Finite Group Databasebeta

← #13140917

Subgroups — #13140917

Order 1728.

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

normal subgroups only

This group has 609 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×#3755⋊C3C211yesmaximal, characteristic, complemented
3#11721141C211yesmaximal, characteristic, complemented
4C2×#88432C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591593—31nomaximal
9#10591588—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#88722—22no—
12#88722—22no—
13C2×C4⋊((C3⋊D3)⋊C3)—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3—21no—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#6198—21no—
19C4⋊#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21#3755⋊C3—22no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88450—21no—
24#88432—22no—
25C2×#3832D311yescharacteristic, complemented
26C2×C3×#239—31no—
27C2×#3755D311yescharacteristic, complemented
28C2×C3×#215—31no—
29#63916—31no—
30#63909—31no—
31C2×#3714—31no—
32C2×#3714—31no—
33C3×#835—31no—
34C3×#835—31no—
35C3×#835—61no—
36#63564—31no—
37#63562—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×(C3⋊D3)⋊C3—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
47C2×#1084—21no—
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—24no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
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#6198—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)C2×D311yescharacteristic
71C2×C4×C3⋊D3—21no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C3×#239—62no—
74C3×#239—62no—
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×C4×D3—31no—
79C2×C3×C4×D3—61no—
80C2×#626—61no—
81C2×#626—61no—
82#3755—22no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×#215—62no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—61no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—61no—
92#3722—61no—
93(C2×D3)⋊Dic3—61no—
94#3714—62no—
95#3714—62no—
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C2×(C3⋊D3)⋊C3—21no—
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—21no—
100C2×(C3⋊D3)⋊C3—21no—
101C2×(C3⋊D3)⋊C3—41no—
102C2×(C3⋊D3)⋊C3—41no—
103C2×(C3⋊D3)⋊C3—41no—
104C2×(C3⋊D3)⋊C3—41no—
105C2×(C3⋊D3)⋊C3—41no—
106C4×H(3,𝔽3)—22no—
107C4×H(3,𝔽3)C2×D411yescharacteristic
108C4×H(3,𝔽3)C2×D411yescharacteristic
109#1084—41no—
110(C3⋊Dic3)⋊C3—21no—
111C2×C3×#41—91no—
112C2×#239—31no—
113C2×#239—91no—
114C2×C3×SD16—91no—
115C2×#215—31no—
116C2×#215—91no—
117#938—91no—
118C2×#213—91no—
119C2×#213—91no—
120#930—91no—
121C2×D24—91no—
122#895—91no—
123C3×#103—91no—
124#835—31no—
125#835—31no—
126#835—61no—
127#832—91no—
128#830—91no—
129#821—91no—
130(C2×D3)⋊C8—91no—
131(C2×D3)⋊C8—91no—
132#809—91no—
133C22×C3⋊D3—21no—
134C22×C3⋊D3—21no—
135C22×C3⋊D3#6711yescharacteristic
136C22×C3×D3—31no—
137C22×C3×D3—61no—
138C22×C3×D3—61no—
139C32×Q8—21no—
140C32×Q8—21no—
141C32×Q8—22no—
142C32×Q8—41no—
143C32×Q8#6712yescomplemented
144C32×Q8#6712yescomplemented
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—41no—
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3#6711yescharacteristic
152C4⋊(C3⋊D3)—24no—
153C4⋊(C3⋊D3)—21no—
154C4⋊(C3⋊D3)—21no—
155C4⋊(C3⋊D3)—21no—
156C4⋊(C3⋊D3)#6712yescomplemented
157C4×C3⋊D3—24no—
158C4×C3⋊D3—21no—
159C4×C3⋊D3—21no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—121no—
