Finite Group Databasebeta

← #13140947

Subgroups — #13140947

Order 1728.

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

normal subgroups only

This group has 731 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×#88748C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic, complemented
3#11721159C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720323C211yesmaximal, characteristic, complemented
8#10591588—31nomaximal
9#10591586—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#88748—22no—
12#88748—22no—
13C2×C4⋊#272—21no—
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272—21no—
16C2×C4×#272C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×#6200—21no—
19#3755⋊C3—22no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88450—21no—
24#88436—22no—
25C2×C3×#239—31no—
26C2×C3×#239—31no—
27C2×C3×#239—61no—
28C2×#3755D311yescharacteristic, complemented
29C2×C3×#215—31no—
30#63909—31no—
31#63908—31no—
32C2×#3714—31no—
33C2×#3713—31no—
34#63662D311yescharacteristic, complemented
35C3×#835—31no—
36#63563—31no—
37#63562—31no—
38(C2×D3)⋊(C3⋊C8)—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×#272—21no—
41C22×#272D411yescharacteristic
42C22×(C3⋊D3)⋊C3—21no—
43Q8×H(3,𝔽3)—21no—
44Q8×H(3,𝔽3)D412yescomplemented
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×#1084D411yescharacteristic
48C4⋊#272—24no—
49C4⋊#272—21no—
50C4⋊#272D412yescomplemented
51C4×#272—24no—
52C4×#272—21no—
53C4×#272—21no—
54C2×(C3⋊Dic3)⋊C3—21no—
55C4⋊((C3⋊D3)⋊C3)—21no—
56C4⋊((C3⋊D3)⋊C3)—21no—
57#6200—42no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60#5238—91no—
61#5172—91no—
62#4810—91no—
63C2×C32×Q8—21no—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C32×Q8C2×D311yescharacteristic
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C3×#239—62no—
68C3×#239—62no—
69C3×#239—64no—
70C3×#239—64no—
71C3×#239—62no—
72C3×#239—62no—
73C2×C3×D12—31no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×C3×D12—61no—
78C2×C3×D12—61no—
79C2×C3×D12—62no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—31no—
82C2×C3×C4×D3—61no—
83C2×C3×C4×D3—61no—
84C2×C3×C4×D3—61no—
85C2×C3×C4×D3—62no—
86C2×#626—61no—
87C2×#625—61no—
88#3755—22no—
89C4⋊(C3⋊Dic3)C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#215—62no—
93C3×C4⋊Dic3—31no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96#3722—61no—
97(C2×D3)⋊Dic3—61no—
98#3714—62no—
99#3713—62no—
100C22×H(3,𝔽3)C2×D411yescharacteristic
101C2×#272—21no—
102C2×#272—22no—
103C2×#272—21no—
104C2×#272—21no—
105C2×#272—41no—
106C2×#272—41no—
107C2×(C3⋊D3)⋊C3—41no—
108C2×(C3⋊D3)⋊C3—41no—
109C2×(C3⋊D3)⋊C3—41no—
110C4×H(3,𝔽3)—22no—
111C4×H(3,𝔽3)C2×D411yescharacteristic
112C4×H(3,𝔽3)C2×D411yescharacteristic
113#1084—21no—
114(C3⋊Dic3)⋊C3—41no—
115C2×C3×#41—91no—
116C2×#239—31no—
117C2×#239—31no—
118C2×#239—61no—
119C2×C3×SD16—91no—
120C2×#215—31no—
121C2×#215—91no—
122#938—91no—
123C2×#213—91no—
124C2×#213—91no—
125#930—91no—
126C2×D24—91no—
127#895—91no—
128C3×#103—91no—
129#835—31no—
130#835—91no—
131#832—91no—
132#830—91no—
133#821—91no—
134(C2×D3)⋊C8—91no—
135(C2×D3)⋊C8—91no—
136#809—91no—
137C22×C3⋊D3—21no—
138C22×C3×D3—31no—
139C22×C3×D3—31no—
140C22×C3×D3—61no—
141C22×C3×D3—62no—
142C22×C3×D3—61no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C32×Q8—22no—
146C32×Q8—21no—
147C32×Q8—21no—
148C32×Q8—22no—
149C32×Q8D1212yescomplemented
150C32×Q8#6712yescomplemented
151C2×C32×C4—22no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C3⋊Dic3—21no—
158C4⋊(C3⋊D3)—21no—
159C4⋊(C3⋊D3)—21no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C3×D12—32no—
165C3×D12—32no—
166C3×D12—62no—
