Finite Group Databasebeta

← C2×#11722852 #13170608

Subgroups — C2×#11722852 #13170608

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×#3799C212yesmaximal, complemented
2C2×#88810C211yesmaximal, characteristic, complemented
3C2×#88793C212yesmaximal, complemented
4C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C2×C3×#3718C211yesmaximal, characteristic, complemented
6#11722852C218yesmaximal, direct product factor
7C2×#63949—32nomaximal
8C2×#63822—31nomaximal
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C2×C32×C4×D3C2211yescharacteristic
11C2×C32×Dic6C2212yes—
12C2×C3×#626—22no—
13C2×C3×#624—22no—
14C3×#3799—24no—
15C3×#3799C2218yescomplemented
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18#88810C2214yescomplemented
19#88793C2218yescomplemented
20D3×C3⋊(C3⋊C8)C2214yescomplemented
21C3×#3718C2214yescomplemented
22C2×C3×#239—32no—
23C2×C3×#235—32no—
24C2×#3799D312yescomplemented
25C2×#3755—32no—
26C22×C3⋊(C3⋊C8)—31no—
27C2×C3×#216—32no—
28#63949—316no—
29C2×#3718D311yescharacteristic, complemented
30C2×#3715—32no—
31C2×#3715—32no—
32C2×#3715—62no—
33C2×#3714—32no—
34#63822—38no—
35C2×#3713—31no—
36C2×D3×C3⋊C8—32no—
37C2×D3×C3⋊C8—61no—
38C22×C3×C3⋊D3—22no—
39C22×C32×D3D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C3×C4⋊(C3⋊D3)—22no—
42C3×C4⋊(C3⋊D3)C2314yes—
43C2×C32×Dic3—22no—
44C2×C32×Dic3D411yescharacteristic
45C32×C4×D3D412yes—
46C32×C4×D3C2312yes—
47C32×Dic6—22no—
48C32×Dic6C2314yes—
49C3×#626—28no—
50C3×#624—28no—
51C3⋊(C3⋊(C3⋊C8))C2312yes—
52C3×C3⋊(C3⋊C8)C2312yes—
53C2×#950—92no—
54C2×#919—91no—
55C2×C32×Q8—32no—
56C22×C32×C4—31no—
57C2×C4⋊(C3⋊D3)C2×D312yes—
58C2×C3×#67—62no—
59C3×#239—38no—
60C3×#239—64no—
61C3×#235—38no—
62C3×#235—64no—
63C2×C3×D12—32no—
64C2×C3×D12—32no—
65C2×C3×D12—62no—
66C2×C3×D12—62no—
67C2×C3×C4×D3—21no—
68C2×C3×C4×D3—62no—
69C2×C3×C4×D3—62no—
70C2×C3×C4×D3C2×D312yes—
71C2×C3×Dic6—22no—
72C2×C3×Dic6C2×D312yes—
73C2×C3×Dic6C2×D312yes—
74C2×#626—22no—
75C2×#624—22no—
76#3799—24no—
77#3799C2×D318yescomplemented
78#3755—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×#216—38no—
87C2×C3×C3⋊C8—32no—
88C2×C3×C3⋊C8—61no—
89#3718C2×D314yescomplemented
90#3715—38no—
91#3715—38no—
92#3715—68no—
93#3714—38no—
94#3713—34no—
95D3×C3⋊C8—38no—
96D3×C3⋊C8—64no—
97C22×C33C2×D411yescharacteristic
98C2×C3×C3⋊D3—24no—
99C2×C3×C3⋊D3—44no—
100C2×C32×D3—22no—
101C2×C32×D3C2×D412yes—
102C33×C4C2×D411yescharacteristic
103C33×C4C2411yescharacteristic
104C32×Dic3—24no—
105C32×Dic3C2×D412yes—
106C2×C3×#41—92no—
107C2×#239—32no—
108C2×#235—32no—
109C2×C3×Q16—91no—
110#950—916no—
111C2×#216—32no—
112C2×#215—92no—
113C2×#215—92no—
114C2×#215—182no—
115C2×#214—92no—
116C2×#213—92no—
117C22×C3⋊C8—92no—
118C22×C3⋊C8—181no—
119#919—98no—
120C2×D24—91no—
121C2×D3×C8—91no—
122C23×C32—31no—
123C22×C3⋊D3—22no—
124C22×C3×D3—21no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C22×C3×D3—122no—