164C3×D12—32no—
165C3×D12—64no—
166C3×D12—61no—
167C3×D12—61no—
168C3×D12—61no—
169C3×C4×D3—64no—
170C3×C4×D3—61no—
171C3×C4×D3—61no—
172#626—122no—
173#626—122no—
174C3⋊(C3⋊C8)—21no—
175C3×C3⋊C8—61no—
176C3×C3⋊C8—61no—
177C3×C3⋊C8—61no—
178C3×C3⋊C8—121no—
179#449—271noSylow
180C2×H(3,𝔽3)#12011yescharacteristic
181C2×H(3,𝔽3)#13511yescharacteristic
182C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
183(C3⋊D3)⋊C3—41no—
184(C3⋊D3)⋊C3—42no—
185(C3⋊D3)⋊C3—81no—
186C3×#41—182no—
187C3×#41—182no—
188C2×C3×Q8—31no—
189C2×C3×Q8—31no—
190C2×C3×Q8—61no—
191C2×C3×Q8D3211yescharacteristic
192C2×C3×D4—91no—
193C2×C3×D4—91no—
194C2×C3×D4—181no—
195C22×C3×C4—91no—
196C22×C3×C4—181no—
197C2×#67—181no—
198C2×#67—181no—
199#239—62no—
200#239—62no—
201#239—182no—
202#239—182no—
203C2×D12—31no—
204C2×D12—31no—
205C2×D12—61no—
206C2×D12—91no—
207C2×D12—91no—
208C2×D12—181no—
209C2×D12—181no—
210C2×C4×D3—31no—
211C2×C4×D3—61no—
212C2×C4×D3—91no—
213C2×C4×D3—181no—
214C3×SD16—182no—
215C2×C3×C8—91no—
216C2×C3×C8—91no—
217C3×C4⋊C4—91no—
218#217—181no—
219#215—62no—
220#215—182no—
221#213—182no—
222#213—182no—
223(C2×D3)⋊C4—181no—
224(C2×D3)⋊C4—181no—
225C4⋊Dic3—31no—
226C4⋊Dic3—31no—
227C4⋊Dic3—61no—
228C2×C3⋊C8—31no—
229C2×C3⋊C8—31no—
230C2×C3⋊C8—31no—
231C2×C3⋊C8—61no—
232C2×C3⋊C8—91no—
233D24—182no—
234C22×C32—21no—
235C22×C32D3×D411yescharacteristic
236C22×C32C2×#6711yescharacteristic
237C2×C3⋊D3—21no—
238C2×C3⋊D3—21no—
239C2×C3⋊D3—21no—
240C2×C3⋊D3—22no—
241C2×C3⋊D3—41no—
242C2×C3⋊D3—41no—
243C2×C3⋊D3—41no—
244C2×C3⋊D3—41no—
245C2×C3⋊D3—41no—
246C2×C3×D3—61no—
247C2×C3×D3—61no—
248C2×C3×D3—62no—
249C2×C3×D3—61no—
250C2×C3×D3—121no—
251C2×C3×D3—121no—
252C2×C3×D3—121no—
253C2×C3×D3—121no—
254C2×C3×D3—121no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—42no—
260C32×C4D3×D411yescharacteristic
261C32×C4C2×#6711yescharacteristic
262C32×C4C2×#6711yescharacteristic
263C32×C4C2×#6711yescharacteristic
264C3⋊Dic3—21no—
265C3×Dic3—61no—
266C3×Dic3—121no—
267C3×Dic3—121no—
268C3×Dic3—241no—
269C2×#41—91no—
270C2×SD16—91no—
271C2×D8—271no—
272#121—271no—
273#103—91no—
274#102—271no—
275C22⋊C8—271no—
276H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
277C23×C3—91no—
278C23×C3—181no—
279C23×C3—181no—
280C22×D3—31no—
281C22×D3—61no—
282C22×D3—61no—
283C22×D3—91no—
284C22×D3—181no—
285C22×D3—181no—
286C3×Q8—21no—
287C3×Q8—32no—
288C3×Q8—32no—
289C3×Q8—61no—
290C3×Q8—61no—
291C3×Q8—62no—
292C3×Q8—121no—
293C3×Q8#62512yes—
294C3×D4—92no—
295C3×D4—181no—
296C3×D4—181no—
297C3×D4—181no—
298C3×D4—184no—
299C2×C3×C4—21no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—91no—
306C2×C3×C4—121no—
307C2×C3×C4—181no—
308C2×C3×C4—181no—
309C2×C3×C4—184no—
310C2×C3×C4—181no—
311C2×C3×C4C2×D3211yescharacteristic
312#67—362no—
313#67—362no—
314C2×Dic3—31no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—91no—
318C2×Dic3—121no—
319D12—32no—
320D12—64no—
321D12—61no—
322D12—61no—
323D12—61no—
324D12—92no—
325D12—181no—
326D12—181no—
327D12—181no—
328D12—184no—
329D12—362no—
330C4×D3—64no—
331C4×D3—61no—
332C4×D3—61no—
333C4×D3—181no—
334C4×D3—181no—
335C4×D3—184no—
336C3×C8—181no—
337C3×C8—181no—
338C3⋊C8—61no—
339C3⋊C8—61no—
340C3⋊C8—61no—
341C3⋊C8—121no—
342C3⋊C8—181no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32#91411yescharacteristic
