Finite Group Databasebeta

← C2×#11720846 #13149824

Subgroups — C2×#11720846 #13149824

Order 1728.

1111 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 171 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×#88654C211yesmaximal, characteristic
2C2×((C3⋊D3)⋊C3)⋊Q8C211yesmaximal, characteristic, complemented
3C2×#88528C211yesmaximal, characteristic, complemented
4C2×#3744⋊C3C211yesmaximal, characteristic, complemented
5C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
6C2×#88438C211yesmaximal, characteristic, complemented
7C2×#88437C211yesmaximal, characteristic, complemented
8#11720846C218yesmaximal, direct product factor
9C2×#63796—31nomaximal
10C2×D3×#204—31nomaximal
11C22×((C3⋊D3)⋊C3)⋊C2—21no—
12C2×C4⋊#272C2211yescharacteristic
13C2×#6229C2211yescharacteristic, complemented
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic, complemented
17C2×#6200—21no—
18C2×((C3⋊D3)⋊C3)⋊C4—21no—
19C2×#6196—21no—
20#88654—22no—
21#88654C2214yes—
22((C3⋊D3)⋊C3)⋊Q8—22no—
23((C3⋊D3)⋊C3)⋊Q8C2214yescomplemented
24C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
25#88528C2214yescomplemented
26C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
27#3744⋊C3C2214yescomplemented
28C8×(C3⋊D3)⋊C3C2214yescomplemented
29#88438C2214yescomplemented
30#88437C2214yescomplemented
31C2×#3802—31no—
32C2×D3×D12—31no—
33C2×#3797—31no—
34C2×D3×Dic6—31no—
35C2×#3744D311yescharacteristic, complemented
36C2×C8×C3⋊D3D311yescharacteristic, complemented
37C2×C3×#204—31no—
38C2×C3×#204—31no—
39C2×C3×#204—31no—
40C2×C3×#204—61no—
41C2×C3×D3×C8—31no—
42C2×#3717—31no—
43C2×#3716—31no—
44C2×#3715—31no—
45C2×#3715—31no—
46#63796—38no—
47D3×#204—38no—
48C22×#272—21no—
49C22×(C3⋊D3)⋊C3—21no—
50C22×(C3⋊D3)⋊C3D411yescharacteristic
51C2×((C3⋊D3)⋊C3)⋊C2—24no—
52C2×((C3⋊D3)⋊C3)⋊C2—44no—
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×#1084—21no—
55C4⋊#272—21no—
56C4⋊#272C2312yes—
57#6229—21no—
58#6229C2312yescomplemented
59C2×(C3⋊Dic3)⋊C3—21no—
60C2×(C3⋊Dic3)⋊C3D411yescharacteristic
61C4⋊((C3⋊D3)⋊C3)—21no—
62C4⋊((C3⋊D3)⋊C3)C2312yes—
63C4×(C3⋊D3)⋊C3D412yes—
64C4×(C3⋊D3)⋊C3C2312yes—
65#634⋊C3—21no—
66#634⋊C3C2312yescomplemented
67#6200—24no—
68((C3⋊D3)⋊C3)⋊C4—24no—
69#6196—24no—
70C8×H(3,𝔽3)C2312yescomplemented
71(C3⋊(C3⋊C8))⋊C3C2312yescomplemented
72C2×D3×SD16—91no—
73C2×D3×SD16—91no—
74C22×#204—91no—
75C22×D32—61no—
76C22×D32—61no—
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C2×C4×C3⋊D3C2×D311yescharacteristic
79C2×#634C2×D311yescharacteristic, complemented
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—61no—
