Finite Group Databasebeta

← C2×#11720839 #13149811

Subgroups — C2×#11720839 #13149811

Order 1728.

1111 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 147 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×#88655C211yesmaximal, characteristic, complemented
2C2×#88649C211yesmaximal, characteristic, complemented
3C2×#88527C211yesmaximal, characteristic, complemented
4C2×#3745⋊C3C211yesmaximal, characteristic, complemented
5C2×#3744⋊C3C211yesmaximal, characteristic, complemented
6C2×#88432C211yesmaximal, characteristic, complemented
7C2×#88431C211yesmaximal, characteristic, complemented
8#11720839C218yesmaximal, direct product factor
9C2×#63795—31nomaximal
10C2×#63794—31nomaximal
11C22×((C3⋊D3)⋊C3)⋊C2—21no—
12C2×C4×#272C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
16C2×#634⋊C3C2211yescharacteristic, complemented
17C2×#6200—21no—
18C2×#6198—21no—
19C2×((C3⋊D3)⋊C3)⋊C4—21no—
20#88655—22no—
21#88655C2214yescomplemented
22#88649—22no—
23#88649C2214yescomplemented
24C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
25C2×#6180C2211yescharacteristic, complemented
26#88527C2214yescomplemented
27#3745⋊C3C2214yescomplemented
28#3744⋊C3C2214yescomplemented
29#88432C2214yescomplemented
30#88431C2214yescomplemented
31C2×D3×D12—31no—
32C2×D3×D12—31no—
33C2×#3799—31no—
34C2×#3795—31no—
35C2×#3745D311yescharacteristic, complemented
36C2×#3744D311yescharacteristic, complemented
37C2×C3×D24—31no—
38C2×C3×#204—31no—
39C2×C3×ΓL(1,𝔽25)—31no—
40C2×C3×ΓL(1,𝔽25)—31no—
41C2×C3×ΓL(1,𝔽25)—61no—
42C2×#3717—31no—
43C2×#3716—31no—
44C2×#3714—31no—
45C2×#3714—31no—
46#63795—38no—
47#63794—38no—
48C22×#272D411yescharacteristic
49C22×(C3⋊D3)⋊C3—21no—
50C22×(C3⋊D3)⋊C3—21no—
51C22×(C3⋊D3)⋊C3—21no—
52C2×((C3⋊D3)⋊C3)⋊C2—24no—
53C2×((C3⋊D3)⋊C3)⋊C2—44no—
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×#1084D411yescharacteristic
56C4×#272D412yes—
57C4×#272C2312yes—
58C2×(C3⋊Dic3)⋊C3—21no—
59C4⋊((C3⋊D3)⋊C3)—21no—
60C4⋊((C3⋊D3)⋊C3)—21no—
61C4⋊((C3⋊D3)⋊C3)—21no—
62C4⋊((C3⋊D3)⋊C3)C2312yes—
63C4⋊((C3⋊D3)⋊C3)C2312yes—
64C4⋊((C3⋊D3)⋊C3)C2312yescomplemented
65#634⋊C3—21no—
66#634⋊C3C2312yescomplemented
67#6200—24no—
68#6198—24no—
69((C3⋊D3)⋊C3)⋊C4—24no—
70C8×H(3,𝔽3)C2312yescomplemented
71#6180C2312yescomplemented
72C2×#914—91no—
73C2×#911—91no—
74C2×#908—91no—
75C22×D32—61no—
76C22×D32—61no—
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
79C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
80C2×#634C2×D311yescharacteristic, complemented
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—61no—
87C2×C3×Dic6—31no—
88C2×#626—61no—
89C2×#626—61no—
90C2×#626—61no—
91C2×#625—61no—
92C2×#624—61no—
93C2×D3×Dic3—61no—
94D3×D12—34no—
95D3×D12—34no—
96D3×D12—62no—
97D3×D12—62no—
98#3799—34no—
99#3799—62no—
100#3795—34no—
101#3795—62no—
102C2×C32×C8—21no—
103C2×C32×C8C2×D311yescharacteristic, complemented
104C2×C32×C8C2×D311yescharacteristic, complemented
105#3745C2×D314yescomplemented
106#3744C2×D314yescomplemented
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—61no—
110C3×D24—34no—
111C3×#204—34no—
112C3×ΓL(1,𝔽25)—34no—
113C3×ΓL(1,𝔽25)—34no—
114C3×ΓL(1,𝔽25)—64no—
115#3717—34no—
116#3716—34no—
117#3714—34no—
