Finite Group Databasebeta

← #13139034

Subgroups — #13139034

Order 1728.

743 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 743 conjugacy classes of subgroups — too many to draw. Only its 77 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×#3745⋊C3C211yesmaximal, characteristic, complemented
3#11721159C211yesmaximal, characteristic, complemented
4C2×#88432C211yesmaximal, characteristic, complemented
5#11720553C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#11720318C211yesmaximal, characteristic, complemented
8#10590947—31nomaximal
9#10590944—31nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2—21no—
11C2×C4×#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
15C2×#6200—21no—
16C2×#6198—21no—
17#88655—22no—
18#88655—22no—
19C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
20C2×#6180C2211yescharacteristic, complemented
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
22#3745⋊C3—22no—
23#88450—21no—
24#88432—22no—
25C2×D3×D12—31no—
26C2×D3×D12—31no—
27C2×#3745D311yescharacteristic, complemented
28C2×C3×D24—31no—
29#63909—31no—
30#63908—31no—
31C2×#3714—31no—
32C2×#3714—31no—
33#63648D311yescharacteristic, complemented
34C3×#821—31no—
35C3×(C2×D3)⋊C8—31no—
36C3×(C2×D3)⋊C8—31no—
37C3×(C2×D3)⋊C8—61no—
38#63564—31no—
39#63563—31no—
40C22×#272D411yescharacteristic
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3—21no—
43C22×(C3⋊D3)⋊C3—21no—
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×#1084D411yescharacteristic
49C4×#272—21no—
50C4×#272—21no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)—21no—
54C4⋊((C3⋊D3)⋊C3)—21no—
55C4⋊((C3⋊D3)⋊C3)—21no—
56C4⋊((C3⋊D3)⋊C3)D412yescomplemented
57C4⋊((C3⋊D3)⋊C3)D412yescomplemented
58#6200—42no—
59#6198—24no—
60C8×H(3,𝔽3)—21no—
61#6180—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
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
70C2×C3×D12—31no—
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—31no—
75C2×C3×C4×D3—61no—
76C2×#626—61no—
77C2×#626—61no—
78C2×#626—61no—
79C2×#625—61no—
80D3×D12—62no—
81D3×D12—62no—
82D3×D12—62no—
83D3×D12—62no—
84C2×C32×C8—21no—
85C2×C32×C8C2×D311yescharacteristic, complemented
86C2×C32×C8C2×D311yescharacteristic, complemented
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
88#3745—22no—
89C3×C4⋊Dic3—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—61no—
93C3×D24—62no—
94#3722—61no—
95(C2×D3)⋊Dic3—61no—
96#3714—62no—
97#3714—62no—
98C22×H(3,𝔽3)C2×D411yescharacteristic
99C2×#272—21no—
100C2×#272—21no—
101C2×#272D812yescomplemented
102C2×(C3⋊D3)⋊C3—22no—
103C2×(C3⋊D3)⋊C3—22no—
104C2×(C3⋊D3)⋊C3—41no—
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)⋊C3—41no—
111((C3⋊D3)⋊C3)⋊C2—48no—
112C4×H(3,𝔽3)C2×D411yescharacteristic
113C4×H(3,𝔽3)C2×D411yescharacteristic
114#1084—21no—
115(C3⋊Dic3)⋊C3—41no—
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—91no—
133#821—91no—
134(C2×D3)⋊C8—31no—
135(C2×D3)⋊C8—31no—
136(C2×D3)⋊C8—61no—
137#809—91no—
138C22×C3⋊D3—21no—
139C22×C3⋊D3—21no—
140C22×C3⋊D3—21no—
141C22×C3×D3—31no—
142C22×C3×D3—31no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C22×C3×D3—61no—
146C22×C3×D3—61no—
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⋊Dic3—21no—
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)—21no—
159C4⋊(C3⋊D3)—21no—
160C4⋊(C3⋊D3)—21no—
161C4⋊(C3⋊D3)D1212yescomplemented
162C4⋊(C3⋊D3)#6712yescomplemented
163C2×C3×Dic3—31no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C3×D12—32no—
168C3×D12—32no—
169C3×D12—61no—
170C3×D12—61no—
171C3×D12—61no—
172C3×D12—61no—
173C3×C4×D3—61no—
174C3×C4×D3—61no—
175C3×C4×D3—61no—
176C3×C4×D3—61no—
177C3×C4×D3—121no—
178C3×C4×D3—121no—
