Finite Group Databasebeta

← C2×#11723711 #13178750

Subgroups — C2×#11723711 #13178750

Order 1728.

1354 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 199 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×#3832C211yesmaximal, characteristic
2C2×#89040C211yesmaximal, characteristic, complemented
3C2×C3×#3756C212yesmaximal, complemented
4#11723711C218yesmaximal, direct product factor
5C2×#88811C212yesmaximal, complemented
6C2×#88809C211yesmaximal, characteristic, complemented
7C2×#63945—31nomaximal
8C2×#63943—32nomaximal
9C2×C33×Q8C2211yescharacteristic
10C3×#3832—22no—
11C3×#3832C2214yes—
12C2×C3×C4⋊(C3⋊D3)—21no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3—21no—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2212yes—
17C2×#6296—21no—
18C2×#6294—21no—
19#89040—22no—
20#89040C2214yescomplemented
21C3×#3756C2218yescomplemented
22C2×C3×C3⋊(C3⋊C8)C2212yes—
23#88811C2218yescomplemented
24#88809C2214yescomplemented
25C2×#3832D311yescharacteristic
26C2×C3×#239—32no—
27C2×C3×#239—61no—
28C2×#3797—31no—
29C2×D3×Dic6—32no—
30C2×#3756D312yescomplemented
31C2×C3×#216—32no—
32C2×C3×#216—32no—
33C2×C3×#216—62no—
34#63945—38no—
35#63943—316no—
36C2×#3717—32no—
37C2×#3716—32no—
38C2×#3715—32no—
39C2×#3712—31no—
40C2×#3711—32no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C33×Q8—21no—
44C33×Q8C2312yes—
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—22no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4⋊(C3⋊D3)—24no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)C2312yes—
52C3×C4×C3⋊D3—24no—
53C3×C4×C3⋊D3D412yes—
54C3×C4×C3⋊D3C2312yes—
55C3×#634—22no—
56C3×#634C2314yes—
57#6296—24no—
58#6294—24no—
59C3×C3⋊(C3⋊C8)C2314yes—
60C2×#951—91no—
61C2×#918—92no—
62C2×C32×Q8—21no—
63C2×C32×Q8—22no—
64C2×C32×Q8—41no—
65C2×C32×Q8C2×D311yescharacteristic
66C2×C32×Q8C2×D312yes—
67#3832—22no—
68#3832C2×D314yes—
69C2×C4⋊(C3⋊D3)—21no—
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×C4×C3⋊D3—21no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×#634C2×D312yes—
74C3×#239—38no—
75C3×#239—64no—
76C3×#239—64no—
77C3×#239—64no—
78C2×C3×D12—32no—
79C2×C3×D12—62no—
80C2×C3×D12—61no—
81C2×C3×D12—62no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—62no—
84C2×C3×C4×D3—61no—
85C2×C3×C4×D3—62no—
86C2×C3×Dic6—32no—
87C2×C3×Dic6—32no—
88C2×C3×Dic6—62no—
89C2×#627—62no—
90C2×#627—61no—
91C2×#624—61no—
92C2×D3×Dic3—62no—
93#3797—34no—
94#3797—62no—
95D3×Dic6—38no—
96D3×Dic6—64no—
97#3756C2×D318yescomplemented
98C2×C3⋊(C3⋊C8)C2×D312yes—
99C3×#216—38no—
100C3×#216—38no—
101C3×#216—68no—
102C2×C3×C3⋊C8—32no—
103C2×C3×C3⋊C8—32no—
104C2×C3×C3⋊C8—62no—
105#3717—38no—
106#3716—38no—
