Finite Group Databasebeta

← #13140963

Subgroups — #13140963

Order 1728.

635 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 83 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×#88722C211yesmaximal, characteristic, complemented
2C2×#88549C211yesmaximal, characteristic, complemented
3#11721156C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720317C211yesmaximal, characteristic, complemented
8#10591593—31nomaximal
9#10591586—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12#88722—22no—
13#88722—22no—
14C2×C4⋊((C3⋊D3)⋊C3)—21no—
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3—21no—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#6200—21no—
19#88549—22no—
20C2×#6180C2211yescharacteristic
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88451—21no—
24#88436—22no—
25C2×#3832D311yescharacteristic, complemented
26C2×C3×#239—31no—
27C2×C3×#215—31no—
28C2×C3×#215—31no—
29C2×C3×#215—61no—
30#63916—31no—
31#63908—31no—
32C2×#3714—31no—
33C2×#3713—31no—
34#63662D311yescharacteristic, complemented
35C3×#835—31no—
36#63564—31no—
37#63563—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×#272—21no—
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3D411yescharacteristic
43Q8×H(3,𝔽3)—21no—
44Q8×H(3,𝔽3)D412yescomplemented
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C4⋊#272—21no—
48C4⋊#272—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3D411yescharacteristic
51C4⋊((C3⋊D3)⋊C3)—24no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)D412yescomplemented
54C4×(C3⋊D3)⋊C3—24no—
55C4×(C3⋊D3)⋊C3—21no—
56C4×(C3⋊D3)⋊C3—21no—
57#6200—42no—
58#6180—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
66#3832—22no—
67#3832—22no—
68C2×C4⋊(C3⋊D3)—21no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×C4×C3⋊D3—21no—
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C3×#239—62no—
73C3×#239—62no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×D12—61no—
79C2×C3×C4×D3—31no—
80C2×C3×C4×D3—61no—
81C2×#626—61no—
82C2×#625—61no—
83C4⋊(C3⋊Dic3)C2×D311yescharacteristic
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C3×#215—64no—
86C3×#215—62no—
87C3×#215—62no—
88C3×C4⋊Dic3—31no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—61no—
93(C2×D3)⋊Dic3—61no—
94(C2×D3)⋊Dic3—61no—
95#3714—62no—
96#3713—62no—
97C22×H(3,𝔽3)C2×D411yescharacteristic
98C2×#272—41no—
99C2×#272—41no—
100C2×#272—41no—
101C2×(C3⋊D3)⋊C3—21no—
102C2×(C3⋊D3)⋊C3—22no—
103C2×(C3⋊D3)⋊C3—21no—
104C2×(C3⋊D3)⋊C3—21no—
105C2×(C3⋊D3)⋊C3—41no—
106C2×(C3⋊D3)⋊C3—41no—
107C4×H(3,𝔽3)—22no—
108C4×H(3,𝔽3)C2×D411yescharacteristic
109C4×H(3,𝔽3)C2×D411yescharacteristic
110(C3⋊Dic3)⋊C3—21no—
111(C3⋊Dic3)⋊C3—41no—
112C2×C3×#41—91no—
113C2×#239—31no—
114C2×#239—91no—
115C2×C3×SD16—91no—
116C2×#215—31no—
117C2×#215—31no—
118C2×#215—61no—
119#938—91no—
120C2×#213—91no—
121C2×#213—91no—
122#930—91no—
123C2×D24—91no—
124#895—91no—
125C3×#103—91no—
126#835—31no—
127#835—91no—
128#832—91no—
129#830—91no—
130#821—91no—
131(C2×D3)⋊C8—91no—
132(C2×D3)⋊C8—91no—
133#809—91no—
134C22×C3⋊D3—21no—
135C22×C3⋊D3#6711yescharacteristic
136C22×C3×D3—31no—
137C22×C3×D3—61no—
138C22×C3×D3—61no—
139C22×C3×D3—61no—
140C22×C3×D3—121no—
141C32×Q8—21no—
142C32×Q8—21no—
143C32×Q8—22no—
144C32×Q8—41no—
145C32×Q8D1212yescomplemented
146C32×Q8#6712yescomplemented
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—41no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3#6711yescharacteristic
155C4⋊(C3⋊D3)—24no—
156C4⋊(C3⋊D3)—21no—