84C2×C3×C4×D3—31no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—61no—
89C2×#627—61no—
90C2×#627—61no—
91C2×#626—61no—
92C2×#625—61no—
93C2×#624—61no—
94C2×D3×Dic3—61no—
95#3802—34no—
96#3802—62no—
97D3×D12—34no—
98D3×D12—62no—
99#3797—34no—
100#3797—62no—
101D3×Dic6—34no—
102D3×Dic6—62no—
103C2×C32×C8—21no—
104C2×C32×C8C2×D311yescharacteristic, complemented
105C2×C32×C8C2×D311yescharacteristic, complemented
106C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
107#3744C2×D314yescomplemented
108C8×C3⋊D3C2×D314yescomplemented
109C2×C3×C3⋊C8—31no—
110C3×#204—34no—
111C3×#204—34no—
112C3×#204—34no—
113C3×#204—64no—
114C3×D3×C8—34no—
115#3717—34no—
116#3716—34no—
117#3715—34no—
118#3715—34no—
119C22×H(3,𝔽3)C2×D411yescharacteristic
120C2×#272—22no—
121C2×#272—42no—
122C2×(C3⋊D3)⋊C3—22no—
123C2×(C3⋊D3)⋊C3—42no—
124C2×(C3⋊D3)⋊C3SD1614yescomplemented
125C2×(C3⋊D3)⋊C3C2×D412yes—
126((C3⋊D3)⋊C3)⋊C2—48no—
127C4×H(3,𝔽3)C2×D411yescharacteristic
128C4×H(3,𝔽3)C2411yescharacteristic
129#1084—22no—
130(C3⋊Dic3)⋊C3—22no—
131(C3⋊Dic3)⋊C3C2×D412yes—
132C2×D3×Q8—91no—
133C2×D3×Q8—91no—
134C2×D3×D4—91no—
135C2×D3×D4—91no—
136C22×D12—91no—
137C22×Dic6—91no—
138C2×C3×SD16—91no—
139C2×C3×SD16—91no—
140C22×C3×C8—91no—
141C2×#215—91no—
142C2×#215—91no—
143C2×#214—91no—
144C2×#214—91no—
145D3×SD16—98no—
146D3×SD16—98no—
147C2×#204—31no—
148C2×#204—31no—
149C2×#204—31no—
150C2×#204—61no—
151C2×#204—91no—
152C2×#204—98no—
153C2×#204—91no—
154C2×#204—91no—
155C2×D3×C8—31no—
156C2×D3×C8—91no—
157C22×C3⋊D3—21no—
158C22×C3⋊D3D1211yescharacteristic
159C22×C3×D3—31no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—121no—
164C2×D32—64no—
165C2×D32—64no—
166C2×D32—124no—
167C2×D32—124no—
168C2×C32×C4—21no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3—21no—
172C2×C3⋊Dic3D1211yescharacteristic
173C4⋊(C3⋊D3)—21no—
174C4⋊(C3⋊D3)C22×D312yes—
175C4×C3⋊D3D1212yes—
176C4×C3⋊D3C22×D312yes—
177#634—21no—
178#634C22×D312yescomplemented
179C2×C3×Dic3—31no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—121no—
184C3×D12—32no—
185C3×D12—32no—
186C3×D12—32no—
187C3×D12—62no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—61no—
191C3×D12—121no—
192C3×C4×D3—32no—
193C3×C4×D3—32no—
194C3×Dic6—32no—
195C3×Dic6—32no—
196C3×Dic6—32no—
197C3×Dic6—62no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—61no—
201C3×Dic6—121no—
202#627—64no—
203#627—64no—
204#626—64no—
205#625—64no—
206#624—64no—
207D3×Dic3—64no—
208C32×C8—22no—
209C32×C8C22×D312yescomplemented
210C32×C8C22×D312yescomplemented
211C3⋊(C3⋊C8)C22×D312yescomplemented
212C3×C3⋊C8—32no—
213C22×SD16—271noSylow
214C2×H(3,𝔽3)C2×SD1612yescomplemented
