Finite Group Databasebeta

← #13140594

Subgroups — #13140594

Order 1728.

556 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 73 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
1#11721850C211yesmaximal, characteristic
2C2×#3754⋊C3C211yesmaximal, characteristic, complemented
3#11721084C211yesmaximal, characteristic, complemented
4C2×#88437C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720304C211yesmaximal, characteristic, complemented
8#10591439—31nomaximal
9#10591438—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12C2×#6235—21no—
13C2×C4⋊#272C2211yescharacteristic
14C2×#634⋊C3C2211yescharacteristic
15C4×#1084C2211yescharacteristic
16#3754⋊C3—22no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#88447—21no—
22#88437—22no—
23C2×#3754D311yescharacteristic, complemented
24C3×#940—31no—
25C3×#940—31no—
26C3×#940—61no—
27C2×C3×#214—31no—
28#63844—31no—
29Dic3×Dic6—31no—
30C2×#3716—31no—
31C2×#3715—31no—
32#63659D311yescharacteristic, complemented
33C3×#832—31no—
34Dic3⋊(C3⋊C8)—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63563—31no—
37#63562—31no—
38C23×H(3,𝔽3)—21no—
39C22×#272—21no—
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43#6235—42no—
44#6235—42no—
45C2×#1084D411yescharacteristic
46C2×#1084D411yescharacteristic
47C4⋊#272—21no—
48C4⋊#272—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3—21no—
51#634⋊C3—21no—
52#634⋊C3D412yescomplemented
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55#5163—91no—
56#5075—91no—
57#4762—91no—
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C2×D311yescharacteristic
61C2×#634C2×D311yescharacteristic
62C2×C3×#67—61no—
63C2×C3×#67—61no—
64C2×C3×#67—61no—
65C2×C3×#67—61no—
66C2×C3×#67—62no—
67C2×C3×#67—62no—
68C2×C3×D12—31no—
69C2×C3×D12—31no—
70C2×C3×D12—61no—
71C2×C3×Dic6—31no—
72#3754—22no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×#214—62no—
77C3×C4⋊Dic3—31no—
78C3×C4×Dic3—31no—
79C3×C4×Dic3—31no—
80C3×C4×Dic3—61no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83#3724—61no—
84Dic3⋊Dic3—61no—
85Dic32—61no—
86Dic32—61no—
87#3716—62no—
88#3715—62no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)—41no—
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C2×#272—41no—
94C2×#272—41no—
95C2×#272—41no—
96C4×H(3,𝔽3)C2×D411yescharacteristic
97C4×H(3,𝔽3)#4111yescharacteristic
98#1084—21no—
99#1084SD1612yescomplemented
100(C3⋊Dic3)⋊C3—22no—
101(C3⋊Dic3)⋊C3—41no—
102C2×C3×SD16—91no—
103C3×#127—91no—
104Q8×Dic3—91no—
105C2×#215—91no—
106#940—31no—
107#940—31no—
108#940—61no—
109C2×#214—31no—
110C2×#214—91no—
111C2×#214—91no—
112C2×#204—91no—
113#887—91no—
114C4×Dic6—91no—
115C3×#102—91no—
116#832—31no—
117#832—91no—
118#832—91no—
119#821—91no—
120Dic3⋊C8—91no—
121Dic3⋊C8—91no—
122#809—91no—
123C4⋊(C3⋊C8)—91no—
124C23×C32—22no—
125C23×C32—21no—
126C23×C32—21no—
127C22×C3×D3—62no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—21no—
134C32×D4—41no—
135C32×D4—41no—
136C2×C32×C4—21no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—21no—
141#634—21no—
142#634#6712yescomplemented
143C3×#67—122no—
144C3×#67—122no—
145C3×#67—124no—
146C3×#67—124no—
147C3×#67—122no—
148C3×#67—122no—
149C2×C3×Dic3—31no—