157C4⋊(C3⋊D3)#6712yescomplemented
158C4×C3⋊D3—24no—
159C4×C3⋊D3—21no—
160C4×C3⋊D3—21no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—61no—
163C3×D12—32no—
164C3×D12—64no—
165C3×D12—62no—
166C3×D12—61no—
167C3×D12—61no—
168C3×D12—61no—
169C3×D12—61no—
170C3×D12—61no—
171C3×D12—121no—
172C3×C4×D3—64no—
173C3×C4×D3—61no—
174C3×C4×D3—61no—
175#626—122no—
176#625—122no—
177C3⋊(C3⋊C8)—21no—
178C3×C3⋊C8—62no—
179C3×C3⋊C8—61no—
180C3×C3⋊C8—61no—
181C3×C3⋊C8—61no—
182#449—271noSylow
183C2×H(3,𝔽3)#12011yescharacteristic
184C2×H(3,𝔽3)#13511yescharacteristic
185C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
186#272—81no—
187(C3⋊D3)⋊C3—41no—
188(C3⋊D3)⋊C3—42no—
189C3×#41—182no—
190C3×#41—182no—
191C2×C3×Q8—31no—
192C2×C3×Q8—31no—
193C2×C3×Q8—61no—
194C2×C3×Q8D3211yescharacteristic
195C2×C3×D4—91no—
196C2×C3×D4—91no—
197C2×C3×D4—181no—
198C22×C3×C4—91no—
199C22×C3×C4—181no—
200C2×#67—181no—
201C2×#67—181no—
202#239—62no—
203#239—62no—
204#239—182no—
205#239—182no—
206C2×D12—31no—
207C2×D12—31no—
208C2×D12—31no—
209C2×D12—61no—
210C2×D12—61no—
211C2×D12—91no—
212C2×D12—181no—
213C2×D12—181no—
214C2×C4×D3—31no—
215C2×C4×D3—61no—
216C2×C4×D3—91no—
217C2×C4×D3—181no—
218C3×SD16—182no—
219C2×C3×C8—91no—
220C2×C3×C8—91no—
221C3×C4⋊C4—91no—
222#217—181no—
223#215—62no—
224#215—62no—
225#215—64no—
226#213—182no—
227#213—182no—
228(C2×D3)⋊C4—181no—
229(C2×D3)⋊C4—181no—
230C4⋊Dic3—31no—
231C4⋊Dic3—91no—
232C2×C3⋊C8—31no—
233C2×C3⋊C8—31no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—61no—
236C2×C3⋊C8—91no—
237D24—182no—
238C22×C32—21no—
239C22×C32D3×D411yescharacteristic
240C22×C32C2×#6711yescharacteristic
241C2×C3⋊D3—21no—
242C2×C3⋊D3—21no—
243C2×C3⋊D3—21no—
244C2×C3⋊D3—22no—
245C2×C3⋊D3—41no—
246C2×C3⋊D3—41no—
247C2×C3×D3—62no—
248C2×C3×D3—61no—
249C2×C3×D3—61no—
250C2×C3×D3—61no—
251C2×C3×D3—121no—
252C2×C3×D3—121no—
253C2×C3×D3—122no—
254C2×C3×D3—121no—
255C2×C3×D3—121no—
256C2×C3×D3—121no—
257C2×C3×D3—121no—
258C2×C3×D3—121no—
259C2×C3×D3—121no—
260C2×C3×D3—241no—
261C2×C3×D3—241no—
262C32×C4—22no—
263C32×C4—22no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—42no—
267C32×C4C2×D1211yescharacteristic
268C32×C4D3×D411yescharacteristic
269C32×C4C2×#6711yescharacteristic
270C32×C4C2×#6711yescharacteristic
271C3⋊Dic3—21no—
272C3⋊Dic3—41no—
273C3×Dic3—61no—
274C3×Dic3—121no—
275C2×#41—91no—
276C2×SD16—91no—
277C2×D8—271no—
278#121—271no—
279#103—91no—
280#102—271no—
281C22⋊C8—271no—
282H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
283C23×C3—91no—
284C23×C3—181no—
285C23×C3—181no—
286C22×D3—31no—
287C22×D3—61no—
288C22×D3—61no—
289C22×D3—61no—
290C22×D3—91no—
291C22×D3—121no—
292C22×D3—181no—
293C3×Q8—21no—
294C3×Q8—32no—
295C3×Q8—32no—
296C3×Q8—61no—
297C3×Q8—61no—
298C3×Q8—62no—
299C3×Q8—121no—
300C3×Q8#62612yes—
301C3×D4—92no—
302C3×D4—181no—
303C3×D4—184no—
304C3×D4—181no—
305C3×D4—181no—
306C2×C3×C4—21no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—91no—
313C2×C3×C4—121no—
314C2×C3×C4—181no—
315C2×C3×C4—181no—
316C2×C3×C4—184no—
317C2×C3×C4—181no—
318C2×C3×C4C2×D3211yescharacteristic
319#67—362no—
320#67—362no—
321C2×Dic3—31no—
322C2×Dic3—61no—
323C2×Dic3—91no—
324C2×Dic3—181no—
325D12—32no—
326D12—64no—
327D12—61no—
328D12—61no—
329D12—61no—
330D12—61no—
331D12—62no—
332D12—61no—
333D12—92no—
334D12—121no—
335D12—181no—
336D12—184no—
337D12—362no—
338C4×D3—64no—
339C4×D3—61no—
340C4×D3—61no—
341C4×D3—181no—
342C4×D3—181no—
343C4×D3—184no—
344C3×C8—181no—