107#3715—38no—
108#3712—34no—
109#3711—38no—
110C22×C33C2×D411yescharacteristic
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3—22no—
113C2×C3×C3⋊D3—42no—
114C2×C3×C3⋊D3C2×D412yes—
115C33×C4—22no—
116C33×C4C2×D411yescharacteristic
117C33×C4C2411yescharacteristic
118C3×C3⋊Dic3—24no—
119C3×C3⋊Dic3C2×D412yes—
120C2×C3×#41—91no—
121C2×#239—32no—
122C2×#239—61no—
123C2×D3×Q8—92no—
124C22×Dic6—91no—
125C2×C3×Q16—92no—
126#951—98no—
127C2×#216—32no—
128C2×#216—32no—
129C2×#216—62no—
130C2×#215—92no—
131C2×#214—92no—
132C2×#208—91no—
133#918—916no—
134C2×#204—92no—
135C2×ΓL(1,𝔽25)—92no—
136C22×C3⋊D3—21no—
137C22×C3⋊D3#6711yescharacteristic
138C22×C3×D3—32no—
139C22×C3×D3—61no—
140C22×C3×D3—62no—
141C22×C3×D3—62no—
142C32×Q8—21no—
143C32×Q8—22no—
144C32×Q8—22no—
145C32×Q8—22no—
146C32×Q8—24no—
147C32×Q8—42no—
148C32×Q8—42no—
149C32×Q8—42no—
150C32×Q8C22×D312yes—
151C32×Q8C22×D314yes—
152C2×C32×C4—21no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—41no—
158C2×C32×C4—42no—
159C2×C32×C4—42no—
160C2×C32×C4C22×D312yes—
161C2×C32×C4C22×D311yescharacteristic
162C2×C3⋊Dic3—22no—
163C2×C3⋊Dic3#6711yescharacteristic
164C4⋊(C3⋊D3)—24no—
165C4⋊(C3⋊D3)—21no—
166C4⋊(C3⋊D3)C22×D312yes—
167C4×C3⋊D3—24no—
168C4×C3⋊D3#6712yes—
169C4×C3⋊D3C22×D312yes—
170#634—22no—
171#634C22×D314yes—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—122no—
177C3×D12—34no—
178C3×D12—62no—
179C3×D12—68no—
180C3×D12—68no—
181C3×D12—62no—
182C3×D12—62no—
183C3×C4×D3—34no—
184C3×C4×D3—34no—
185C3×C4×D3—62no—
186C3×C4×D3—62no—
187C3×C4×D3—68no—
188C3×C4×D3—68no—
189C3×Dic6—34no—
190C3×Dic6—34no—
191C3×Dic6—64no—
192C3×Dic6—62no—
193C3×Dic6—62no—
194C3×Dic6—122no—
195#627—68no—
196#627—64no—
197#624—64no—
198D3×Dic3—68no—
199C3⋊(C3⋊C8)C22×D314yes—
200C3×C3⋊C8—34no—
201C3×C3⋊C8—34no—
202C3×C3⋊C8—64no—
203C2×#137—271noSylow
204C2×C33#13712yescomplemented
205C2×C33C22×D411yescharacteristic
206C3×C3⋊D3—22no—
207C3×C3⋊D3—42no—
208C3×#41—94no—
209C3×#41—182no—
210C2×C3×Q8—21no—
211C2×C3×Q8—22no—
212C2×C3×Q8—41no—
213C2×C3×Q8—92no—
214C2×C3×Q8D3211yescharacteristic
215C2×C3×Q8D3212yes—
216C2×C3×D4—91no—
217C2×C3×D4—181no—
218C22×C3×C4—91no—
219C22×C3×C4—181no—
220C22×Dic3—181no—
221#239—38no—
222#239—64no—
223#239—64no—
224#239—64no—
225D3×Q8—98no—
226D3×Q8—184no—
227C2×D12—32no—
228C2×D12—61no—
229C2×D12—62no—
230C2×D12—62no—
231C2×C4×D3—32no—
232C2×C4×D3—61no—
233C2×C4×D3—62no—
234C2×C4×D3—62no—
235C2×C4×D3—182no—
236C2×Dic6—32no—
237C2×Dic6—32no—
238C2×Dic6—62no—
239C2×Dic6—94no—
240C2×Dic6—182no—
241C2×Dic6—182no—
242C2×Dic6—181no—
243C3×Q16—98no—
244C2×C3×C8—92no—
245#216—38no—
246#216—38no—
247#216—68no—
248#215—98no—
249#214—98no—