150C2×C3×Dic3—31no—
151C2×C3×Dic3—31no—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—61no—
157C3×D12—61no—
158C3×D12—61no—
159C3×D12—61no—
160C3×D12—61no—
161C3×D12—121no—
162C3×D12—121no—
163C3×Dic6—32no—
164C3×Dic6—61no—
165C3⋊(C3⋊C8)—21no—
166C3⋊(C3⋊C8)—21no—
167C3×C3⋊C8—61no—
168C3×C3⋊C8—61no—
169#460—271noSylow
170C2×H(3,𝔽3)—81no—
171C2×H(3,𝔽3)#12111yescharacteristic
172C2×H(3,𝔽3)C2×SD1611yescharacteristic
173C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
174#272—81no—
175C2×C3×Q8—91no—
176C2×C3×D4—31no—
177C2×C3×D4—31no—
178C2×C3×D4—61no—
179C2×C3×D4—91no—
180C2×C3×D4—181no—
181C2×C3×D4—181no—
182C2×C3×D4D3211yescharacteristic
183C2×#67—61no—
184C2×#67—61no—
185C2×#67—61no—
186C2×#67—61no—
187C2×#67—62no—
188C2×#67—62no—
189C2×D12—31no—
190C2×D12—31no—
191C2×D12—61no—
192C2×Dic6—31no—
193C2×Dic6—91no—
194C3×SD16—182no—
195C2×C3×C8—91no—
196C2×C3×C8—91no—
197C3×C4⋊C4—91no—
198C3×C42—91no—
199#215—182no—
200#214—62no—
201#214—182no—
202#214—182no—
203C4⋊Dic3—31no—
204C4⋊Dic3—91no—
205C4⋊Dic3—181no—
206Dic3⋊C4—181no—
207Dic3⋊C4—181no—
208C4×Dic3—31no—
209C4×Dic3—31no—
210C4×Dic3—61no—
211C4×Dic3—181no—
212C4×Dic3—181no—
213C4×Dic3—181no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—31no—
216C2×C3⋊C8—91no—
217C2×C3⋊C8—91no—
218#204—182no—
219C22×C32—21no—
220C22×C32—42no—
221C22×C32—42no—
222C22×C32—42no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32—41no—
229C22×C32D3×D411yescharacteristic
230C22×C32D3×D411yescharacteristic
231C2×C3×D3—122no—
232C2×C3×D3—122no—
233C2×C3×D3—122no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4#23511yescharacteristic
243C32×C4D3×D411yescharacteristic
244C32×C4#23711yescharacteristic
245C32×C4C2×#6711yescharacteristic
246C3⋊Dic3—22no—
247C3⋊Dic3—41no—
248C3×Dic3—32no—
249C3×Dic3—32no—
250C3×Dic3—62no—
251C3×Dic3—62no—
252C3×Dic3—62no—
253C3×Dic3—61no—
254C3×Dic3—61no—
255C3×Dic3—121no—
256C2×SD16—91no—
257C2×SD16—271no—
258#127—91no—
259C4×Q8—271no—
260C4⋊C8—271no—
261#102—91no—
262#102—271no—
263H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
264C23×C3—21no—
265C23×C3—61no—
266C23×C3—61no—
267C23×C3—62no—
268C23×C3—181no—
269C22×D3—61no—
270C22×D3—61no—
271C22×D3—62no—
272C3×Q8—92no—
273C3×Q8—181no—
274C3×D4—21no—
275C3×D4—21no—
276C3×D4—61no—
277C3×D4—61no—
278C3×D4—61no—
279C3×D4—61no—
280C3×D4—121no—
281C3×D4—121no—
282C3×D4—181no—
283C3×D4—181no—
284C3×D4—362no—
285C3×D4—362no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—61no—
289C2×C3×C4—91no—
290C2×C3×C4—91no—
291C2×C3×C4—181no—
292C2×C3×C4—181no—
293C2×C3×C4C2×D3211yescharacteristic
294#67—122no—
295#67—122no—
296#67—124no—
297#67—122no—
298#67—122no—
299#67—124no—
300C2×Dic3—31no—
301C2×Dic3—31no—
302C2×Dic3—31no—
303C2×Dic3—31no—
304C2×Dic3—61no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—181no—
309C2×Dic3—181no—
310D12—61no—
311D12—61no—
312D12—61no—
313D12—61no—
314D12—121no—
315D12—121no—
316Dic6—32no—
317Dic6—61no—
318Dic6—92no—
319Dic6—181no—
320C3×C8—181no—
321C3×C8—181no—
322C3⋊C8—61no—
323C3⋊C8—61no—
324C3⋊C8—181no—
325C3⋊C8—181no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—82no—
330C2×C32—81no—
331C2×C32—81no—
332C2×C32#88411yescharacteristic