118#3714—34no—
119C22×H(3,𝔽3)C2×D411yescharacteristic
120C2×#272—22no—
121C2×#272C2×D412yes—
122C2×(C3⋊D3)⋊C3—22no—
123C2×(C3⋊D3)⋊C3—22no—
124C2×(C3⋊D3)⋊C3—22no—
125C2×(C3⋊D3)⋊C3—42no—
126C2×(C3⋊D3)⋊C3—42no—
127C2×(C3⋊D3)⋊C3—42no—
128((C3⋊D3)⋊C3)⋊C2—48no—
129C4×H(3,𝔽3)C2×D411yescharacteristic
130C4×H(3,𝔽3)C2411yescharacteristic
131#1084C2×D412yes—
132(C3⋊Dic3)⋊C3—22no—
133C2×#239—91no—
134C2×#237—91no—
135C2×D3×D4—91no—
136C2×D3×D4—91no—
137C2×#235—91no—
138C22×D12—91no—
139C2×C3×SD16—91no—
140C2×C3×D8—91no—
141C2×C3×M16—91no—
142C2×#215—91no—
143C2×#214—91no—
144C2×#213—91no—
145C2×#213—91no—
146#914—98no—
147#911—98no—
148#908—98no—
149C2×D24—31no—
150C2×D24—91no—
151C2×D24—91no—
152C2×#204—31no—
153C2×#204—91no—
154C2×#204—91no—
155C2×ΓL(1,𝔽25)—31no—
156C2×ΓL(1,𝔽25)—31no—
157C2×ΓL(1,𝔽25)—61no—
158C22×C3⋊D3—21no—
159C22×C3⋊D3—21no—
160C22×C3⋊D3—21no—
161C22×C3×D3—31no—
162C22×C3×D3—31no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C22×C3×D3—61no—
167C2×D32—64no—
168C2×D32—64no—
169C2×D32—124no—
170C2×D32—124no—
171C2×C32×C4—21no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—21no—
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)C22×D312yes—
179C4⋊(C3⋊D3)C22×D312yes—
180C4⋊(C3⋊D3)C22×D312yescomplemented
181#634—21no—
182#634C22×D312yescomplemented
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C3×D12—32no—
188C3×D12—32no—
189C3×D12—32no—
190C3×D12—61no—
191C3×D12—61no—
192C3×D12—61no—
193C3×C4×D3—32no—
194C3×C4×D3—32no—
195C3×C4×D3—32no—
196C3×C4×D3—32no—
197C3×C4×D3—62no—
198C3×C4×D3—62no—
199C3×Dic6—32no—
200C3×Dic6—61no—
201#626—64no—
202#626—64no—
203#626—64no—
204#625—64no—
205#624—64no—
206D3×Dic3—64no—
207C32×C8—22no—
208C32×C8C22×D312yescomplemented
209C32×C8C22×D312yescomplemented
210C3×C3⋊C8—32no—
211C3×C3⋊C8—32no—
212C3×C3⋊C8—62no—
213C2×AGL(1,ℤ/8)—271noSylow
214C2×H(3,𝔽3)AGL(1,ℤ/8)12yescomplemented
215C2×H(3,𝔽3)C22×D411yescharacteristic
216#272—22no—
217(C3⋊D3)⋊C3—42no—
218(C3⋊D3)⋊C3—42no—
219(C3⋊D3)⋊C3—42no—
220C23×D3—181no—
221C23×D3—181no—
222C23×D3—181no—
223C2×C3×Q8—91no—
224C2×C3×D4—91no—
225C2×C3×D4—91no—
226C2×C3×D4—91no—
227C22×C3×C4—91no—
228C2×#67—181no—
229C2×#67—181no—
230C2×#67—181no—
231C2×#67—181no—
232C22×Dic3—181no—
233#239—94no—
234#239—182no—
235#237—94no—
236#237—182no—
237D3×D4—94no—
238D3×D4—94no—
239D3×D4—182no—
240D3×D4—182no—
241#235—94no—
242#235—182no—
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—31no—
246C2×D12—91no—
247C2×D12—91no—
248C2×D12—91no—
249C2×D12—94no—
250C2×D12—182no—
251C2×D12—181no—
252C2×D12—181no—
253C2×C4×D3—31no—
254C2×C4×D3—31no—
255C2×C4×D3—61no—
256C2×C4×D3—181no—
257C2×C4×D3—181no—
258C2×Dic6—31no—
259C2×Dic6—91no—
260C3×SD16—94no—
261C3×D8—94no—
262C3×M16—94no—
263C2×C3×C8—31no—
264C2×C3×C8—31no—
265C2×C3×C8—61no—
266C2×C3×C8—91no—
267C2×C3×C8D3211yescharacteristic
268#215—94no—
269#214—94no—
270#213—94no—
271#213—94no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—61no—
275D24—34no—
276D24—94no—
277D24—94no—
278#204—34no—
279#204—94no—
280#204—94no—
281ΓL(1,𝔽25)—34no—
282ΓL(1,𝔽25)—34no—
283ΓL(1,𝔽25)—64no—
284C22×C32—21no—
285C22×C32D3×D411yescharacteristic
286C22×C32D3×D411yescharacteristic
