Finite Group Databasebeta

← #13140598

Subgroups — #13140598

Order 1728.

566 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 75 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×#88545C211yesmaximal, characteristic, complemented
2#64149⋊C3C211yesmaximal, characteristic, complemented
3#11721149C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10591433—31nomaximal
9#10591428—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#638⋊C3—21no—
13C2×#638⋊C3—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×#6199—21no—
16#88545—22no—
17C2×#6180C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88451—21no—
22#88436—22no—
23#64149D311yescharacteristic, complemented
24C3×#940—31no—
25C2×C3×#213—31no—
26C2×C3×#213—31no—
27C2×C3×#213—61no—
28#63898—31no—
29Dic3×D12—31no—
30C2×#3714—31no—
31C2×#3713—31no—
32#63659D311yescharacteristic, complemented
33C3×#832—31no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63564—31no—
37#63563—31no—
38C23×H(3,𝔽3)—21no—
39C22×#272—21no—
40C22×(C3⋊D3)⋊C3—21no—
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C4⋊#272—21no—
45C4⋊#272D412yescomplemented
46#638⋊C3—42no—
47#638⋊C3—42no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3D411yescharacteristic
50C2×(C3⋊Dic3)⋊C3D411yescharacteristic
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53#6199—42no—
54#6180—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5150—91no—
57#5072—91no—
58#4786—91no—
59C2×C32×D4—21no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C2×D311yescharacteristic
62C2×#638—21no—
63C2×#638—21no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C3×#67—61no—
66C2×C3×#67—61no—
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—31no—
70C2×C3×D12—61no—
71C2×D3×Dic3—61no—
72C2×D3×Dic3—61no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×#213—62no—
77C3×#213—62no—
78C3×#213—64no—
79C3×C4⋊Dic3—31no—
80C3×C4×Dic3—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—61no—
85(C2×D3)⋊Dic3—61no—
86(C2×D3)⋊Dic3—61no—
87#3714—62no—
88#3713—62no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)—41no—
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C2×#272—22no—
94C2×#272—41no—
95C2×#272—41no—
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3—41no—
98C2×(C3⋊D3)⋊C3—41no—
99C4×H(3,𝔽3)C2×D411yescharacteristic
100C4×H(3,𝔽3)#4111yescharacteristic
101(C3⋊Dic3)⋊C3—21no—
102(C3⋊Dic3)⋊C3—41no—
103(C3⋊Dic3)⋊C3D812yescomplemented
104C2×C3×D8—91no—
105C3×#127—91no—
106#940—31no—
107#940—91no—
108D4×Dic3—91no—
109C2×#213—31no—
110C2×#213—31no—
111C2×#213—61no—
112C2×#213—91no—
113C2×#213—91no—
114C2×D24—91no—
115#893—91no—
116C4×D12—91no—
117C3×#102—91no—
118#832—31no—
119#832—91no—
120#832—91no—
121#821—91no—
122Dic3⋊C8—91no—
123Dic3⋊C8—91no—
124#809—91no—
125C4⋊(C3⋊C8)—91no—
126C23×C32—21no—
127C23×C32—21no—
128C23×C32—41no—
129C22×C3⋊D3—21no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—61no—
133C22×C3×D3—121no—
134C32×D4—22no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—21no—
138C32×D4—21no—
139C32×D4—41no—
140C2×C32×C4—21no—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143#638—42no—
144#638—42no—
145C2×C3⋊Dic3—21no—
146C2×C3⋊Dic3#6711yescharacteristic
147C2×C3⋊Dic3#6711yescharacteristic
148C4⋊(C3⋊D3)—21no—
149C4⋊(C3⋊D3)—21no—
150C3×#67—122no—
151C3×#67—122no—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—31no—