215C2×H(3,𝔽3)C22×D411yescharacteristic
216#272—42no—
217(C3⋊D3)⋊C3—42no—
218(C3⋊D3)⋊C3C2×SD1614yescomplemented
219C23×D3—181no—
220C23×D3—181no—
221C23×D3—181no—
222C2×C3×Q8—91no—
223C2×C3×Q8—91no—
224C2×C3×D4—91no—
225C2×C3×D4—91no—
226C22×C3×C4—91no—
227C2×#67—181no—
228C2×#67—181no—
229C22×Dic3—181no—
230D3×Q8—94no—
231D3×Q8—94no—
232D3×Q8—182no—
233D3×Q8—182no—
234D3×D4—94no—
235D3×D4—94no—
236D3×D4—182no—
237D3×D4—182no—
238C2×D12—31no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—61no—
242C2×D12—91no—
243C2×D12—94no—
244C2×D12—182no—
245C2×D12—181no—
246C2×C4×D3—31no—
247C2×C4×D3—91no—
248C2×C4×D3—181no—
249C2×C4×D3—181no—
250C2×Dic6—31no—
251C2×Dic6—31no—
252C2×Dic6—31no—
253C2×Dic6—61no—
254C2×Dic6—91no—
255C2×Dic6—94no—
256C2×Dic6—182no—
257C2×Dic6—181no—
258C2×Dic6—181no—
259C2×Dic6—181no—
260C3×SD16—94no—
261C3×SD16—94no—
262C2×C3×C8—31no—
263C2×C3×C8—31no—
264C2×C3×C8—61no—
265C2×C3×C8—94no—
266C2×C3×C8—91no—
267C2×C3×C8D3211yescharacteristic
268#215—94no—
269#215—94no—
270#214—94no—
271#214—94no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—91no—
274#204—34no—
275#204—34no—
276#204—34no—
277#204—64no—
278#204—94no—
279#204—94no—
280#204—94no—
281D3×C8—34no—
282D3×C8—94no—
283C22×C32—21no—
284C22×C32C2×D1211yescharacteristic
285C22×C32D3×D411yescharacteristic
286C2×C3⋊D3—22no—
287C2×C3⋊D3—42no—
288C2×C3⋊D3#20414yescomplemented
289C2×C3⋊D3C2×D1212yes—
290C2×C3×D3—34no—
291C2×C3×D3—32no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C2×C3×D3—62no—
295C2×C3×D3—122no—
296C2×C3×D3—122no—
297C2×C3×D3—122no—
298C2×C3×D3—122no—
299C2×C3×D3—242no—
300D32—128no—
301D32—128no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4C2×D1211yescharacteristic
305C32×C4D3×D411yescharacteristic
306C32×C4C23×D311yescharacteristic
307C32×C4C23×D311yescharacteristic
308C3⋊Dic3—22no—
309C3⋊Dic3C2×D1212yes—
310C3×Dic3—32no—
311C3×Dic3—62no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—122no—
315C22×Q8—271no—
316C22×D4—271no—
317C2×SD16—91no—
318C2×SD16—91no—
319C2×SD16—278no—
320C2×SD16—271no—
321C2×SD16—271no—
322C22×C8—91no—
323H(3,𝔽3)C22×SD1611yescharacteristic, complemented, Sylow
324C23×C3—91no—
325C23×C3—181no—
326C23×C3—181no—
327C22×D3—31no—
328C22×D3—61no—
329C22×D3—61no—
330C22×D3—61no—
331C22×D3—91no—
332C22×D3—121no—
333C22×D3—181no—
334C22×D3—184no—
335C22×D3—184no—
336C22×D3—184no—
337C22×D3—364no—
338C22×D3—364no—
339C22×D3—364no—
340C3×Q8—92no—
341C3×Q8—92no—
342C3×Q8—181no—
343C3×Q8—181no—
344C3×D4—92no—
345C3×D4—92no—
346C3×D4—181no—
347C3×D4—181no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—61no—
