Finite Group Databasebeta

← #13139048

Subgroups — #13139048

Order 1728.

743 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 743 conjugacy classes of subgroups — too many to draw. Only its 79 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, complemented
2C2×#3745⋊C3C211yesmaximal, characteristic, complemented
3#11721141C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#11720554C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590947—31nomaximal
9#10590945—31nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2—21no—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#6200—21no—
16C2×#6198—21no—
17#88654—22no—
18#88654—22no—
19C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
20C4⋊#1084C2211yescharacteristic, complemented
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
22#3745⋊C3—22no—
23#88450—21no—
24#88436—22no—
25C2×#3802—31no—
26C2×D3×D12—31no—
27C2×#3745D311yescharacteristic, complemented
28C2×C3×D24—31no—
29#63916—31no—
30#63909—31no—
31C2×#3714—31no—
32C2×#3713—31no—
33#63647D311yescharacteristic, complemented
34C3×#821—31no—
35C3×#821—31no—
36C3×#821—61no—
37C3×(C2×D3)⋊C8—31no—
38#63564—31no—
39#63562—31no—
40C22×#272—21no—
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3—21no—
43C22×(C3⋊D3)⋊C3D411yescharacteristic
44C2×((C3⋊D3)⋊C3)⋊C2—24no—
45C2×((C3⋊D3)⋊C3)⋊C2—42no—
46C2×((C3⋊D3)⋊C3)⋊C2—42no—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×#1084—21no—
49C4⋊#272—21no—
50C4⋊#272D412yescomplemented
51C2×(C3⋊Dic3)⋊C3D411yescharacteristic
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)—21no—
54C4⋊((C3⋊D3)⋊C3)—21no—
55C4⋊((C3⋊D3)⋊C3)D412yescomplemented
56C4×(C3⋊D3)⋊C3—21no—
57C4×(C3⋊D3)⋊C3—21no—
58#6200—24no—
59#6198—42no—
60C8×H(3,𝔽3)—21no—
61(C3⋊(C3⋊C8))⋊C3—21no—
62#5156—91no—
63#4773—91no—
64#4732—91no—
65C22×D32—61no—
66C22×D32—61no—
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×C3×D12—31no—
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×C4×D3—31no—
76C2×#626—61no—
77C2×#626—61no—
78C2×#626—61no—
79C2×#625—61no—
80#3802—62no—
81#3802—62no—
82D3×D12—62no—
83D3×D12—62no—
84C2×C32×C8—21no—
85C2×C32×C8C2×D311yescharacteristic, complemented
86C2×C32×C8C2×D311yescharacteristic, complemented
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
88#3745—22no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C2×C3×C3⋊C8—31no—
93C3×D24—62no—
94#3722—61no—
95(C2×D3)⋊Dic3—61no—
96#3714—62no—
97#3713—62no—
98C22×H(3,𝔽3)C2×D411yescharacteristic
99C2×#272—22no—
100C2×#272—41no—
101C2×#272—41no—
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3—21no—
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3—41no—
106C2×(C3⋊D3)⋊C3—41no—
107C2×(C3⋊D3)⋊C3—41no—
108C2×(C3⋊D3)⋊C3—41no—
109C2×(C3⋊D3)⋊C3—41no—
110C2×(C3⋊D3)⋊C3D812yescomplemented
111((C3⋊D3)⋊C3)⋊C2—48no—
112C4×H(3,𝔽3)C2×D411yescharacteristic
113C4×H(3,𝔽3)C2×D411yescharacteristic
114#1084—41no—
115(C3⋊Dic3)⋊C3—21no—
