Finite Group Databasebeta

← #13140965

Subgroups — #13140965

Order 1728.

488 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 71 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×#88549C211yesmaximal, characteristic, complemented
2#64155⋊C3C211yesmaximal, characteristic, complemented
3#11721156C211yesmaximal, characteristic
4C2×#88430C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720315C211yesmaximal, characteristic, complemented
8#10591590—31nomaximal
9#10591587—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#6200—21no—
14#88549—22no—
15C2×#6180C2211yescharacteristic
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3750⋊C3—21no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88451—21no—
22#88430—22no—
23#64155D311yescharacteristic, complemented
24C3×#946—31no—
25C2×C3×#215—31no—
26C2×C3×#215—31no—
27C2×C3×#215—61no—
28#63916—31no—
29#63908—31no—
30C2×#3712—31no—
31C2×#3711—31no—
32#63662D311yescharacteristic, complemented
33C3×#835—31no—
34#63575—31no—
35#63575—31no—
36#63563—31no—
37#63562—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C4⋊#272—21no—
45C4⋊#272—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4×(C3⋊D3)⋊C3—21no—
50C4×(C3⋊D3)⋊C3—21no—
51#6200—42no—
52#6180D412yescomplemented
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5174—91no—
55#5049—91no—
56#4812—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C2×D311yescharacteristic
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—61no—
64C2×C3×C4×D3—31no—
65C2×#626—61no—
66C2×#625—61no—
67#3752—21no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70#3750—21no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×#215—62no—
73C3×#215—62no—
74C3×#215—64no—
75C3×(C2×D3)⋊C4—61no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×Dic3⋊C4—61no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—61no—
83(C2×D3)⋊Dic3—61no—
84(C2×D3)⋊Dic3—61no—
85#3712—62no—
86#3711—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—41no—
89C2×#272—41no—
90C2×#272—41no—
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—21no—
94C4×H(3,𝔽3)—41no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×SD16—91no—
101C3×C22⋊Q8—91no—
102#946—31no—
103#946—91no—
104C2×#215—31no—
105C2×#215—31no—
106C2×#215—61no—
107#938—91no—
108#930—91no—
109C2×#208—91no—
110C2×ΓL(1,𝔽25)—91no—
111C2×ΓL(1,𝔽25)—91no—
112#895—91no—
113C3×#103—91no—
114#835—31no—
115#835—91no—
116#832—91no—
117#821—91no—
118#818—91no—
119#809—91no—
120#807—91no—
121#807—91no—
122C22×C3⋊D3#6711yescharacteristic
123C22×C3×D3—31no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C22×C3×D3—121no—
127C32×Q8—22no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—41no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—41no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3#6711yescharacteristic
142C4×C3⋊D3—21no—
143C4×C3⋊D3—21no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C3×D12—62no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—121no—
153C3×C4×D3—61no—
154C3×C4×D3—61no—
155#626—122no—
156#625—122no—
157C3⋊(C3⋊C8)—21no—
158C3×C3⋊C8—32no—
159C3×C3⋊C8—32no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—61no—
162#468—271noSylow
163C2×H(3,𝔽3)#12111yescharacteristic
164C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
165C2×H(3,𝔽3)#13711yescharacteristic
166#272—81no—
167(C3⋊D3)⋊C3—41no—
168C2×C3×Q8—31no—
169C2×C3×Q8—31no—
170C2×C3×Q8—61no—
171C2×C3×Q8D3211yescharacteristic
172C2×C3×D4—91no—
173C22×C3×C4—91no—
174C2×#67—181no—
175C2×#67—181no—
176C2×D12—31no—
177C2×D12—31no—
178C2×D12—61no—
179C2×D12—181no—
180C2×C4×D3—31no—
181C2×C4×D3—91no—
182C3×SD16—182no—
183C2×C3×C8—91no—
184C2×C3×C8—91no—
185C3×C4⋊C4—91no—
186C3×C4⋊C4—91no—
187C3×C4⋊C4—181no—
188C3×C22⋊C4—181no—
189#217—181no—
190#215—62no—
191#215—62no—
192#215—64no—
193(C2×D3)⋊C4—61no—
194(C2×D3)⋊C4—181no—
195(C2×D3)⋊C4—181no—
196(C2×D3)⋊C4—181no—
197C4⋊Dic3—31no—
198C4⋊Dic3—31no—
199C4⋊Dic3—91no—
200C4⋊Dic3—91no—
201Dic3⋊C4—61no—
202Dic3⋊C4—181no—
203#208—182no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—61no—
208C2×C3⋊C8—91no—
209ΓL(1,𝔽25)—182no—
210ΓL(1,𝔽25)—182no—
211C22×C32—21no—
212C22×C32D3×D411yescharacteristic
213C22×C32C2×#6711yescharacteristic
214C2×C3⋊D3—21no—
215C2×C3⋊D3—21no—
216C2×C3⋊D3—21no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—122no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C2×C3×D3—241no—