287C2×C3⋊D3—22no—
288C2×C3⋊D3—22no—
289C2×C3⋊D3—22no—
290C2×C3⋊D3—42no—
291C2×C3⋊D3—42no—
292C2×C3⋊D3—42no—
293C2×C3×D3—32no—
294C2×C3×D3—32no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—122no—
302C2×C3×D3—122no—
303C2×C3×D3—122no—
304C2×C3×D3—122no—
305D32—128no—
306D32—128no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4D3×D411yescharacteristic
310C32×C4D3×D411yescharacteristic
311C32×C4C23×D311yescharacteristic
312C32×C4C23×D311yescharacteristic
313C3⋊Dic3—22no—
314C3×Dic3—32no—
315C3×Dic3—32no—
316C3×Dic3—62no—
317C3×Dic3—62no—
318C2×#41—271no—
319C22×D4—271no—
320AGL(1,ℤ/8)—278no—
321C2×SD16—91no—
322C2×SD16—271no—
323C2×D8—91no—
324C2×D8—271no—
325C2×M16—91no—
326H(3,𝔽3)C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
327C23×C3—91no—
328C23×C3—181no—
329C23×C3—181no—
330C23×C3—181no—
331C22×D3—31no—
332C22×D3—31no—
333C22×D3—61no—
334C22×D3—61no—
335C22×D3—61no—
336C22×D3—61no—
337C22×D3—181no—
338C22×D3—181no—
339C22×D3—181no—
340C22×D3—184no—
341C22×D3—184no—
342C22×D3—184no—
343C22×D3—364no—
344C22×D3—364no—
345C22×D3—364no—
346C3×Q8—92no—
347C3×Q8—181no—
348C3×D4—92no—
349C3×D4—92no—
350C3×D4—92no—
351C3×D4—181no—
352C3×D4—181no—
353C3×D4—181no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—61no—
357C2×C3×C4—91no—
358C2×C3×C4—92no—
359C2×C3×C4—92no—
360C2×C3×C4—181no—
361C2×C3×C4C2×D3211yescharacteristic
362#67—184no—
363#67—184no—
364#67—184no—
365#67—184no—
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—181no—
371C2×Dic3—184no—
372D12—32no—
373D12—32no—
374D12—32no—
375D12—61no—
376D12—61no—
377D12—61no—
378D12—92no—
379D12—92no—
380D12—92no—
381D12—181no—
382D12—181no—
383D12—181no—
384D12—184no—
385D12—184no—
386C4×D3—32no—
387C4×D3—32no—
388C4×D3—32no—
389C4×D3—32no—
390C4×D3—62no—
391C4×D3—62no—
392C4×D3—184no—
393C4×D3—184no—
394Dic6—32no—
395Dic6—61no—
396Dic6—92no—
397Dic6—181no—
398C3×C8—32no—
399C3×C8—32no—
400C3×C8—62no—
401C3×C8—92no—
402C3×C8C2×D3212yes—
403C3⋊C8—32no—
404C3⋊C8—32no—
405C3⋊C8—62no—
406C2×C32—21no—
407C2×C32—22no—
408C2×C32#91112yescomplemented
409C2×C32#91412yescomplemented
410C2×C32C2×D3×D411yescharacteristic
411C2×C32C2×D3×D411yescharacteristic
412C3⋊D3—42no—
413C3⋊D3—42no—
414C3⋊D3—42no—
415C3×D3—62no—
416C3×D3—62no—
417C3×D3—122no—
418C3×D3—122no—
419C3×D3—122no—
420C3×D3—122no—
421C24—541no—
422#41—274no—
423#41—542no—
424C2×Q8—91no—
425C2×D4—91no—
426C2×D4—91no—
427C2×D4—91no—
428C2×D4—274no—
429C2×D4—542no—
430C2×D4—541no—
431C2×D4—541no—
432C22×C4—91no—
433C22×C4—541no—
434SD16—94no—
435SD16—274no—
436D8—94no—
437D8—274no—
438M16—94no—
439C2×C8—91no—
440C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
441C22×C3—31no—
442C22×C3—31no—
443C22×C3—61no—
444C22×C3—92no—
445C22×C3—182no—
446C22×C3—182no—
447C22×C3—182no—
448C22×C3—182no—
449C22×C3—362no—
450C22×C3—362no—
451C22×C3—362no—
452C22×C3#380211yescharacteristic
453C2×D3—32no—
454C2×D3—32no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—62no—
458C2×D3—62no—
459C2×D3—62no—
460C2×D3—62no—
461C2×D3—122no—
462C2×D3—122no—
463C2×D3—122no—
464C2×D3—122no—
465C2×D3—182no—
466C2×D3—182no—
467C2×D3—182no—
468C2×D3—362no—
469C2×D3—362no—
470C2×D3—362no—
471C2×D3—368no—