154C2×C3×Dic3—61no—
155C3×D12—32no—
156C3×D12—32no—
157C3×D12—62no—
158C3×D12—61no—
159C3×D12—61no—
160C3×D12—61no—
161C3×D12—61no—
162C3×D12—121no—
163D3×Dic3—122no—
164D3×Dic3—122no—
165C3⋊(C3⋊C8)—21no—
166C3×C3⋊C8—62no—
167C3×C3⋊C8—61no—
168C3×C3⋊C8—61no—
169C3×C3⋊C8—61no—
170#459—271noSylow
171C2×H(3,𝔽3)—81no—
172C2×H(3,𝔽3)#12111yescharacteristic
173C2×H(3,𝔽3)C2×D811yescharacteristic
174C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
175#272—42no—
176(C3⋊D3)⋊C3—81no—
177C2×C3×D4—31no—
178C2×C3×D4—31no—
179C2×C3×D4—61no—
180C2×C3×D4—91no—
181C2×C3×D4—91no—
182C2×C3×D4—181no—
183C2×C3×D4—181no—
184C2×C3×D4D3211yescharacteristic
185C2×#67—61no—
186C2×#67—61no—
187C2×#67—181no—
188C2×#67—181no—
189C22×Dic3—181no—
190C2×D12—31no—
191C2×D12—31no—
192C2×D12—31no—
193C2×D12—61no—
194C2×D12—91no—
195C2×C4×D3—181no—
196C2×C4×D3—181no—
197C3×D8—182no—
198C2×C3×C8—91no—
199C2×C3×C8—91no—
200C3×C4⋊C4—91no—
201C3×C42—91no—
202#217—181no—
203#213—62no—
204#213—62no—
205#213—64no—
206#213—182no—
207#213—182no—
208(C2×D3)⋊C4—181no—
209(C2×D3)⋊C4—181no—
210C4⋊Dic3—31no—
211C4⋊Dic3—91no—
212C4×Dic3—31no—
213C4×Dic3—91no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—31no—
216C2×C3⋊C8—31no—
217C2×C3⋊C8—61no—
218C2×C3⋊C8—91no—
219D24—182no—
220C22×C32—21no—
221C22×C32—42no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32—81no—
229C22×C32—81no—
230C22×C32D3×D411yescharacteristic
231C22×C32C2×#6711yescharacteristic
232C2×C3⋊D3—41no—
233C2×C3⋊D3—41no—
234C2×C3⋊D3—41no—
235C2×C3×D3—62no—
236C2×C3×D3—62no—
237C2×C3×D3—122no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C2×C3×D3—121no—
241C2×C3×D3—121no—
242C2×C3×D3—121no—
243C2×C3×D3—121no—
244C2×C3×D3—121no—
245C2×C3×D3—241no—
246C2×C3×D3—241no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4#23511yescharacteristic
250C32×C4D3×D411yescharacteristic
251C32×C4D3×D411yescharacteristic
252C32×C4#23711yescharacteristic
253C3⋊Dic3—21no—
254C3⋊Dic3—41no—
255C3⋊Dic3#21312yescomplemented
256C3×Dic3—32no—
257C3×Dic3—61no—
258C3×Dic3—121no—
259C2×D8—91no—
260C2×D8—271no—
261#127—91no—
262C4×D4—271no—
263C4⋊C8—271no—
264#102—91no—
265#102—271no—
266H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
267C23×C3—21no—
268C23×C3—61no—
269C23×C3—61no—
270C23×C3—121no—
271C23×C3—181no—
272C23×C3—181no—
273C22×D3—61no—
274C22×D3—61no—
275C22×D3—61no—
276C22×D3—121no—
277C22×D3—181no—
278C3×D4—21no—
279C3×D4—21no—
280C3×D4—61no—
281C3×D4—61no—
282C3×D4—61no—
283C3×D4—61no—
284C3×D4—62no—
285C3×D4—92no—
286C3×D4—121no—
287C3×D4—181no—
288C3×D4—181no—
289C3×D4—181no—
290C3×D4—362no—
291C3×D4—362no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—61no—
295C2×C3×C4—91no—
296C2×C3×C4—91no—
297C2×C3×C4—181no—
298C2×C3×C4C2×D3211yescharacteristic
299#67—122no—
300#67—122no—
301#67—362no—
302#67—362no—
303C2×Dic3—31no—
304C2×Dic3—31no—
305C2×Dic3—61no—
306C2×Dic3—91no—
307C2×Dic3—91no—
308C2×Dic3—181no—
309C2×Dic3—362no—
310D12—32no—
311D12—32no—
312D12—62no—
313D12—61no—
314D12—61no—
315D12—61no—
316D12—61no—
317D12—121no—
318D12—181no—
319D12—181no—
320C4×D3—362no—
321C4×D3—362no—
322C3×C8—181no—
323C3×C8—181no—
324C3⋊C8—61no—
325C3⋊C8—61no—
326C3⋊C8—62no—
327C3⋊C8—61no—
328C3⋊C8—181no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—82no—
333C2×C32—81no—
334C2×C32—81no—