250#208—94no—
251C2×C3⋊C8—32no—
252C2×C3⋊C8—32no—
253C2×C3⋊C8—62no—
254#204—98no—
255ΓL(1,𝔽25)—98no—
256C22×C32—22no—
257C22×C32—21no—
258C22×C32—41no—
259C22×C32D3×D412yes—
260C22×C32C2×#6711yescharacteristic
261C2×C3⋊D3—22no—
262C2×C3⋊D3—22no—
263C2×C3⋊D3—42no—
264C2×C3⋊D3C2×#6712yes—
265C2×C3×D3—34no—
266C2×C3×D3—62no—
267C2×C3×D3—64no—
268C2×C3×D3—64no—
269C2×C3×D3—64no—
270C2×C3×D3—122no—
271C2×C3×D3—124no—
272C2×C3×D3—124no—
273C32×C4—22no—
274C32×C4—24no—
275C32×C4—24no—
276C32×C4—22no—
277C32×C4—21no—
278C32×C4—22no—
279C32×C4—21no—
280C32×C4—41no—
281C32×C4—41no—
282C32×C4—44no—
283C32×C4—44no—
284C32×C4D3×D412yes—
285C32×C4C2×#6711yescharacteristic
286C32×C4C23×D312yes—
287C32×C4C23×D311yescharacteristic
288C3⋊Dic3—24no—
289C3⋊Dic3C2×#6712yes—
290C3×Dic3—34no—
291C3×Dic3—62no—
292C3×Dic3—64no—
293C3×Dic3—64no—
294C3×Dic3—124no—
295C2×#41—91no—
296C22×Q8—271no—
297#137—278no—
298C2×Q16—92no—
299C2×SD16—272no—
300C2×M16—271no—
301C33C2×#13711yescharacteristic, complemented, Sylow
302C23×C3—91no—
303C23×C3—181no—
304C22×D3—32no—
305C22×D3—61no—
306C22×D3—62no—
307C22×D3—62no—
308C3×Q8—22no—
309C3×Q8—22no—
310C3×Q8—21no—
311C3×Q8—22no—
312C3×Q8—24no—
313C3×Q8—42no—
314C3×Q8—42no—
315C3×Q8—42no—
316C3×Q8—94no—
317C3×Q8—182no—
318C3×Q8C2×D3212yes—
319C3×Q8C2×D3214yes—
320C3×D4—92no—
321C3×D4—184no—
322C3×D4—181no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—91no—
332C2×C3×C4—92no—
333C2×C3×C4—92no—
334C2×C3×C4—184no—
335C2×C3×C4—182no—
336C2×C3×C4C2×D3212yes—
337C2×C3×C4C2×D3211yescharacteristic
338C2×Dic3—32no—
339C2×Dic3—61no—
340C2×Dic3—62no—
341C2×Dic3—62no—
342C2×Dic3—122no—
343C2×Dic3—184no—
344D12—34no—
345D12—62no—
346D12—68no—
347D12—68no—
348D12—62no—
349D12—62no—
350C4×D3—34no—
351C4×D3—34no—
352C4×D3—62no—
353C4×D3—62no—
354C4×D3—68no—
355C4×D3—68no—
356C4×D3—188no—
357Dic6—34no—
358Dic6—34no—
359Dic6—64no—
360Dic6—62no—
361Dic6—62no—
362Dic6—122no—
363Dic6—188no—
364Dic6—184no—
365C3×C8—94no—
366C3⋊C8—34no—
367C3⋊C8—34no—
368C3⋊C8—64no—
369C2×C32—22no—
370C2×C32—21no—
371C2×C32—24no—
372C2×C32—22no—
373C2×C32—42no—
374C2×C32—41no—
375C2×C32#91814yescomplemented
376C2×C32#95112yescomplemented
377C2×C32C2×D3×D412yes—
378C2×C32C22×#6711yescharacteristic
379C3⋊D3—22no—
380C3⋊D3—42no—
381C3×D3—64no—
382C3×D3—122no—
383C3×D3—124no—
384C3×D3—124no—
385#41—94no—
386#41—182no—
387C2×Q8—92no—
388C2×Q8—274no—
389C2×Q8—541no—
390C2×Q8—542no—
391C2×Q8#111311yescharacteristic
392C2×D4—91no—
393C2×D4—181no—
394C22×C4—91no—
395C22×C4—181no—
396C22×C4—541no—
397Q16—98no—
398SD16—278no—
399M16—274no—
400C2×C8—92no—
401C22×C3—21no—
402C22×C3—22no—
403C22×C3—41no—
404C22×C3—92no—