128C22×C3×D3#6712yes—
129C32×Q8—34no—
130C32×Q8—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×C4—62no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D312yes—
140C4⋊(C3⋊D3)—22no—
141C4⋊(C3⋊D3)C22×D314yes—
142C3×#67—68no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—42no—
147C2×C3×Dic3#6712yes—
148C3×D12—34no—
149C3×D12—34no—
150C3×D12—64no—
151C3×D12—68no—
152C3×D12—62no—
153C3×D12—62no—
154C3×D12—122no—
155C3×C4×D3—22no—
156C3×C4×D3—22no—
157C3×C4×D3—68no—
158C3×C4×D3—68no—
159C3×C4×D3#6714yes—
160C3×C4×D3C22×D314yes—
161C3×Dic6—22no—
162C3×Dic6—22no—
163C3×Dic6—24no—
164C3×Dic6—42no—
165C3×Dic6C22×D314yes—
166C3×Dic6C22×D314yes—
167#626—28no—
168#624—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
180C3×C3⋊D3—44no—
181C32×D3—22no—
182C3×#41—98no—
183C3×#41—184no—
184C2×C3×Q8—32no—
185C2×C3×Q8—32no—
186C2×C3×Q8—62no—
187C2×C3×D4—92no—
188C2×C3×D4—182no—
189C22×C3×C4—32no—
190C22×C3×C4—61no—
191C22×C3×C4—182no—
192C2×#67—62no—
193#239—38no—
194#239—64no—
195#235—38no—
196#235—64no—
197C2×D12—32no—
198C2×D12—32no—
199C2×D12—62no—
200C2×D12—62no—
201C2×C4×D3—62no—
202C2×C4×D3—62no—
203C2×C4×D3D3211yescharacteristic
204C2×Dic6D3212yes—
205C3×Q16—94no—
206C2×C3×C8—91no—
207#216—38no—
208#215—98no—
209#215—98no—
210#215—188no—
211#214—98no—
212#213—98no—
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—
222D24—94no—
223D3×C8—94no—
224C22×C32—21no—
225C22×C32—22no—
226C22×C32—32no—
227C22×C32—41no—
228C22×C32—62no—
229C22×C32D3×D411yescharacteristic
230C22×C32C2×#6712yes—
231C2×C3⋊D3—24no—
232C2×C3⋊D3—44no—
233C2×C3×D3—24no—
234C2×C3×D3—24no—
235C2×C3×D3—22no—
236C2×C3×D3—64no—
237C2×C3×D3—64no—
238C2×C3×D3—124no—
239C2×C3×D3—124no—
240C2×C3×D3—124no—
241C2×C3×D3—244no—
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×C4—64no—
251C32×C4D3×D411yescharacteristic
252C32×C4C2×#6712yes—
253C32×C4C23×D311yescharacteristic
254C32×C4C23×D312yes—
255C3×Dic3—24no—
256C3×Dic3—24no—
257C3×Dic3—22no—
258C3×Dic3—44no—
259C3×Dic3C2×#6714yes—
260C2×#41—92no—
261#135—278no—
262C2×Q16—91no—
263C2×SD16—272no—
264C2×D8—271no—
265C22×C8—271no—
266C33C2×#13511yescharacteristic, complemented, Sylow
267C23×C3—32no—
268C23×C3—61no—
269C23×C3—182no—
270C22×D3—62no—
271C22×D3—62no—
272C22×D3—122no—
273C22×D3#62511yescharacteristic
274C3×Q8—34no—
275C3×Q8—34no—
276C3×Q8—64no—
277C3×Q8—62no—
278C3×Q8—62no—
279C3×Q8—122no—
280C3×D4—94no—
281C3×D4—188no—
282C3×D4—182no—
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—62no—
293C2×C3×C4—62no—
294C2×C3×C4—122no—
295C2×C3×C4—188no—
296C2×C3×C4C2×D3211yescharacteristic
297C2×C3×C4C2×D3212yes—
298#67—68no—
299C2×Dic3—22no—
300C2×Dic3#62511yescharacteristic
301D12—34no—
302D12—34no—
303D12—64no—
304D12—68no—
305D12—62no—
306D12—62no—
307D12—122no—
308C4×D3—68no—
309C4×D3—68no—
310C4×D3#62512yes—
311C4×D3C2×D3212yes—
312Dic6—22no—
313Dic6C2×D3214yes—