351C2×C3×C4—92no—
352C2×C3×C4—92no—
353C2×C3×C4—91no—
354C2×C3×C4—181no—
355C2×C3×C4—181no—
356C2×C3×C4C2×D3211yescharacteristic
357#67—184no—
358#67—184no—
359C2×Dic3—31no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—91no—
364C2×Dic3—121no—
365C2×Dic3—181no—
366C2×Dic3—184no—
367D12—32no—
368D12—32no—
369D12—32no—
370D12—62no—
371D12—61no—
372D12—61no—
373D12—61no—
374D12—92no—
375D12—121no—
376D12—181no—
377D12—184no—
378C4×D3—32no—
379C4×D3—32no—
380C4×D3—92no—
381C4×D3—92no—
382C4×D3—184no—
383C4×D3—184no—
384Dic6—32no—
385Dic6—32no—
386Dic6—32no—
387Dic6—62no—
388Dic6—61no—
389Dic6—61no—
390Dic6—61no—
391Dic6—92no—
392Dic6—121no—
393Dic6—181no—
394Dic6—184no—
395Dic6—184no—
396Dic6—184no—
397C3×C8—32no—
398C3×C8—32no—
399C3×C8—62no—
400C3×C8—92no—
401C3×C8C2×D3212yes—
402C3⋊C8—32no—
403C3⋊C8—92no—
404C2×C32—21no—
405C2×C32—22no—
406C2×C32C2×#20412yescomplemented
407C2×C32D3×SD1612yescomplemented
408C2×C32C22×D1211yescharacteristic
409C2×C32C2×D3×D411yescharacteristic
410C3⋊D3—42no—
411C3⋊D3C2×#20414yescomplemented
412C3×D3—34no—
413C3×D3—122no—
414C3×D3—122no—
415C3×D3—122no—
416C3×D3—242no—
417C24—541no—
418C2×Q8—91no—
419C2×Q8—91no—
420C2×Q8—274no—
421C2×Q8—541no—
422C2×Q8—542no—
423C2×D4—91no—
424C2×D4—91no—
425C2×D4—274no—
426C2×D4—542no—
427C2×D4—541no—
428C22×C4—91no—
429C22×C4—541no—
430SD16—94no—
431SD16—94no—
432SD16—274no—
433SD16—274no—
434C2×C8—91no—
435C2×C8—94no—
436C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
437C22×C3—31no—
438C22×C3—31no—
439C22×C3—61no—
440C22×C3—92no—
441C22×C3—94no—
442C22×C3—182no—
443C22×C3—182no—
444C22×C3—362no—
445C22×C3—362no—
446C22×C3D3×D1211yescharacteristic
447C2×D3—32no—
448C2×D3—34no—
449C2×D3—62no—
450C2×D3—62no—
451C2×D3—62no—
452C2×D3—92no—
453C2×D3—94no—
454C2×D3—122no—
455C2×D3—122no—
456C2×D3—122no—
457C2×D3—122no—
458C2×D3—182no—
459C2×D3—242no—
460C2×D3—362no—
461C2×D3—368no—
462C2×D3—368no—
463C2×D3—368no—
464C3×C4—31no—
465C3×C4—31no—
466C3×C4—31no—
467C3×C4—31no—
468C3×C4—61no—
469C3×C4—61no—
470C3×C4—92no—
471C3×C4—182no—
472C3×C4—182no—
473C3×C4D3×D1211yescharacteristic
474C3×C4C22×D3211yescharacteristic
475Dic3—32no—
476Dic3—62no—
477Dic3—62no—
478Dic3—62no—
479Dic3—92no—
480Dic3—122no—
481Dic3—182no—
482C32—21no—
483C32C22×#20411yescharacteristic, complemented
484C32C2×D3×SD1611yescharacteristic, complemented
485C23—91no—
486C23—181no—
487C23—181no—
488C23—544no—
489C23—1084no—
490Q8—92no—
491Q8—92no—
492Q8—181no—
493Q8—181no—
494Q8—544no—
495D4—92no—
496D4—92no—
497D4—181no—
498D4—181no—
499D4—544no—
500C2×C4—91no—
501C2×C4—92no—
502C2×C4—92no—
503C2×C4—181no—