167C3×D12—68no—
168C3×D12—64no—
169C3×D12—64no—
170C3×D12—62no—
171C3×D12—61no—
172C3×D12—61no—
173C3×D12—61no—
174C3×D12—61no—
175C3×C4×D3—68no—
176C3×C4×D3—64no—
177C3×C4×D3—64no—
178C3×C4×D3—62no—
179C3×C4×D3—62no—
180C3×C4×D3—61no—
181C3×C4×D3—61no—
182C3×C4×D3—61no—
183C3×C4×D3—61no—
184#626—122no—
185#625—122no—
186C3⋊(C3⋊C8)—21no—
187C3⋊(C3⋊C8)—21no—
188C3×C3⋊C8—61no—
189C3×C3⋊C8—61no—
190#449—271noSylow
191C2×H(3,𝔽3)#12011yescharacteristic
192C2×H(3,𝔽3)#13511yescharacteristic
193C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
194#272—41no—
195#272—42no—
196(C3⋊D3)⋊C3—81no—
197C3×#41—182no—
198C3×#41—182no—
199C2×C3×Q8—31no—
200C2×C3×Q8—31no—
201C2×C3×Q8—61no—
202C2×C3×Q8D3211yescharacteristic
203C2×C3×D4—91no—
204C2×C3×D4—91no—
205C2×C3×D4—181no—
206C22×C3×C4—91no—
207C22×C3×C4—181no—
208C2×#67—181no—
209C2×#67—181no—
210#239—62no—
211#239—62no—
212#239—62no—
213#239—62no—
214#239—64no—
215#239—64no—
216C2×D12—31no—
217C2×D12—31no—
218C2×D12—31no—
219C2×D12—61no—
220C2×D12—61no—
221C2×D12—61no—
222C2×D12—62no—
223C2×D12—91no—
224C2×D12—181no—
225C2×C4×D3—31no—
226C2×C4×D3—31no—
227C2×C4×D3—61no—
228C2×C4×D3—61no—
229C2×C4×D3—61no—
230C2×C4×D3—62no—
231C3×SD16—182no—
232C2×C3×C8—91no—
233C2×C3×C8—91no—
234C3×C4⋊C4—91no—
235#217—181no—
236#215—62no—
237#215—182no—
238#213—182no—
239#213—182no—
240(C2×D3)⋊C4—181no—
241(C2×D3)⋊C4—181no—
242C4⋊Dic3—31no—
243C4⋊Dic3—91no—
244C2×C3⋊C8—31no—
245C2×C3⋊C8—31no—
246C2×C3⋊C8—91no—
247C2×C3⋊C8—91no—
248D24—182no—
249C22×C32—21no—
250C22×C32D3×D411yescharacteristic
251C22×C32D3×D411yescharacteristic
252C2×C3⋊D3—41no—
253C2×C3⋊D3—41no—
254C2×C3⋊D3—41no—
255C2×C3×D3—64no—
256C2×C3×D3—62no—
257C2×C3×D3—62no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C2×C3×D3—62no—
264C2×C3×D3—61no—
265C2×C3×D3—121no—
266C2×C3×D3—121no—
267C2×C3×D3—122no—
268C2×C3×D3—122no—
269C2×C3×D3—121no—
270C2×C3×D3—121no—
271C2×C3×D3—121no—
272C2×C3×D3—121no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275C2×C3×D3—121no—
276C32×C4—24no—
277C32×C4—22no—
278C32×C4—22no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4C2×D1211yescharacteristic
282C32×C4D3×D411yescharacteristic
283C32×C4D3×D411yescharacteristic
284C32×C4C2×#6711yescharacteristic
285C3⋊Dic3—41no—
286C3×Dic3—62no—
287C3×Dic3—61no—
288C3×Dic3—61no—
289C3×Dic3—121no—
290C2×#41—91no—
291C2×SD16—91no—
292C2×D8—271no—
293#121—271no—
294#103—91no—
295#102—271no—
296C22⋊C8—271no—
297H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
298C23×C3—91no—
299C23×C3—181no—
300C23×C3—181no—
301C22×D3—31no—
302C22×D3—31no—
303C22×D3—61no—
304C22×D3—61no—
305C22×D3—61no—
306C22×D3—62no—
307C22×D3—61no—
308C22×D3—181no—
309C3×Q8—21no—
310C3×Q8—32no—
311C3×Q8—32no—
312C3×Q8—61no—
313C3×Q8—61no—
314C3×Q8—62no—
315C3×Q8—62no—
316C3×Q8#62612yes—
317C3×D4—92no—
318C3×D4—184no—
319C3×D4—181no—
320C3×D4—181no—
321C3×D4—181no—
322C2×C3×C4—21no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—62no—
328C2×C3×C4—61no—
329C2×C3×C4—91no—
330C2×C3×C4—184no—
331C2×C3×C4—181no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4C2×D3211yescharacteristic
335#67—362no—
336#67—362no—
337C2×Dic3—31no—
338C2×Dic3—31no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—181no—
342D12—32no—
343D12—32no—
344D12—62no—
345D12—61no—
346D12—61no—
347D12—64no—
348D12—64no—
349D12—68no—
350D12—62no—
351D12—61no—
352D12—61no—
353D12—181no—
354D12—181no—
355D12—362no—
356C4×D3—61no—
357C4×D3—61no—
358C4×D3—61no—
359C4×D3—61no—
360C4×D3—64no—
361C4×D3—64no—
362C4×D3—68no—
363C4×D3—62no—
364C4×D3—62no—
365C3×C8—181no—
366C3×C8—181no—
367C3⋊C8—61no—
368C3⋊C8—61no—