347C2×C32#91611yescharacteristic
348C2×C32#93711yescharacteristic
349C2×C32#94911yescharacteristic
350C2×C32#95011yescharacteristic
351C2×C32#95311yescharacteristic
352C3⋊D3—41no—
353C3⋊D3—42no—
354C3⋊D3—81no—
355C3×D3—121no—
356C3×D3—122no—
357C3×D3—241no—
358#41—182no—
359#41—182no—
360C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
361C2×D4—91no—
362C2×D4—91no—
363C2×D4—181no—
364C2×D4—541no—
365C22×C4—91no—
366C22×C4—181no—
367SD16—182no—
368D8—542no—
369C2×C8—91no—
370C2×C8—91no—
371C4⋊C4—91no—
372C22⋊C4—541no—
373C22×C3—31no—
374C22×C3—31no—
375C22×C3—61no—
376C22×C3—181no—
377C22×C3—182no—
378C22×C3—181no—
379C22×C3—181no—
380C22×C3—361no—
381C22×C3—361no—
382C22×C3—361no—
383C22×C3—361no—
384C22×C3—361no—
385C22×C3D3×#6711yescharacteristic
386C2×D3—61no—
387C2×D3—61no—
388C2×D3—62no—
389C2×D3—61no—
390C2×D3—121no—
391C2×D3—121no—
392C2×D3—121no—
393C2×D3—121no—
394C2×D3—121no—
395C2×D3—182no—
396C2×D3—181no—
397C2×D3—181no—
398C2×D3—181no—
399C2×D3—361no—
400C2×D3—361no—
401C2×D3—361no—
402C2×D3—361no—
403C2×D3—361no—
404C3×C4—22no—
405C3×C4—31no—
406C3×C4—31no—
407C3×C4—31no—
408C3×C4—31no—
409C3×C4—62no—
410C3×C4—62no—
411C3×C4—61no—
412C3×C4—61no—
413C3×C4—122no—
414C3×C4—181no—
415C3×C4—361no—
416C3×C4C2×#62511yescharacteristic
417C3×C4D3×#6711yescharacteristic
418Dic3—61no—
419Dic3—121no—
420Dic3—121no—
421Dic3—181no—
422Dic3—241no—
423C32—21no—
424C32#517211yescharacteristic, complemented
425C32#523811yescharacteristic, complemented
426C23—91no—
427C23—181no—
428C23—181no—
429Q8—21no—
430Q8#619812yescomplemented
431D4—92no—
432D4—184no—
433D4—181no—
434D4—181no—
435D4—181no—
436D4—1082no—
437C2×C4—21no—
438C2×C4—91no—
439C2×C4—184no—
440C2×C4—181no—
441C2×C4—181no—
442C2×C4—181no—
443C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
444C8—181no—
445C8—181no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—31no—
449C2×C3—31no—
450C2×C3—31no—
451C2×C3—31no—
452C2×C3—61no—
453C2×C3—61no—
454C2×C3—61no—
455C2×C3—361no—
456C2×C3—362no—
457C2×C3—721no—
458C2×C3#6393911yescharacteristic
459C2×C3#6394711yescharacteristic
460C2×C3#6397611yescharacteristic
461D3—121no—
462D3—122no—
463D3—241no—
464D3—361no—
465D3—362no—
466D3—721no—
467C22—181no—
468C22—181no—
469C22—182no—
470C22—181no—
471C22—361no—
472C22—361no—
473C22—361no—
474C22—361no—
475C22—361no—
476C22#8867611yescharacteristic
477C4—22no—
478C4—181no—
479C4—361no—
480C4C2×#619811yescharacteristic
481C4#8867611yescharacteristic
482C3—31no—
483C3—31no—
484C3—61no—
485C3#1059158611yescharacteristic
486C2—361no—
487C2—362no—
488C2—721no—
489C2#1172123311yescharacteristic
490C2#1172123911yescharacteristic
491C2#1172131611yescharacteristic
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×#3755⋊C3C2semidirect
3. #11721141C2semidirect
4. C2×#88432C2semidirect
5. #11720583C2semidirect
6. #11720317C2semidirect
7. #11720308C2semidirect
25. C2×#3832D3semidirect
27. C2×#3755D3semidirect
44. Q8×H(3,𝔽3)D4semidirect
53. C4⋊((C3⋊D3)⋊C3)D4semidirect
143. C32×Q8#67semidirect
144. C32×Q8#67semidirect
156. C4⋊(C3⋊D3)#67semidirect
276. H(3,𝔽3)#449semidirect
424. C32#5172semidirect
425. C32#5238semidirect
430. Q8#6198semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
C2×#3755⋊C32
#117211412
C2×#884322
#117205832
#117203172
#117203082
#105915933
#105915883