228C2×C3×D3—241no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—42no—
232C32×C4—41no—
233C32×C4—41no—
234C32×C4#23511yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4D3×D411yescharacteristic
237C32×C4#23711yescharacteristic
238C3⋊Dic3—21no—
239C3⋊Dic3—41no—
240C3⋊Dic3—41no—
241C3×Dic3—61no—
242C3×Dic3—121no—
243C3×Dic3—121no—
244C2×SD16—91no—
245C2×M16—271no—
246C22⋊Q8—91no—
247#121—271no—
248#107—271no—
249#103—91no—
250#102—271no—
251H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
252C23×C3—91no—
253C23×C3—181no—
254C22×D3—31no—
255C22×D3—61no—
256C22×D3—61no—
257C22×D3—91no—
258C22×D3—121no—
259C3×Q8—21no—
260C3×Q8—21no—
261C3×Q8—61no—
262C3×Q8—61no—
263C3×Q8—61no—
264C3×Q8—61no—
265C3×Q8—62no—
266C3×Q8—121no—
267C3×D4—181no—
268C3×D4—181no—
269C2×C3×C4—21no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—91no—
276C2×C3×C4—121no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4—181no—
281C2×C3×C4C2×D3211yescharacteristic
282#67—362no—
283#67—362no—
284C2×Dic3—31no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—91no—
288C2×Dic3—181no—
289C2×Dic3—181no—
290D12—61no—
291D12—61no—
292D12—61no—
293D12—61no—
294D12—62no—
295D12—121no—
296D12—362no—
297C4×D3—61no—
298C4×D3—61no—
299C4×D3—181no—
300C4×D3—181no—
301C3×C8—92no—
302C3×C8—181no—
303C3⋊C8—32no—
304C3⋊C8—32no—
305C3⋊C8—62no—
306C3⋊C8—61no—
307C3⋊C8—181no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32#88411yescharacteristic
312C2×C32#91411yescharacteristic
313C2×C32#91811yescharacteristic
314C2×C32#93911yescharacteristic
315C2×C32#94211yescharacteristic
316C2×C32#94911yescharacteristic
317C3⋊D3—41no—
318C3×D3—121no—
319C3×D3—242no—
320C3×D3—241no—
321C3×D3—241no—
322C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
323C2×D4—91no—
324C2×D4—541no—
325C22×C4—91no—
326SD16—182no—
327M16—542no—
328C2×C8—91no—
329C2×C8—91no—
330C4⋊C4—91no—
331C4⋊C4—91no—
332C4⋊C4—181no—
333C22⋊C4—181no—
334C22⋊C4—541no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—61no—
338C22×C3—181no—
339C22×C3—181no—
340C22×C3—181no—
341C22×C3—361no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3D3×#6711yescharacteristic
345C2×D3—61no—
346C2×D3—61no—
347C2×D3—61no—
348C2×D3—121no—
349C2×D3—121no—
350C2×D3—121no—
351C2×D3—121no—
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—122no—
355C2×D3—181no—
356C2×D3—181no—
357C2×D3—181no—
358C2×D3—241no—
359C2×D3—241no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—41no—
365C3×C4—61no—
366C3×C4—61no—
367C3×C4—121no—
368C3×C4—121no—
369C3×C4—122no—
370C3×C4—181no—
371C3×C4—361no—
372C3×C4—361no—
373C3×C4#380411yescharacteristic
374C3×C4#381111yescharacteristic
375Dic3—61no—
376Dic3—121no—
377Dic3—121no—
378Dic3—181no—
379Dic3—361no—
380Dic3—361no—
381C32—21no—
382C32#481211yescharacteristic, complemented
383C32#504911yescharacteristic, complemented
384C23—91no—
385C23—181no—
386Q8—21no—
387Q8—21no—
388D4—181no—
389D4—181no—
390D4—1082no—
391C2×C4—21no—
392C2×C4—91no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
398C8—92no—
399C8—181no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—61no—
407C2×C3—61no—
408C2×C3—61no—
409C2×C3—361no—
410C2×C3—721no—
411C2×C3#6393911yescharacteristic
412C2×C3#6394811yescharacteristic
413C2×C3#6397111yescharacteristic
414D3—121no—
415D3—241no—
416D3—241no—
417D3—242no—
418D3—361no—
419C22—181no—
420C22—181no—
421C22—181no—
422C22—361no—
423C22—361no—
424C22—361no—
425C22#8867611yescharacteristic
426C4—41no—
427C4—181no—
428C4—361no—
429C4—361no—
430C4#8867111yescharacteristic
431C4#8867711yescharacteristic
432C3—31no—
433C3—31no—
434C3—61no—
435C3#1059158911yescharacteristic
436C2—361no—
437C2—721no—
438C2#1172123311yescharacteristic
439C2#1172123411yescharacteristic
440C2#1172131211yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#11721156C2
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#468
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×#88549C2semidirect
2. #64155⋊C3C2semidirect
4. C2×#88430C2semidirect
5. #63662⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720315C2semidirect
23. #64155D3semidirect
32. #63662D3semidirect
52. #6180D4semidirect
251. H(3,𝔽3)#468semidirect
382. C32#4812semidirect
383. C32#5049semidirect

Maximal subgroups

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

subgroupindex
C2×#885492
#64155⋊C32
#117211562
C2×#884302
#63662⋊C32
#117203242
#117203152
#105915903
#105915873