Finite Group Databasebeta

← #13140607

Subgroups — #13140607

Order 1728.

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

normal subgroups only

This group has 643 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×#3830⋊C3C211yesmaximal, characteristic, complemented
2C2×#88545C211yesmaximal, characteristic, complemented
3#11721156C211yesmaximal, characteristic, complemented
4C2×#88437C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10591457—31nomaximal
9#10591450—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#638⋊C3—21no—
13C22×(C3⋊Dic3)⋊C3—21no—
14#3830⋊C3—22no—
15#3830⋊C3—22no—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×#6200—21no—
19#88545—22no—
20C2×#6180C2211yescharacteristic
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88451—21no—
24#88437—22no—
25C2×#3830D311yescharacteristic, complemented
26C2×C3×#237—31no—
27C2×C3×#213—31no—
28C2×C3×#213—31no—
29C2×C3×#213—61no—
30#63916—31no—
31#63908—31no—
32C2×#3716—31no—
33C2×#3715—31no—
34#63659D311yescharacteristic, complemented
35C3×#832—31no—
36#63567—31no—
37#63566—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C23×H(3,𝔽3)—21no—
41C22×#272—21no—
42C22×(C3⋊D3)⋊C3D411yescharacteristic
43D4×H(3,𝔽3)—21no—
44D4×H(3,𝔽3)D412yescomplemented
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C4⋊#272—21no—
47C4⋊#272—21no—
48#638⋊C3—24no—
49C2×(C3⋊Dic3)⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C2×(C3⋊Dic3)⋊C3D411yescharacteristic
53C4×(C3⋊D3)⋊C3—21no—
54C4×(C3⋊D3)⋊C3—21no—
55#634⋊C3—21no—
56#634⋊C3D412yescomplemented
57#6200—42no—
58#6180—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60#5238—91no—
61#5158—91no—
62#4781—91no—
63C2×C32×D4—21no—
64C2×C32×D4C2×D311yescharacteristic
65C2×C32×D4C2×D311yescharacteristic
66C2×#638—21no—
67C22×C3⋊Dic3—21no—
68#3830—22no—
69#3830—22no—
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C2×#634C2×D311yescharacteristic
72C2×C3×#67—61no—
73C22×C3×Dic3—61no—
74C3×#237—62no—
75C3×#237—62no—
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—61no—
79C2×C3×C4×D3—31no—
80C2×C3×Dic6—31no—
81C2×#626—61no—
82C2×#625—61no—
83C4⋊(C3⋊Dic3)C2×D311yescharacteristic
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C3×#213—64no—
86C3×#213—62no—
87C3×#213—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#3716—62no—
96#3715—62no—
97C22×H(3,𝔽3)—22no—
98C22×H(3,𝔽3)—41no—
99C22×H(3,𝔽3)—41no—
100C22×H(3,𝔽3)C2×D411yescharacteristic
101C2×#272—41no—
102C2×#272—41no—
103C2×#272—41no—
104C2×(C3⋊D3)⋊C3—21no—
105C2×(C3⋊D3)⋊C3—21no—
106C2×(C3⋊D3)⋊C3—21no—
107C4×H(3,𝔽3)C2×D411yescharacteristic
108C4×H(3,𝔽3)C2×D411yescharacteristic
109(C3⋊Dic3)⋊C3—22no—
110(C3⋊Dic3)⋊C3—21no—
111(C3⋊Dic3)⋊C3—41no—
112C2×C3×#41—91no—
113C2×#237—31no—
114C2×#237—91no—
115C2×C3×D8—91no—
116C2×#215—91no—
117#938—91no—
118C2×#214—91no—
119C2×#213—31no—
120C2×#213—31no—
121C2×#213—61no—
122#930—91no—
123C2×#204—91no—
124#895—91no—
125C3×#102—91no—
126#835—91no—
127#832—31no—
128#832—91no—
129#830—91no—
130(C2×D3)⋊C8—91no—
131(C2×D3)⋊C8—91no—
132#816—91no—
133#810—91no—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—41no—
137C22×C3⋊D3#6711yescharacteristic
138C22×C3×D3—31no—
139C22×C3×D3—61no—
140C22×C3×D3—61no—
141C22×C3×D3—121no—
142C32×D4—21no—
143C32×D4—21no—
144C32×D4—22no—
145C32×D4—41no—
146C32×D4D1212yescomplemented
147C32×D4#6712yescomplemented
148C2×C32×C4—21no—
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151#638—24no—
152C2×C3⋊Dic3—24no—
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3#6711yescharacteristic