335C2×C32#88411yescharacteristic
336C2×C32D3×D811yescharacteristic
337C2×C32#91411yescharacteristic
338C2×C32C2×#21311yescharacteristic
339C2×C32#93211yescharacteristic
340C2×C32#93811yescharacteristic
341C3⋊D3—81no—
342C3×D3—122no—
343C3×D3—122no—
344C3×D3—242no—
345C3×D3—241no—
346C2×D4—91no—
347C2×D4—91no—
348C2×D4—181no—
349C2×D4—181no—
350C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
351C22×C4—541no—
352D8—182no—
353D8—542no—
354C2×C8—91no—
355C2×C8—91no—
356C4⋊C4—91no—
357C22⋊C4—541no—
358C42—91no—
359C22×C3—31no—
360C22×C3—31no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—61no—
365C22×C3—121no—
366C22×C3—121no—
367C22×C3—121no—
368C22×C3—121no—
369C22×C3—121no—
370C22×C3—121no—
371C22×C3—122no—
372C22×C3—182no—
373C22×C3—241no—
374C22×C3—241no—
375C22×C3—361no—
376C22×C3—361no—
377C22×C3—361no—
378C22×C3—361no—
379C22×C3—361no—
380C22×C3D3×#6711yescharacteristic
381C2×D3—62no—
382C2×D3—62no—
383C2×D3—122no—
384C2×D3—121no—
385C2×D3—121no—
386C2×D3—121no—
387C2×D3—121no—
388C2×D3—121no—
389C2×D3—121no—
390C2×D3—121no—
391C2×D3—241no—
392C2×D3—241no—
393C2×D3—361no—
394C2×D3—361no—
395C2×D3—361no—
396C3×C4—31no—
397C3×C4—31no—
398C3×C4—31no—
399C3×C4—31no—
400C3×C4—61no—
401C3×C4—61no—
402C3×C4—92no—
403C3×C4—181no—
404C3×C4—361no—
405C3×C4#380411yescharacteristic
406C3×C4#381111yescharacteristic
407Dic3—32no—
408Dic3—61no—
409Dic3—92no—
410Dic3—121no—
411Dic3—181no—
412Dic3—361no—
413C32—21no—
414C32#478611yescharacteristic, complemented
415C32#507211yescharacteristic, complemented
416C23—21no—
417C23—181no—
418C23—181no—
419D4—21no—
420D4—21no—
421D4—92no—
422D4—181no—
423D4—181no—
424D4—181no—
425D4—362no—
426D4—362no—
427C2×C4—91no—
428C2×C4—91no—
429C2×C4—181no—
430C2×C4—1082no—
431C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
432C8—181no—
433C8—181no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—61no—
441C2×C3—61no—
442C2×C3—61no—
443C2×C3—81no—
444C2×C3—241no—
445C2×C3—241no—
446C2×C3—242no—
447C2×C3—362no—
448C2×C3—721no—
449C2×C3D3×#21311yescharacteristic
450C2×C3#6393411yescharacteristic
451C2×C3#6395811yescharacteristic
452D3—122no—
453D3—122no—
454D3—242no—
455D3—241no—
456D3—721no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—182no—
461C22—361no—
462C22—361no—
463C22—361no—
464C22—361no—
465C22—361no—
466C22#8867611yescharacteristic
467C4—92no—
468C4—181no—
469C4—361no—
470C4#8867111yescharacteristic
471C4#8867711yescharacteristic
472C3—31no—
473C3—31no—
474C3—61no—
475C3#1059142911yescharacteristic
476C2—81no—
477C2—362no—
478C2—721no—
479C2#1172119411yescharacteristic
480C2#1172119611yescharacteristic
481C2#1172130311yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#459
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88545C2semidirect
2. #64149⋊C3C2semidirect
3. #11721149C2semidirect
4. C2×#88436C2semidirect
5. #63659⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720311C2semidirect
23. #64149D3semidirect
32. #63659D3semidirect
45. C4⋊#272D4semidirect
103. (C3⋊Dic3)⋊C3D8semidirect
255. C3⋊Dic3#213semidirect
266. H(3,𝔽3)#459semidirect
414. C32#4786semidirect
415. C32#5072semidirect

Maximal subgroups

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

subgroupindex
C2×#885452
#64149⋊C32
#117211492
C2×#884362
#63659⋊C32
#117203182
#117203112
#105914333
#105914283