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#91912yescomplemented
330C2×C32#95014yescomplemented
331C2×C32C2×D3×D411yescharacteristic
332C2×C32C22×#6712yes—
333C3⋊D3—44no—
334C3×D3—24no—
335C3×D3—24no—
336C3×D3—124no—
337C3×D3—124no—
338C3×D3—244no—
339#41—98no—
340#41—184no—
341C2×Q8—32no—
342C2×D4—92no—
343C2×D4—182no—
344C22×C4—31no—
345C22×C4—182no—
346Q16—94no—
347SD16—278no—
348D8—274no—
349C2×C8—91no—
350C2×C8—274no—
351C2×C8—271no—
352C22×C3—21no—
353C22×C3—22no—
354C22×C3—34no—
355C22×C3—41no—
356C22×C3—62no—
357C22×C3—64no—
358C22×C3—64no—
359C22×C3—184no—
360C22×C3—364no—
361C22×C3C2×#62511yescharacteristic
362C22×C3D3×#6712yes—
363C2×D3—22no—
364C2×D3—64no—
365C2×D3—64no—
366C2×D3—124no—
367C2×D3—124no—
368C2×D3—124no—
369C2×D3—244no—
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—64no—
380C3×C4—64no—
381C3×C4—124no—
382C3×C4C2×#62511yescharacteristic
383C3×C4D3×#6712yes—
384C3×C4C22×D3211yescharacteristic
385C3×C4C22×D3212yes—
386Dic3—24no—
387Dic3C2×#62512yes—
388C32—22no—
389C32—21no—
390C32—41no—
391C32C2×#91911yescharacteristic, complemented
392C32C2×#95012yescomplemented
393C23—31no—
394C23—182no—
395Q8—34no—
396Q8—62no—
397D4—94no—
398D4—188no—
399D4—182no—
400C2×C4—31no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—62no—
404C2×C4—188no—
405C2×C4D3311yescharacteristic
406C8—92no—
407C8—272no—
408C2×C3—24no—
409C2×C3—22no—
410C2×C3—21no—
411C2×C3—22no—
412C2×C3—42no—
413C2×C3—41no—
414C2×C3—64no—
415C2×C3—64no—
416C2×C3—364no—
417C2×C3#6382212yescomplemented
418C2×C3#6394914yescomplemented
419C2×C3C22×#62511yescharacteristic
420C2×C3C2×D3×#6712yes—
421D3—22no—
422D3—124no—
423D3—124no—
424D3—244no—
425C22—32no—
426C22—62no—
427C22—184no—
428C22—364no—
429C22D3×#62511yescharacteristic
430C4—32no—
431C4—64no—
432C4D3×#62511yescharacteristic
433C4C2×D3311yescharacteristic
434C3—22no—
435C3—21no—
436C3—41no—
437C3C2×#6382211yescharacteristic, complemented
438C3C2×#6394912yescomplemented
439C2—62no—
440C2—364no—
441C2#1172285212yesdirect 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×#3799C2semidirect
2. C2×#88810C2semidirect
3. C2×#88793C2semidirect
4. C2×D3×C3⋊(C3⋊C8)C2semidirect
5. C2×C3×#3718C2semidirect
6. #11722852C2direct (N × H)
15. C3×#3799C22semidirect
18. #88810C22semidirect
19. #88793C22semidirect
20. D3×C3⋊(C3⋊C8)C22semidirect
21. C3×#3718C22semidirect
24. C2×#3799D3semidirect
29. C2×#3718D3semidirect
77. #3799C2×D3semidirect
89. #3718C2×D3semidirect
178. C2×C33#135semidirect
266. C33C2×#135semidirect
329. C2×C32#919semidirect
330. C2×C32#950semidirect
391. C32C2×#919semidirect
392. C32C2×#950semidirect
417. C2×C3#63822semidirect
418. C2×C3#63949semidirect
437. C3C2×#63822semidirect
438. C3C2×#63949semidirect
441. C2#11722852direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37992
C2×#888102
C2×#887932
C2×D3×C3⋊(C3⋊C8)2
C2×C3×#37182
#117228522
C2×#639493
C2×#638223