156C4×C3⋊D3—21no—
157C4×C3⋊D3—21no—
158#634—21no—
159#634#6712yescomplemented
160C3×#67—64no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—64no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C3×D12—62no—
166C3×D12—61no—
167C3×D12—61no—
168C3×D12—61no—
169C3×D12—61no—
170C3×D12—121no—
171C3×C4×D3—61no—
172C3×C4×D3—61no—
173C3×Dic6—32no—
174C3×Dic6—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)—42no—
184C2×H(3,𝔽3)#12011yescharacteristic
185C2×H(3,𝔽3)#13511yescharacteristic
186C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
187#272—81no—
188(C3⋊D3)⋊C3—41no—
189C3×#41—182no—
190C3×#41—182no—
191C2×C3×Q8—91no—
192C2×C3×D4—31no—
193C2×C3×D4—31no—
194C2×C3×D4—61no—
195C2×C3×D4—91no—
196C2×C3×D4—181no—
197C2×C3×D4D3211yescharacteristic
198C22×C3×C4—91no—
199C22×C3×C4—181no—
200C2×#67—61no—
201C2×#67—181no—
202C2×#67—181no—
203C2×#67—181no—
204C22×Dic3—61no—
205C22×Dic3—181no—
206#237—62no—
207#237—62no—
208#237—182no—
209#237—182no—
210C2×D12—31no—
211C2×D12—31no—
212C2×D12—61no—
213C2×D12—181no—
214C2×C4×D3—31no—
215C2×C4×D3—91no—
216C2×Dic6—31no—
217C2×Dic6—91no—
218C3×D8—182no—
219C2×C3×C8—91no—
220C2×C3×C8—91no—
221C3×C4⋊C4—91no—
222#217—181no—
223#215—182no—
224#214—182no—
225#213—62no—
226#213—62no—
227#213—64no—
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—
237#204—182no—
238C22×C32—22no—
239C22×C32—22no—
240C22×C32—21no—
241C22×C32—42no—
242C22×C32—41no—
243C22×C32—41no—
244C22×C32—41no—
245C22×C32—41no—
246C22×C32—81no—
247C22×C32—81no—
248C22×C32D3×D411yescharacteristic
249C22×C32C2×#6711yescharacteristic
250C2×C3⋊D3—21no—
251C2×C3⋊D3—21no—
252C2×C3⋊D3—21no—
253C2×C3×D3—61no—
254C2×C3×D3—61no—
255C2×C3×D3—61no—
256C2×C3×D3—122no—
257C2×C3×D3—121no—
258C2×C3×D3—121no—
259C2×C3×D3—121no—
260C2×C3×D3—121no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263C2×C3×D3—241no—
264C2×C3×D3—241no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4C2×D1211yescharacteristic
268C32×C4D3×D411yescharacteristic
269C32×C4C2×#6711yescharacteristic
270C32×C4C2×#6711yescharacteristic
271C3⋊Dic3—22no—
272C3⋊Dic3—21no—
273C3⋊Dic3—41no—
274C3×Dic3—62no—
275C3×Dic3—61no—
276C3×Dic3—121no—
277C2×#41—91no—
278C2×SD16—271no—
279C2×D8—91no—
280#121—271no—
281#103—271no—
282#102—91no—
283C22⋊C8—271no—
284H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
285C23×C3—21no—
286C23×C3—61no—
287C23×C3—61no—
288C23×C3—91no—
289C23×C3—121no—
290C23×C3—181no—
291C22×D3—31no—
292C22×D3—61no—
293C22×D3—61no—
294C22×D3—91no—
295C22×D3—121no—
296C3×Q8—92no—
297C3×Q8—181no—
298C3×D4—21no—
299C3×D4—32no—
300C3×D4—32no—
301C3×D4—61no—
302C3×D4—61no—
303C3×D4—62no—
304C3×D4—121no—
305C3×D4—184no—
306C3×D4—181no—
307C3×D4—181no—
308C3×D4#62612yes—
309C2×C3×C4—31no—
310C2×C3×C4—31no—
311C2×C3×C4—61no—
312C2×C3×C4—91no—
313C2×C3×C4—181no—
314C2×C3×C4—181no—
315C2×C3×C4—181no—
316C2×C3×C4—184no—
317C2×C3×C4—181no—
318C2×C3×C4C2×D3211yescharacteristic
319#67—64no—
320#67—184no—
321#67—362no—
322#67—362no—
323C2×Dic3—31no—
324C2×Dic3—64no—
325C2×Dic3—61no—
326C2×Dic3—61no—
327C2×Dic3—91no—
328C2×Dic3—181no—
329C2×Dic3—184no—
330C2×Dic3—181no—
331D12—61no—
332D12—61no—
333D12—61no—
334D12—61no—
335D12—62no—
336D12—121no—
337D12—362no—
338C4×D3—61no—
339C4×D3—61no—
340C4×D3—181no—
341C4×D3—181no—
342Dic6—32no—
343Dic6—61no—
344Dic6—92no—
345Dic6—181no—
346C3×C8—181no—
347C3×C8—181no—
348C3⋊C8—61no—
349C3⋊C8—61no—
350C3⋊C8—62no—
351C3⋊C8—61no—
352C3⋊C8—181no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—42no—
357C2×C32—42no—
358C2×C32—82no—