116C2×D3×D4—91no—
117C2×D3×D4—91no—
118C22×D12—91no—
119C2×C3×D8—91no—
120#938—91no—
121C2×#213—91no—
122C2×#213—91no—
123#930—91no—
124C2×D24—31no—
125C2×D24—91no—
126C2×D24—91no—
127#895—91no—
128C3×#102—91no—
129C3×C22⋊C8—91no—
130#832—91no—
131#821—31no—
132#821—31no—
133#821—61no—
134#821—91no—
135(C2×D3)⋊C8—31no—
136(C2×D3)⋊C8—91no—
137#809—91no—
138C22×C3⋊D3—21no—
139C22×C3⋊D3—21no—
140C22×C3⋊D3D1211yescharacteristic
141C22×C3×D3—31no—
142C22×C3×D3—61no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C22×C3×D3—61no—
146C22×C3×D3—121no—
147C2×D32—64no—
148C2×D32—64no—
149C2×D32—122no—
150C2×D32—122no—
151C2×D32—122no—
152C2×D32—122no—
153C2×C32×C4—21no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3D1211yescharacteristic
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)—21no—
159C4⋊(C3⋊D3)—21no—
160C4⋊(C3⋊D3)#6712yescomplemented
161C4×C3⋊D3—21no—
162C4×C3⋊D3—21no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—121no—
167C3×D12—32no—
168C3×D12—32no—
169C3×D12—32no—
170C3×D12—62no—
171C3×D12—61no—
172C3×D12—61no—
173C3×D12—61no—
174C3×D12—61no—
175C3×D12—61no—
176C3×D12—121no—
177C3×C4×D3—61no—
178C3×C4×D3—61no—
179#626—64no—
180#626—122no—
181#626—122no—
182#625—64no—
183C32×C8—21no—
184C32×C8—21no—
185C32×C8—41no—
186C3⋊(C3⋊C8)—21no—
187C3×C3⋊C8—61no—
188#447—271noSylow
189C2×H(3,𝔽3)#12011yescharacteristic
190C2×H(3,𝔽3)C2×D811yescharacteristic
191C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
192#272—42no—
193(C3⋊D3)⋊C3—22no—
194(C3⋊D3)⋊C3—42no—
195(C3⋊D3)⋊C3—81no—
196C23×D3—181no—
197C23×D3—181no—
198C23×D3—181no—
199C2×C3×D4—91no—
200C2×C3×D4—91no—
201C2×C3×D4—91no—
202C22×C3×C4—91no—
203C2×#67—181no—
204C2×#67—181no—
205C2×#67—181no—
206C2×#67—181no—
207D3×D4—182no—
208D3×D4—182no—
209D3×D4—182no—
210D3×D4—182no—
211C2×D12—31no—
212C2×D12—31no—
213C2×D12—31no—
214C2×D12—31no—
215C2×D12—61no—
216C2×D12—91no—
217C2×D12—91no—
218C2×D12—182no—
219C2×D12—182no—
220C2×D12—181no—
221C2×D12—181no—
222C2×C4×D3—31no—
223C2×C4×D3—91no—
224C3×D8—182no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—61no—
228C2×C3×C8—91no—
229C2×C3×C8D3211yescharacteristic
230C3×C4⋊C4—91no—
231#217—181no—
232#213—182no—
233#213—182no—
234(C2×D3)⋊C4—181no—
235(C2×D3)⋊C4—181no—
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238C4⋊Dic3—61no—
239C2×C3⋊C8—31no—
240C2×C3⋊C8—91no—
241D24—62no—
242D24—182no—
243D24—182no—
244C22×C32—21no—
245C22×C32C2×D1211yescharacteristic
246C22×C32D3×D411yescharacteristic
247C2×C3⋊D3—21no—
248C2×C3⋊D3—21no—
249C2×C3⋊D3—22no—
250C2×C3⋊D3—41no—
251C2×C3⋊D3—41no—
252C2×C3⋊D3—41no—
253C2×C3⋊D3—41no—
254C2×C3⋊D3—41no—
255C2×C3⋊D3D2412yescomplemented
256C2×C3×D3—32no—
257C2×C3×D3—61no—
258C2×C3×D3—61no—
259C2×C3×D3—62no—
260C2×C3×D3—62no—
261C2×C3×D3—62no—
262C2×C3×D3—122no—
263C2×C3×D3—121no—
264C2×C3×D3—121no—
265C2×C3×D3—121no—
266C2×C3×D3—121no—
267C2×C3×D3—121no—
268C2×C3×D3—121no—