179#626—64no—
180#626—64no—
181#626—122no—
182#625—122no—
183C32×C8—21no—
184C32×C8—21no—
185C32×C8—41no—
186C3×C3⋊C8—61no—
187C3×C3⋊C8—61no—
188C3×C3⋊C8—121no—
189#447—271noSylow
190C2×H(3,𝔽3)#12011yescharacteristic
191C2×H(3,𝔽3)C2×D811yescharacteristic
192C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
193#272—22no—
194(C3⋊D3)⋊C3—42no—
195(C3⋊D3)⋊C3—42no—
196(C3⋊D3)⋊C3—81no—
197C23×D3—181no—
198C23×D3—181no—
199C23×D3—181no—
200C2×C3×D4—91no—
201C2×C3×D4—91no—
202C2×C3×D4—91no—
203C22×C3×C4—91no—
204C2×#67—181no—
205C2×#67—181no—
206C2×#67—181no—
207C2×#67—181no—
208D3×D4—182no—
209D3×D4—182no—
210D3×D4—182no—
211D3×D4—182no—
212C2×D12—31no—
213C2×D12—31no—
214C2×D12—31no—
215C2×D12—91no—
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—31no—
224C2×C4×D3—61no—
225C3×D8—182no—
226C2×C3×C8—31no—
227C2×C3×C8—31no—
228C2×C3×C8—61no—
229C2×C3×C8—91no—
230C2×C3×C8D3211yescharacteristic
231C3×C4⋊C4—91no—
232#217—181no—
233#213—182no—
234#213—182no—
235(C2×D3)⋊C4—181no—
236(C2×D3)⋊C4—181no—
237C4⋊Dic3—31no—
238C4⋊Dic3—91no—
239C2×C3⋊C8—31no—
240C2×C3⋊C8—31no—
241C2×C3⋊C8—61no—
242D24—62no—
243D24—182no—
244D24—182no—
245C22×C32—21no—
246C22×C32D3×D411yescharacteristic
247C22×C32D3×D411yescharacteristic
248C2×C3⋊D3—22no—
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⋊D3—41no—
256C2×C3⋊D3—41no—
257C2×C3×D3—32no—
258C2×C3×D3—32no—
259C2×C3×D3—62no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C2×C3×D3—61no—
264C2×C3×D3—62no—
265C2×C3×D3—62no—
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—121no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275D32—128no—
276D32—128no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4C2×D1211yescharacteristic
280C32×C4D3×D411yescharacteristic
281C32×C4D3×D411yescharacteristic
282C32×C4C2×#6711yescharacteristic
283C3⋊Dic3—41no—
284C3×Dic3—61no—
285C3×Dic3—61no—
286C3×Dic3—121no—
287C3×Dic3—121no—
288C22×D4—271no—
289C2×D8—91no—
290C2×D8—271no—
291#121—271no—
292#102—91no—
293#102—271no—
294C22⋊C8—91no—
295H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
296C23×C3—91no—
297C23×C3—181no—
298C23×C3—181no—
299C23×C3—181no—
300C22×D3—31no—
301C22×D3—31no—
302C22×D3—61no—
303C22×D3—61no—
304C22×D3—61no—
305C22×D3—61no—
306C22×D3—181no—
307C22×D3—181no—
308C22×D3—181no—
309C22×D3—184no—
310C22×D3—184no—
311C22×D3—184no—
312C22×D3—362no—
313C22×D3—362no—
314C22×D3—362no—
315C22×D3—362no—
316C22×D3—362no—
317C22×D3—362no—
318C3×D4—92no—
319C3×D4—92no—
320C3×D4—181no—
321C3×D4—181no—
322C3×D4—181no—
323C3×D4—181no—
324C2×C3×C4—31no—
325C2×C3×C4—31no—
326C2×C3×C4—61no—
327C2×C3×C4—91no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4C2×D3211yescharacteristic
332#67—184no—
333#67—184no—
334#67—362no—
335#67—362no—
336C2×Dic3—31no—
337C2×Dic3—31no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—181no—
341D12—32no—
342D12—32no—
343D12—61no—
344D12—61no—
345D12—61no—
346D12—61no—
347D12—92no—
348D12—92no—
349D12—181no—
350D12—181no—
351D12—181no—
352D12—181no—
353D12—184no—
354D12—362no—
355C4×D3—61no—
356C4×D3—61no—
357C4×D3—61no—
358C4×D3—61no—
359C4×D3—121no—
360C4×D3—121no—
361C3×C8—21no—
362C3×C8—61no—
363C3×C8—61no—
364C3×C8—121no—
365C3×C8—181no—
366C3⋊C8—61no—
367C3⋊C8—61no—
368C3⋊C8—121no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32#88211yescharacteristic
373C2×C32D3×D811yescharacteristic
374C2×C32D3×D811yescharacteristic
375C2×C32#91111yescharacteristic
376C2×C32#91411yescharacteristic
377C2×C32#93711yescharacteristic
378C3⋊D3—42no—
379C3⋊D3—42no—
380C3⋊D3—81no—
381C3×D3—62no—
382C3×D3—62no—
383C3×D3—122no—
384C3×D3—122no—
385C3×D3—122no—
386C3×D3—241no—
387C24—541no—
388C2×D4—91no—