359C2×C32#88211yescharacteristic
360C2×C32#91111yescharacteristic
361C2×C32#91611yescharacteristic
362C2×C32#94911yescharacteristic
363C2×C32#95011yescharacteristic
364C2×C32#95311yescharacteristic
365C3⋊D3—41no—
366C3×D3—121no—
367C3×D3—242no—
368C3×D3—241no—
369C3×D3—241no—
370#41—182no—
371#41—182no—
372C2×Q8—91no—
373C2×D4—91no—
374C2×D4—181no—
375C2×D4—541no—
376C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
377C22×C4—91no—
378C22×C4—181no—
379SD16—542no—
380D8—182no—
381C2×C8—91no—
382C2×C8—91no—
383C4⋊C4—91no—
384C22⋊C4—541no—
385C22×C3—22no—
386C22×C3—31no—
387C22×C3—31no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—62no—
391C22×C3—62no—
392C22×C3—61no—
393C22×C3—122no—
394C22×C3—121no—
395C22×C3—121no—
396C22×C3—121no—
397C22×C3—121no—
398C22×C3—181no—
399C22×C3—181no—
400C22×C3—181no—
401C22×C3—241no—
402C22×C3—241no—
403C22×C3—361no—
404C22×C3—361no—
405C22×C3—361no—
406C22×C3D3×#6711yescharacteristic
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—61no—
410C2×D3—121no—
411C2×D3—121no—
412C2×D3—121no—
413C2×D3—121no—
414C2×D3—121no—
415C2×D3—121no—
416C2×D3—122no—
417C2×D3—181no—
418C2×D3—181no—
419C2×D3—181no—
420C2×D3—241no—
421C2×D3—241no—
422C3×C4—31no—
423C3×C4—31no—
424C3×C4—31no—
425C3×C4—31no—
426C3×C4—61no—
427C3×C4—61no—
428C3×C4—181no—
429C3×C4—182no—
430C3×C4—361no—
431C3×C4C2×#62611yescharacteristic
432C3×C4D3×#6711yescharacteristic
433Dic3—62no—
434Dic3—61no—
435Dic3—121no—
436Dic3—182no—
437Dic3—181no—
438Dic3—361no—
439C32—21no—
440C32#478111yescharacteristic, complemented
441C32#523811yescharacteristic, complemented
442C23—21no—
443C23—91no—
444C23—181no—
445Q8—92no—
446Q8—181no—
447D4—21no—
448D4—184no—
449D4—181no—
450D4—181no—
451D4—1082no—
452D4#620012yescomplemented
453C2×C4—91no—
454C2×C4—184no—
455C2×C4—181no—
456C2×C4—181no—
457C2×C4—181no—
458C2×C4—181no—
459C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
460C8—181no—
461C8—181no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—31no—
465C2×C3—31no—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—42no—
469C2×C3—61no—
470C2×C3—61no—
471C2×C3—61no—
472C2×C3—122no—
473C2×C3—122no—
474C2×C3—242no—
475C2×C3—361no—
476C2×C3—721no—
477C2×C3#6392511yescharacteristic
478C2×C3#6393511yescharacteristic
479C2×C3#6397711yescharacteristic
480D3—121no—
481D3—241no—
482D3—241no—
483D3—242no—
484D3—361no—
485C22—22no—
486C22—41no—
487C22—41no—
488C22—181no—
489C22—181no—
490C22—181no—
491C22—361no—
492C22—361no—
493C22—361no—
494C22#8867611yescharacteristic
495C4—181no—
496C4—182no—
497C4—361no—
498C4C2×#620011yescharacteristic
499C4#8867611yescharacteristic
500C3—31no—
501C3—31no—
502C3—61no—
503C3#1059145211yescharacteristic
504C2—42no—
505C2—361no—
506C2—721no—
507C2#1172119511yescharacteristic
508C2#1172119711yescharacteristic
509C2#1172132111yescharacteristic
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)#449
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×#3830⋊C3C2semidirect
2. C2×#88545C2semidirect
3. #11721156C2semidirect
4. C2×#88437C2semidirect
5. #63659⋊C3C2semidirect
6. #11720317C2semidirect
7. #11720313C2semidirect
25. C2×#3830D3semidirect
34. #63659D3semidirect
44. D4×H(3,𝔽3)D4semidirect
56. #634⋊C3D4semidirect
146. C32×D4D12semidirect
147. C32×D4#67semidirect
159. #634#67semidirect
284. H(3,𝔽3)#449semidirect
440. C32#4781semidirect
441. C32#5238semidirect
452. D4#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
C2×#885452
#117211562
C2×#884372
#63659⋊C32
#117203172
#117203132
#105914573
#105914503