269C2×C3×D3—121no—
270C2×C3×D3—121no—
271C2×C3×D3—121no—
272C2×C3×D3—241no—
273C2×C3×D3—241no—
274D32—128no—
275D32—128no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4D3×D411yescharacteristic
279C32×C4D3×D411yescharacteristic
280C32×C4D3×D411yescharacteristic
281C32×C4C2×#6711yescharacteristic
282C3⋊Dic3—21no—
283C3×Dic3—61no—
284C3×Dic3—121no—
285C3×Dic3—121no—
286C3×Dic3—241no—
287C22×D4—271no—
288C2×D8—91no—
289C2×D8—271no—
290#121—271no—
291#102—91no—
292#102—271no—
293C22⋊C8—91no—
294H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
295C23×C3—91no—
296C23×C3—181no—
297C23×C3—181no—
298C23×C3—181no—
299C22×D3—31no—
300C22×D3—61no—
301C22×D3—61no—
302C22×D3—61no—
303C22×D3—61no—
304C22×D3—91no—
305C22×D3—121no—
306C22×D3—181no—
307C22×D3—181no—
308C22×D3—184no—
309C22×D3—184no—
310C22×D3—184no—
311C22×D3—362no—
312C22×D3—362no—
313C22×D3—362no—
314C22×D3—362no—
315C22×D3—362no—
316C22×D3—362no—
317C3×D4—92no—
318C3×D4—92no—
319C3×D4—181no—
320C3×D4—181no—
321C3×D4—181no—
322C3×D4—181no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—61no—
326C2×C3×C4—91no—
327C2×C3×C4—181no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4C2×D3211yescharacteristic
331#67—184no—
332#67—184no—
333#67—362no—
334#67—362no—
335C2×Dic3—31no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—91no—
339C2×Dic3—121no—
340D12—32no—
341D12—32no—
342D12—32no—
343D12—62no—
344D12—61no—
345D12—61no—
346D12—61no—
347D12—61no—
348D12—61no—
349D12—92no—
350D12—121no—
351D12—181no—
352D12—181no—
353D12—181no—
354D12—184no—
355D12—362no—
356C4×D3—61no—
357C4×D3—61no—
358C4×D3—181no—
359C4×D3—181no—
360C3×C8—21no—
361C3×C8—61no—
362C3×C8—61no—
363C3×C8—121no—
364C3×C8—181no—
365C3⋊C8—61no—
366C3⋊C8—181no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32#88211yescharacteristic
371C2×C32C2×D2411yescharacteristic
372C2×C32#90811yescharacteristic
373C2×C32D3×D811yescharacteristic
374C2×C32#91111yescharacteristic
375C2×C32#93711yescharacteristic
376C3⋊D3—22no—
377C3⋊D3—42no—
378C3⋊D3—81no—
379C3×D3—62no—
380C3×D3—122no—
381C3×D3—122no—
382C3×D3—122no—
383C3×D3—242no—
384C3×D3—241no—
385C24—541no—
386C2×D4—91no—
387C2×D4—91no—
388C2×D4—91no—
389C2×D4—541no—
390C2×D4—542no—
391C2×D4—542no—
392C2×D4—541no—
393C22×C4—91no—
394D8—182no—
395D8—542no—
396C2×C8—91no—
397C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
398C4⋊C4—91no—
399C22⋊C4—541no—
400C22×C3—31no—
401C22×C3—31no—
402C22×C3—61no—
403C22×C3—92no—
404C22×C3—181no—
405C22×C3—181no—
406C22×C3—182no—
407C22×C3—182no—
408C22×C3—361no—
409C22×C3—361no—
410C22×C3—361no—
411C22×C3—361no—
412C22×C3—361no—
413C22×C3—361no—
414C22×C3—361no—
415C22×C3D3×D1211yescharacteristic
416C2×D3—32no—
417C2×D3—61no—
418C2×D3—61no—
419C2×D3—62no—
420C2×D3—62no—
421C2×D3—62no—
422C2×D3—92no—
423C2×D3—122no—
424C2×D3—121no—
425C2×D3—121no—
426C2×D3—121no—
427C2×D3—121no—
428C2×D3—121no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3—181no—