405C22×C3—182no—
406C22×C3—182no—
407C22×C3—362no—
408C22×C3D3×#6712yes—
409C22×C3#381111yescharacteristic
410C2×D3—34no—
411C2×D3—62no—
412C2×D3—64no—
413C2×D3—64no—
414C2×D3—64no—
415C2×D3—122no—
416C2×D3—124no—
417C2×D3—124no—
418C3×C4—24no—
419C3×C4—24no—
420C3×C4—22no—
421C3×C4—22no—
422C3×C4—21no—
423C3×C4—22no—
424C3×C4—21no—
425C3×C4—44no—
426C3×C4—44no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—92no—
430C3×C4—184no—
431C3×C4D3×#6712yes—
432C3×C4#381111yescharacteristic
433C3×C4C22×D3212yes—
434C3×C4C22×D3211yescharacteristic
435Dic3—34no—
436Dic3—62no—
437Dic3—64no—
438Dic3—64no—
439Dic3—124no—
440C32—21no—
441C32—22no—
442C32—41no—
443C32C2×#91812yescomplemented
444C32C2×#95111yescharacteristic, complemented
445C23—91no—
446C23—181no—
447Q8—21no—
448Q8—94no—
449Q8—182no—
450Q8—544no—
451Q8C2×#111312yes—
452D4—92no—
453D4—184no—
454D4—181no—
455C2×C4—21no—
456C2×C4—91no—
457C2×C4—92no—
458C2×C4—92no—
459C2×C4—184no—
460C2×C4—182no—
461C2×C4—544no—
462C2×C4C2×#111311yescharacteristic
463C8—94no—
464C2×C3—21no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3—24no—
468C2×C3—41no—
469C2×C3—42no—
470C2×C3—182no—
471C2×C3—362no—
472C2×C3#6394314yescomplemented
473C2×C3#6394512yescomplemented
474C2×C3C2×D3×#6712yes—
475C2×C3C2×#381111yescharacteristic
476D3—64no—
477D3—122no—
478D3—124no—
479D3—124no—
480C22—92no—
481C22—182no—
482C22—182no—
483C22—362no—
484C22#8904911yescharacteristic
485C4—22no—
486C4—92no—
487C4—184no—
488C4#8904911yescharacteristic
489C4C22×#111311yescharacteristic
490C3—21no—
491C3—22no—
492C3—41no—
493C3C2×#6394312yescomplemented
494C3C2×#6394511yescharacteristic, complemented
495C2—182no—
496C2—362no—
497C2#1172371112yesdirect product factor
498C2C2×#8904911yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC33×C4C24
FrattiniC4C22×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×#3832C2
hypercenterC2×Q8#1113
hypocenterC33C2×#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#89040C2semidirect
3. C2×C3×#3756C2semidirect
4. #11723711C2direct (N × H)
5. C2×#88811C2semidirect
6. C2×#88809C2semidirect
20. #89040C22semidirect
21. C3×#3756C22semidirect
23. #88811C22semidirect
24. #88809C22semidirect
30. C2×#3756D3semidirect
97. #3756C2×D3semidirect
204. C2×C33#137semidirect
301. C33C2×#137semidirect
375. C2×C32#918semidirect
376. C2×C32#951semidirect
443. C32C2×#918semidirect
444. C32C2×#951semidirect
472. C2×C3#63943semidirect
473. C2×C3#63945semidirect
493. C3C2×#63943semidirect
494. C3C2×#63945semidirect
497. C2#11723711direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×#890402
C2×C3×#37562
#117237112
C2×#888112
C2×#888092
C2×#639453
C2×#639433