472C2×D3—368no—
473C2×D3—368no—
474C3×C4—31no—
475C3×C4—31no—
476C3×C4—31no—
477C3×C4—31no—
478C3×C4—61no—
479C3×C4—61no—
480C3×C4—92no—
481C3×C4—182no—
482C3×C4#380211yescharacteristic
483C3×C4C22×D3211yescharacteristic
484Dic3—32no—
485Dic3—32no—
486Dic3—62no—
487Dic3—62no—
488Dic3—182no—
489C32—21no—
490C32C2×#91111yescharacteristic, complemented
491C32C2×#91411yescharacteristic, complemented
492C23—91no—
493C23—181no—
494C23—181no—
495C23—181no—
496C23—544no—
497C23—1084no—
498Q8—92no—
499Q8—181no—
500D4—92no—
501D4—92no—
502D4—92no—
503D4—181no—
504D4—181no—
505D4—181no—
506D4—544no—
507D4—544no—
508C2×C4—91no—
509C2×C4—92no—
510C2×C4—92no—
511C2×C4—181no—
512C2×C4—544no—
513C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
514C8—92no—
515C8C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
516C2×C3—32no—
517C2×C3—32no—
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—62no—
521C2×C3—61no—
522C2×C3—182no—
523C2×C3—362no—
524C2×C3—362no—
525C2×C3—362no—
526C2×C3#6379812yes—
527C2×C3C2×#380211yescharacteristic
528D3—62no—
529D3—62no—
530D3—122no—
531D3—122no—
532D3—122no—
533D3—122no—
534D3—362no—
535D3—362no—
536D3—362no—
537C22—92no—
538C22—182no—
539C22—182no—
540C22—182no—
541C22—182no—
542C22—362no—
543C22—362no—
544C22—362no—
545C22—1088no—
546C22#8865511yescharacteristic
547C4—92no—
548C4—182no—
549C4#8865511yescharacteristic
550C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
551C3—31no—
552C3—31no—
553C3—61no—
554C3C2×#6379811yescharacteristic
555C2—182no—
556C2—362no—
557C2—362no—
558C2—362no—
559C2#1172083912yesdirect product factor
560C2C2×#8865511yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×AGL(1,ℤ/8)
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
1. C2×#88655C2semidirect
2. C2×#88649C2semidirect
3. C2×#88527C2semidirect
4. C2×#3745⋊C3C2semidirect
5. C2×#3744⋊C3C2semidirect
6. C2×#88432C2semidirect
7. C2×#88431C2semidirect
8. #11720839C2direct (N × H)
15. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
16. C2×#634⋊C3C22semidirect
21. #88655C22semidirect
23. #88649C22semidirect
24. C2×C8×H(3,𝔽3)C22semidirect
25. C2×#6180C22semidirect
26. #88527C22semidirect
27. #3745⋊C3C22semidirect
28. #3744⋊C3C22semidirect
29. #88432C22semidirect
30. #88431C22semidirect
35. C2×#3745D3semidirect
36. C2×#3744D3semidirect
64. C4⋊((C3⋊D3)⋊C3)C23semidirect
66. #634⋊C3C23semidirect
70. C8×H(3,𝔽3)C23semidirect
71. #6180C23semidirect
79. C2×C4⋊(C3⋊D3)C2×D3semidirect
80. C2×#634C2×D3semidirect
103. C2×C32×C8C2×D3semidirect
104. C2×C32×C8C2×D3semidirect
105. #3745C2×D3semidirect
106. #3744C2×D3semidirect
180. C4⋊(C3⋊D3)C22×D3semidirect
182. #634C22×D3semidirect
208. C32×C8C22×D3semidirect
209. C32×C8C22×D3semidirect
214. C2×H(3,𝔽3)AGL(1,ℤ/8)semidirect
326. H(3,𝔽3)C2×AGL(1,ℤ/8)semidirect
408. C2×C32#911semidirect
409. C2×C32#914semidirect
440. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
490. C32C2×#911semidirect
491. C32C2×#914semidirect
515. C8C2×((C3⋊D3)⋊C3)⋊C2semidirect
559. C2#11720839direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886552
C2×#886492
C2×#885272
C2×#3745⋊C32
C2×#3744⋊C32
C2×#884322
C2×#884312
#117208392
C2×#637953
C2×#637943