504C2×C4—181no—
505C2×C4—544no—
506C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
507C8—92no—
508C8C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
509C2×C3—32no—
510C2×C3—32no—
511C2×C3—31no—
512C2×C3—31no—
513C2×C3—62no—
514C2×C3—61no—
515C2×C3—94no—
516C2×C3—362no—
517C2×C3—362no—
518C2×C3D3×#20412yes—
519C2×C3C2×D3×D1211yescharacteristic
520D3—34no—
521D3—94no—
522D3—122no—
523D3—122no—
524D3—122no—
525D3—242no—
526D3—362no—
527C22—94no—
528C22—92no—
529C22—182no—
530C22—182no—
531C22—362no—
532C22—362no—
533C22—1088no—
534C22#8865411yescharacteristic
535C4—92no—
536C4—182no—
537C4—182no—
538C4#8865411yescharacteristic
539C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
540C3—31no—
541C3—31no—
542C3—61no—
543C3C2×D3×#20411yescharacteristic
544C2—94no—
545C2—362no—
546C2—362no—
547C2#1172084612yesdirect product factor
548C2C2×#8865411yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×#88654C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×SD16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×((C3⋊D3)⋊C3)⋊Q8C2semidirect
3. C2×#88528C2semidirect
4. C2×#3744⋊C3C2semidirect
5. C2×C8×(C3⋊D3)⋊C3C2semidirect
6. C2×#88438C2semidirect
7. C2×#88437C2semidirect
8. #11720846C2direct (N × H)
13. C2×#6229C22semidirect
16. C2×#634⋊C3C22semidirect
23. ((C3⋊D3)⋊C3)⋊Q8C22semidirect
24. C2×C8×H(3,𝔽3)C22semidirect
25. #88528C22semidirect
26. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
27. #3744⋊C3C22semidirect
28. C8×(C3⋊D3)⋊C3C22semidirect
29. #88438C22semidirect
30. #88437C22semidirect
35. C2×#3744D3semidirect
36. C2×C8×C3⋊D3D3semidirect
58. #6229C23semidirect
66. #634⋊C3C23semidirect
70. C8×H(3,𝔽3)C23semidirect
71. (C3⋊(C3⋊C8))⋊C3C23semidirect
79. C2×#634C2×D3semidirect
104. C2×C32×C8C2×D3semidirect
105. C2×C32×C8C2×D3semidirect
106. C2×C3⋊(C3⋊C8)C2×D3semidirect
107. #3744C2×D3semidirect
108. C8×C3⋊D3C2×D3semidirect
124. C2×(C3⋊D3)⋊C3SD16semidirect
178. #634C22×D3semidirect
209. C32×C8C22×D3semidirect
210. C32×C8C22×D3semidirect
211. C3⋊(C3⋊C8)C22×D3semidirect
214. C2×H(3,𝔽3)C2×SD16semidirect
218. (C3⋊D3)⋊C3C2×SD16semidirect
288. C2×C3⋊D3#204semidirect
323. H(3,𝔽3)C22×SD16semidirect
406. C2×C32C2×#204semidirect
407. C2×C32D3×SD16semidirect
411. C3⋊D3C2×#204semidirect
436. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
483. C32C22×#204semidirect
484. C32C2×D3×SD16semidirect
508. C8C2×((C3⋊D3)⋊C3)⋊C2semidirect
547. C2#11720846direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886542
C2×((C3⋊D3)⋊C3)⋊Q82
C2×#885282
C2×#3744⋊C32
C2×C8×(C3⋊D3)⋊C32
C2×#884382
C2×#884372
#117208462
C2×#637963
C2×D3×#2043