345C3×C8—181no—
346C3⋊C8—61no—
347C3⋊C8—61no—
348C3⋊C8—62no—
349C3⋊C8—61no—
350C3⋊C8—181no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32#88211yescharacteristic
355C2×C32#91411yescharacteristic
356C2×C32#91911yescharacteristic
357C2×C32#94911yescharacteristic
358C2×C32#95011yescharacteristic
359C2×C32#95311yescharacteristic
360C3⋊D3—41no—
361C3⋊D3—42no—
362C3×D3—121no—
363C3×D3—122no—
364C3×D3—242no—
365C3×D3—241no—
366C3×D3—241no—
367#41—182no—
368#41—182no—
369C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
370C2×D4—91no—
371C2×D4—91no—
372C2×D4—181no—
373C2×D4—541no—
374C22×C4—91no—
375C22×C4—181no—
376SD16—182no—
377D8—542no—
378C2×C8—91no—
379C2×C8—91no—
380C4⋊C4—91no—
381C22⋊C4—541no—
382C22×C3—31no—
383C22×C3—31no—
384C22×C3—61no—
385C22×C3—181no—
386C22×C3—182no—
387C22×C3—181no—
388C22×C3—181no—
389C22×C3—361no—
390C22×C3—361no—
391C22×C3—361no—
392C22×C3—361no—
393C22×C3—361no—
394C22×C3D3×#6711yescharacteristic
395C2×D3—61no—
396C2×D3—61no—
397C2×D3—62no—
398C2×D3—61no—
399C2×D3—121no—
400C2×D3—121no—
401C2×D3—121no—
402C2×D3—121no—
403C2×D3—121no—
404C2×D3—121no—
405C2×D3—121no—
406C2×D3—121no—
407C2×D3—122no—
408C2×D3—182no—
409C2×D3—181no—
410C2×D3—181no—
411C2×D3—181no—
412C2×D3—241no—
413C2×D3—241no—
414C2×D3—361no—
415C2×D3—361no—
416C3×C4—22no—
417C3×C4—31no—
418C3×C4—31no—
419C3×C4—31no—
420C3×C4—31no—
421C3×C4—62no—
422C3×C4—62no—
423C3×C4—61no—
424C3×C4—61no—
425C3×C4—122no—
426C3×C4—181no—
427C3×C4—361no—
428C3×C4C2×#62611yescharacteristic
429C3×C4D3×#6711yescharacteristic
430Dic3—61no—
431Dic3—121no—
432Dic3—181no—
433Dic3—361no—
434C32—21no—
435C32#481011yescharacteristic, complemented
436C32#523811yescharacteristic, complemented
437C23—91no—
438C23—181no—
439C23—181no—
440Q8—21no—
441Q8#620012yescomplemented
442D4—92no—
443D4—184no—
444D4—181no—
445D4—181no—
446D4—181no—
447D4—1082no—
448C2×C4—21no—
449C2×C4—91no—
450C2×C4—184no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
455C8—181no—
456C8—181no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—61no—
464C2×C3—61no—
465C2×C3—61no—
466C2×C3—361no—
467C2×C3—362no—
468C2×C3—721no—
469C2×C3#6393911yescharacteristic
470C2×C3#6394911yescharacteristic
471C2×C3#6397711yescharacteristic
472D3—121no—
473D3—122no—
474D3—241no—
475D3—241no—
476D3—242no—
477D3—361no—
478D3—362no—
479C22—181no—
480C22—181no—
481C22—182no—
482C22—181no—
483C22—361no—
484C22—361no—
485C22—361no—
486C22—361no—
487C22—361no—
488C22#8867611yescharacteristic
489C4—22no—
490C4—181no—
491C4—361no—
492C4C2×#620011yescharacteristic
493C4#8867611yescharacteristic
494C3—31no—
495C3—31no—
496C3—61no—
497C3#1059158811yescharacteristic
498C2—361no—
499C2—362no—
500C2—721no—
501C2#1172123311yescharacteristic
502C2#1172123511yescharacteristic
503C2#1172132111yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
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×#88722C2semidirect
2. C2×#88549C2semidirect
3. #11721156C2semidirect
4. C2×#88436C2semidirect
5. #63662⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720317C2semidirect
25. C2×#3832D3semidirect
34. #63662D3semidirect
44. Q8×H(3,𝔽3)D4semidirect
53. C4⋊((C3⋊D3)⋊C3)D4semidirect
145. C32×Q8D12semidirect
146. C32×Q8#67semidirect
157. C4⋊(C3⋊D3)#67semidirect
282. H(3,𝔽3)#449semidirect
435. C32#4810semidirect
436. C32#5238semidirect
441. Q8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
C2×#885492
#117211562
C2×#884362
#63662⋊C32
#117203182
#117203172
#105915933
#105915863