389C2×D4—91no—
390C2×D4—91no—
391C2×D4—541no—
392C2×D4—542no—
393C2×D4—542no—
394C2×D4—541no—
395C22×C4—91no—
396D8—182no—
397D8—542no—
398C2×C8—91no—
399C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
400C4⋊C4—91no—
401C22⋊C4—541no—
402C22×C3—31no—
403C22×C3—31no—
404C22×C3—61no—
405C22×C3—92no—
406C22×C3—182no—
407C22×C3—182no—
408C22×C3—181no—
409C22×C3—181no—
410C22×C3—361no—
411C22×C3—361no—
412C22×C3—361no—
413C22×C3—361no—
414C22×C3—361no—
415C22×C3—361no—
416C22×C3—361no—
417C22×C3#380211yescharacteristic
418C2×D3—32no—
419C2×D3—32no—
420C2×D3—62no—
421C2×D3—61no—
422C2×D3—61no—
423C2×D3—61no—
424C2×D3—61no—
425C2×D3—62no—
426C2×D3—62no—
427C2×D3—121no—
428C2×D3—121no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3—121no—
434C2×D3—121no—
435C2×D3—121no—
436C2×D3—182no—
437C2×D3—182no—
438C2×D3—361no—
439C2×D3—361no—
440C2×D3—361no—
441C2×D3—361no—
442C2×D3—361no—
443C2×D3—361no—
444C2×D3—361no—
445C2×D3—368no—
446C2×D3—368no—
447C2×D3—368no—
448C3×C4—31no—
449C3×C4—31no—
450C3×C4—31no—
451C3×C4—31no—
452C3×C4—61no—
453C3×C4—61no—
454C3×C4—181no—
455C3×C4—361no—
456C3×C4D3×D1211yescharacteristic
457C3×C4D3×#6711yescharacteristic
458Dic3—61no—
459Dic3—61no—
460Dic3—121no—
461Dic3—121no—
462Dic3—361no—
463C32—21no—
464C32#477311yescharacteristic, complemented
465C32#515611yescharacteristic, complemented
466C23—91no—
467C23—181no—
468C23—181no—
469C23—181no—
470C23—544no—
471C23—1082no—
472C23—1082no—
473D4—92no—
474D4—92no—
475D4—181no—
476D4—181no—
477D4—181no—
478D4—181no—
479D4—544no—
480D4—1082no—
481C2×C4—91no—
482C2×C4—181no—
483C2×C4—181no—
484C2×C4—181no—
485C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
486C8—21no—
487C8—181no—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—31no—
494C2×C3—61no—
495C2×C3—61no—
496C2×C3—61no—
497C2×C3—182no—
498C2×C3—362no—
499C2×C3—362no—
500C2×C3—721no—
501C2×C3#6379711yescharacteristic
502C2×C3#6379811yescharacteristic
503C2×C3#6392211yescharacteristic
504D3—62no—
505D3—62no—
506D3—122no—
507D3—122no—
508D3—122no—
509D3—241no—
510D3—362no—
511D3—362no—
512D3—721no—
513C22—92no—
514C22—181no—
515C22—181no—
516C22—182no—
517C22—182no—
518C22—361no—
519C22—361no—
520C22—361no—
521C22—361no—
522C22—361no—
523C22—361no—
524C22—361no—
525C22—1088no—
526C22#8865511yescharacteristic
527C4—181no—
528C4—361no—
529C4#8865411yescharacteristic
530C4#8867611yescharacteristic
531C3—31no—
532C3—31no—
533C3—61no—
534C3#1059094511yescharacteristic
535C2—182no—
536C2—362no—
537C2—362no—
538C2—721no—
539C2#1172083811yescharacteristic
540C2#1172083911yescharacteristic
541C2#1172116211yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
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×#88655C2semidirect
2. C2×#3745⋊C3C2semidirect
3. #11721159C2semidirect
4. C2×#88432C2semidirect
5. #11720553C2semidirect
6. #63648⋊C3C2semidirect
7. #11720318C2semidirect
14. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
19. C2×C8×H(3,𝔽3)C22semidirect
20. C2×#6180C22semidirect
21. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
27. C2×#3745D3semidirect
33. #63648D3semidirect
56. C4⋊((C3⋊D3)⋊C3)D4semidirect
57. C4⋊((C3⋊D3)⋊C3)D4semidirect
69. C2×C4⋊(C3⋊D3)C2×D3semidirect
85. C2×C32×C8C2×D3semidirect
86. C2×C32×C8C2×D3semidirect
87. C4⋊(C3⋊Dic3)C2×D3semidirect
101. C2×#272D8semidirect
161. C4⋊(C3⋊D3)D12semidirect
162. C4⋊(C3⋊D3)#67semidirect
295. H(3,𝔽3)#447semidirect
399. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
464. C32#4773semidirect
465. C32#5156semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
C2×#3745⋊C32
#117211592
C2×#884322
#117205532
#63648⋊C32
#117203182
#105909473
#105909443