434C2×D3—181no—
435C2×D3—182no—
436C2×D3—241no—
437C2×D3—241no—
438C2×D3—361no—
439C2×D3—361no—
440C2×D3—361no—
441C2×D3—361no—
442C2×D3—361no—
443C2×D3—368no—
444C2×D3—368no—
445C2×D3—368no—
446C3×C4—31no—
447C3×C4—31no—
448C3×C4—31no—
449C3×C4—31no—
450C3×C4—61no—
451C3×C4—61no—
452C3×C4—181no—
453C3×C4—361no—
454C3×C4#380211yescharacteristic
455C3×C4D3×#6711yescharacteristic
456Dic3—61no—
457Dic3—121no—
458Dic3—121no—
459Dic3—181no—
460Dic3—241no—
461C32—21no—
462C32#473211yescharacteristic, complemented
463C32#515611yescharacteristic, complemented
464C23—91no—
465C23—181no—
466C23—181no—
467C23—181no—
468C23—544no—
469C23—1082no—
470C23—1082no—
471D4—92no—
472D4—92no—
473D4—181no—
474D4—181no—
475D4—181no—
476D4—181no—
477D4—544no—
478D4—1082no—
479C2×C4—91no—
480C2×C4—181no—
481C2×C4—181no—
482C2×C4—181no—
483C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
484C8—21no—
485C8—181no—
486C2×C3—31no—
487C2×C3—31no—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—61no—
493C2×C3—61no—
494C2×C3—61no—
495C2×C3—182no—
496C2×C3—362no—
497C2×C3—362no—
498C2×C3—721no—
499C2×C3D3×D2411yescharacteristic
500C2×C3#6379511yescharacteristic
501C2×C3#6392211yescharacteristic
502D3—62no—
503D3—122no—
504D3—122no—
505D3—122no—
506D3—182no—
507D3—242no—
508D3—241no—
509D3—362no—
510D3—721no—
511C22—92no—
512C22—181no—
513C22—181no—
514C22—182no—
515C22—182no—
516C22—361no—
517C22—361no—
518C22—361no—
519C22—361no—
520C22—361no—
521C22—361no—
522C22—361no—
523C22—1088no—
524C22#8865411yescharacteristic
525C4—181no—
526C4—361no—
527C4#8865511yescharacteristic
528C4#8867611yescharacteristic
529C3—31no—
530C3—31no—
531C3—61no—
532C3#1059094411yescharacteristic
533C2—182no—
534C2—362no—
535C2—362no—
536C2—721no—
537C2#1172084711yescharacteristic
538C2#1172084911yescharacteristic
539C2#1172116211yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#447
LCSG ▷ C2×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×#88654C2semidirect
2. C2×#3745⋊C3C2semidirect
3. #11721141C2semidirect
4. C2×#88436C2semidirect
5. #11720554C2semidirect
6. #63647⋊C3C2semidirect
7. #11720308C2semidirect
13. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
19. C2×C8×H(3,𝔽3)C22semidirect
20. C4⋊#1084C22semidirect
21. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
27. C2×#3745D3semidirect
33. #63647D3semidirect
50. C4⋊#272D4semidirect
55. C4⋊((C3⋊D3)⋊C3)D4semidirect
68. C2×C4⋊(C3⋊D3)C2×D3semidirect
85. C2×C32×C8C2×D3semidirect
86. C2×C32×C8C2×D3semidirect
87. C2×C3⋊(C3⋊C8)C2×D3semidirect
110. C2×(C3⋊D3)⋊C3D8semidirect
160. C4⋊(C3⋊D3)#67semidirect
255. C2×C3⋊D3D24semidirect
294. H(3,𝔽3)#447semidirect
397. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
462. C32#4732semidirect
463. C32#5156semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
C2×#3745⋊C32
#117211412
C2×#884362
#117205542
#63647⋊C32
#117203082
#105909473
#105909453