35. C2×#3713D3semidirect
77. #3795C2×D3semidirect
94. #3713C2×D3semidirect
178. C2×C33#135semidirect
266. C33C2×#135semidirect
330. C2×C32#912semidirect
331. C2×C32#950semidirect
391. C32C2×#912semidirect
392. C32C2×#950semidirect
419. C2×C3#63822semidirect
420. C2×C3#63933semidirect
437. C3C2×#63822semidirect
438. C3C2×#63933semidirect
441. C2#11722850direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37952
C2×#888122
C2×#887922
C2×D3×C3⋊(C3⋊C8)2
C2×C3×#37132
#117228502
C2×#639333
C2×#638223