369C3⋊C8—181no—
370C3⋊C8—181no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32#88211yescharacteristic
375C2×C32#91411yescharacteristic
376C2×C32#91411yescharacteristic
377C2×C32#91611yescharacteristic
378C2×C32#91911yescharacteristic
379C2×C32#93711yescharacteristic
380C3⋊D3—81no—
381C3×D3—122no—
382C3×D3—124no—
383C3×D3—121no—
384C3×D3—122no—
385C3×D3—121no—
386C3×D3—122no—
387C3×D3—241no—
388#41—182no—
389#41—182no—
390C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
391C2×D4—91no—
392C2×D4—91no—
393C2×D4—181no—
394C2×D4—541no—
395C22×C4—91no—
396C22×C4—181no—
397SD16—182no—
398D8—542no—
399C2×C8—91no—
400C2×C8—91no—
401C4⋊C4—91no—
402C22⋊C4—541no—
403C22×C3—31no—
404C22×C3—31no—
405C22×C3—61no—
406C22×C3—181no—
407C22×C3—181no—
408C22×C3—182no—
409C22×C3—181no—
410C22×C3—361no—
411C22×C3—361no—
412C22×C3—361no—
413C22×C3—361no—
414C22×C3—361no—
415C22×C3#381111yescharacteristic
416C2×D3—62no—
417C2×D3—62no—
418C2×D3—61no—
419C2×D3—61no—
420C2×D3—61no—
421C2×D3—61no—
422C2×D3—61no—
423C2×D3—61no—
424C2×D3—64no—
425C2×D3—62no—
426C2×D3—121no—
427C2×D3—121no—
428C2×D3—121no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—121no—
432C2×D3—122no—
433C2×D3—122no—
434C2×D3—121no—
435C2×D3—121no—
436C2×D3—121no—
437C2×D3—361no—
438C2×D3—361no—
439C2×D3—361no—
440C3×C4—22no—
441C3×C4—31no—
442C3×C4—31no—
443C3×C4—31no—
444C3×C4—31no—
445C3×C4—62no—
446C3×C4—62no—
447C3×C4—64no—
448C3×C4—61no—
449C3×C4—61no—
450C3×C4—181no—
451C3×C4—361no—
452C3×C4C2×#62611yescharacteristic
453C3×C4#381111yescharacteristic
454Dic3—61no—
455Dic3—61no—
456Dic3—62no—
457Dic3—121no—
458Dic3—361no—
459C32—21no—
460C32#481011yescharacteristic, complemented
461C32#517211yescharacteristic, complemented
462C23—91no—
463C23—181no—
464C23—181no—
465Q8—21no—
466Q8#620012yescomplemented
467D4—92no—
468D4—184no—
469D4—181no—
470D4—181no—
471D4—181no—
472D4—1082no—
473C2×C4—21no—
474C2×C4—91no—
475C2×C4—184no—
476C2×C4—181no—
477C2×C4—181no—
478C2×C4—181no—
479C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
480C8—181no—
481C8—181no—
482C2×C3—31no—
483C2×C3—31no—
484C2×C3—31no—
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—31no—
488C2×C3—61no—
489C2×C3—61no—
490C2×C3—61no—
491C2×C3—361no—
492C2×C3—362no—
493C2×C3—721no—
494C2×C3#6394111yescharacteristic
495C2×C3#6395011yescharacteristic
496C2×C3#6398111yescharacteristic
497D3—121no—
498D3—121no—
499D3—122no—
500D3—122no—
501D3—122no—
502D3—124no—
503D3—241no—
504D3—721no—
505C22—181no—
506C22—181no—
507C22—182no—
508C22—181no—
509C22—361no—
510C22—361no—
511C22—361no—
512C22—361no—
513C22—361no—
514C22#8867711yescharacteristic
515C4—22no—
516C4—181no—
517C4—361no—
518C4C2×#620011yescharacteristic
519C4#8867711yescharacteristic
520C3—31no—
521C3—31no—
522C3—61no—
523C3#1059159311yescharacteristic
524C2—361no—
525C2—362no—
526C2—721no—
527C2#1172122911yescharacteristic
528C2#1172123011yescharacteristic
529C2#1172132311yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#449
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88748C2semidirect
2. C2×#3755⋊C3C2semidirect
3. #11721159C2semidirect
4. C2×#88436C2semidirect
5. #63662⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720323C2semidirect
28. C2×#3755D3semidirect
34. #63662D3semidirect
44. Q8×H(3,𝔽3)D4semidirect
50. C4⋊#272D4semidirect
149. C32×Q8D12semidirect
150. C32×Q8#67semidirect
297. H(3,𝔽3)#449semidirect
460. C32#4810semidirect
461. C32#5172semidirect
466. Q8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#887482
C2×#3755⋊C32
#117211592
C2×#884362
#63662⋊C32
#117203242
#117203232
#105915883
#105915863