333C2×C32#91111yescharacteristic
334C2×C32D3×SD1611yescharacteristic
335C2×C32D3×SD1611yescharacteristic
336C2×C32#91411yescharacteristic
337C2×C32#93911yescharacteristic
338C3×D3—242no—
339C3×D3—241no—
340C3×D3—241no—
341C2×Q8—91no—
342C2×D4—91no—
343C2×D4—181no—
344C2×D4—181no—
345C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
346SD16—182no—
347SD16—542no—
348C2×C8—91no—
349C2×C8—91no—
350C4⋊C4—91no—
351C4⋊C4—541no—
352C42—91no—
353C42—541no—
354C22×C3—31no—
355C22×C3—31no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—61no—
360C22×C3—121no—
361C22×C3—121no—
362C22×C3—121no—
363C22×C3—121no—
364C22×C3—121no—
365C22×C3—121no—
366C22×C3—122no—
367C22×C3—122no—
368C22×C3—122no—
369C22×C3—361no—
370C22×C3—361no—
371C22×C3—361no—
372C22×C3#381111yescharacteristic
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—121no—
376C2×D3—121no—
377C2×D3—121no—
378C2×D3—121no—
379C2×D3—122no—
380C2×D3—122no—
381C2×D3—122no—
382C3×C4—31no—
383C3×C4—31no—
384C3×C4—31no—
385C3×C4—31no—
386C3×C4—61no—
387C3×C4—61no—
388C3×C4—92no—
389C3×C4—182no—
390C3×C4—181no—
391C3×C4—361no—
392C3×C4#380411yescharacteristic
393C3×C4D3×#6711yescharacteristic
394Dic3—32no—
395Dic3—32no—
396Dic3—62no—
397Dic3—61no—
398Dic3—61no—
399Dic3—62no—
400Dic3—62no—
401Dic3—121no—
402Dic3—182no—
403Dic3—361no—
404C32—21no—
405C32#476211yescharacteristic, complemented
406C32#516311yescharacteristic, complemented
407C23—21no—
408C23—181no—
409Q8—92no—
410Q8—181no—
411D4—21no—
412D4—21no—
413D4—181no—
414D4—181no—
415D4—362no—
416D4—362no—
417C2×C4—91no—
418C2×C4—91no—
419C2×C4—181no—
420C2×C4—181no—
421C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
422C8—181no—
423C8—181no—
424C2×C3—31no—
425C2×C3—31no—
426C2×C3—31no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—61no—
431C2×C3—61no—
432C2×C3—61no—
433C2×C3—81no—
434C2×C3—241no—
435C2×C3—241no—
436C2×C3—242no—
437C2×C3—721no—
438C2×C3#6393011yescharacteristic
439C2×C3#6393711yescharacteristic
440C2×C3#6397111yescharacteristic
441D3—241no—
442D3—241no—
443D3—242no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—361no—
448C22—361no—
449C22—361no—
450C22#8867711yescharacteristic
451C4—92no—
452C4—182no—
453C4—181no—
454C4—361no—
455C4#8867111yescharacteristic
456C4#8867611yescharacteristic
457C3—31no—
458C3—31no—
459C3—61no—
460C3#1059144411yescharacteristic
461C2—81no—
462C2—721no—
463C2#1172118811yescharacteristic
464C2#1172119311yescharacteristic
465C2#1172131211yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721850C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#460
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
2. C2×#3754⋊C3C2semidirect
3. #11721084C2semidirect
4. C2×#88437C2semidirect
5. #63659⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720304C2semidirect
23. C2×#3754D3semidirect
32. #63659D3semidirect
52. #634⋊C3D4semidirect
99. #1084SD16semidirect
142. #634#67semidirect
263. H(3,𝔽3)#460semidirect
405. C32#4762semidirect
406. C32#5163semidirect

Maximal subgroups

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

subgroupindex
#117218502
C2×#3754⋊C32
#117210842
C2×#884372
#63659⋊C32
#117203242
#117203042
#105914393
#105914383