Finite Group Databasebeta

← #13140958

Subgroups — #13140958

Order 1728.

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

normal subgroups only

This group has 502 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#64156⋊C3C211yesmaximal, characteristic, complemented
3#11721149C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10591570—31nomaximal
9#10591567—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#6199—21no—
14#88549—22no—
15C2×#6180C2211yescharacteristic
16#3752⋊C3—21no—
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88451—21no—
22#88436—22no—
23#64156D311yescharacteristic, complemented
24C3×#947—31no—
25C2×C3×#215—31no—
26C2×C3×#215—31no—
27C2×C3×#215—61no—
28#63898—31no—
29Dic3×D12—31no—
30C2×#3714—31no—
31C2×#3713—31no—
32#63662D311yescharacteristic, complemented
33C3×#835—31no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63564—31no—
37#63563—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3—21no—
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⋊#272D412yes—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#6199—42no—
52#6180—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5165—91no—
55#5057—91no—
56#4819—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×D12—61no—
65C2×D3×Dic3—61no—
66C2×D3×Dic3—61no—
67#3752—21no—
68#3752—21no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×#215—62no—
73C3×#215—62no—
74C3×#215—64no—
75C3×(C2×D3)⋊C4—61no—
76C3×(C2×D3)⋊C4—61no—
77C3×C4⋊Dic3—31no—
78C3×C4×Dic3—31no—
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#3714—62no—
86#3713—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—22no—
89C2×#272—41no—
90C2×#272—41no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
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—21no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×SD16—91no—
101C3×#124—91no—
102#947—31no—
103#947—91no—
104C2×#215—31no—
105C2×#215—31no—
106C2×#215—61no—
107D4×Dic3—91no—
108C2×#213—91no—
109C2×#213—91no—
110C2×D24—91no—
111#893—91no—
112C4×D12—91no—
113C3×#103—91no—
114#835—31no—
115#835—91no—
116#832—91no—
117#821—91no—
118Dic3⋊C8—91no—
119Dic3⋊C8—91no—
120#809—91no—
121C4⋊(C3⋊C8)—91no—
122C22×C3⋊D3—21no—
123C22×C3×D3—61no—
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#6711yescharacteristic
141C2×C3⋊Dic3#6711yescharacteristic
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)—21no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—31no—
146C2×C3×Dic3—61no—
147C3×D12—32no—
148C3×D12—32no—
149C3×D12—62no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—61no—
154C3×D12—121no—
155D3×Dic3—122no—
156D3×Dic3—122no—
157C3⋊(C3⋊C8)—21no—
158C3×C3⋊C8—62no—
159C3×C3⋊C8—61no—
160C3×C3⋊C8—61no—
161C3×C3⋊C8—61no—
162#463—271noSylow
163C2×H(3,𝔽3)#12111yescharacteristic
164C2×H(3,𝔽3)#13511yescharacteristic
165C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
166#272—42no—
167(C3⋊D3)⋊C3—81no—
168C2×C3×Q8—31no—
169C2×C3×Q8—31no—
170C2×C3×Q8—61no—
171C2×C3×Q8D3211yescharacteristic
172C2×C3×D4—91no—
173C2×C3×D4—91no—
174C22×Dic3—181no—
175C2×D12—31no—
176C2×D12—31no—
177C2×D12—31no—
178C2×D12—61no—
179C2×D12—91no—
180C2×C4×D3—181no—
181C2×C4×D3—181no—
182C3×SD16—182no—
183C2×C3×C8—91no—
184C2×C3×C8—91no—
185C3×C4⋊C4—91no—
186C3×C22⋊C4—181no—
187C3×C22⋊C4—181no—
188C3×C42—91no—
189#217—181no—
190#215—62no—
191#215—62no—
192#215—64no—
193#213—182no—
194#213—182no—
195(C2×D3)⋊C4—61no—
196(C2×D3)⋊C4—61no—
197(C2×D3)⋊C4—181no—
198(C2×D3)⋊C4—181no—
199(C2×D3)⋊C4—181no—
200(C2×D3)⋊C4—181no—
201C4⋊Dic3—31no—
202C4⋊Dic3—91no—
203C4×Dic3—31no—
204C4×Dic3—91no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—61no—
209C2×C3⋊C8—91no—
210D24—182no—
211C22×C32—21no—
212C22×C32D3×D411yescharacteristic
213C22×C32C2×#6711yescharacteristic
214C2×C3⋊D3—41no—
215C2×C3⋊D3—41no—
216C2×C3⋊D3—41no—
217C2×C3×D3—62no—
218C2×C3×D3—62no—
219C2×C3×D3—122no—
220C2×C3×D3—121no—
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—21no—
240C3⋊Dic3—41no—
241C3×Dic3—61no—
242C3×Dic3—61no—
243C3×Dic3—121no—
244C2×SD16—91no—
245C2×D8—271no—
246#124—91no—
247C4×D4—271no—
248C4⋊C8—271no—
249#103—91no—
250#102—271no—
251H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
252C23×C3—181no—
253C23×C3—181no—
254C22×D3—61no—
255C22×D3—61no—
256C22×D3—61no—
257C22×D3—121no—
258C22×D3—181no—
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—92no—
268C3×D4—181no—
269C3×D4—181no—
270C3×D4—181no—
271C2×C3×C4—21no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—91no—
278C2×C3×C4—91no—
279C2×C3×C4—121no—
280C2×C3×C4—181no—
281C2×C3×C4C2×D3211yescharacteristic
282C2×Dic3—31no—
283C2×Dic3—31no—
284C2×Dic3—61no—
285C2×Dic3—91no—
286C2×Dic3—91no—
287C2×Dic3—181no—
288C2×Dic3—362no—
289D12—32no—
290D12—32no—
291D12—62no—
292D12—61no—
293D12—61no—
294D12—61no—
295D12—61no—
296D12—121no—
297D12—181no—
298D12—181no—
299C4×D3—362no—
300C4×D3—362no—
301C3×C8—181no—
302C3×C8—181no—
303C3⋊C8—61no—
304C3⋊C8—61no—
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#91911yescharacteristic
314C2×C32#93811yescharacteristic
315C2×C32#94911yescharacteristic
316C2×C32#95011yescharacteristic
317C3⋊D3—81no—
318C3×D3—122no—
319C3×D3—122no—
320C3×D3—242no—
321C3×D3—241no—
322C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
323C2×D4—91no—
324C2×D4—91no—
325C22×C4—541no—
326SD16—182no—
327D8—542no—
328C2×C8—91no—
329C2×C8—91no—
330C4⋊C4—91no—
331C22⋊C4—181no—
332C22⋊C4—181no—
333C22⋊C4—541no—
334C42—91no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—61no—
338C22×C3—182no—
339C22×C3—361no—
340C22×C3—361no—
341C22×C3—361no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3D3×#6711yescharacteristic
345C2×D3—62no—
346C2×D3—62no—
347C2×D3—122no—
348C2×D3—121no—
349C2×D3—121no—
350C2×D3—121no—
351C2×D3—121no—
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—241no—
356C2×D3—241no—
357C2×D3—361no—
358C2×D3—361no—
359C2×D3—361no—
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—181no—
372C3×C4—361no—
373C3×C4#380411yescharacteristic
374C3×C4#381111yescharacteristic
375Dic3—61no—
376Dic3—61no—
377Dic3—121no—
378Dic3—181no—
379Dic3—181no—
380Dic3—361no—
381C32—21no—
382C32#481911yescharacteristic, complemented
383C32#505711yescharacteristic, complemented
384C23—181no—
385C23—181no—
386Q8—21no—
387Q8—21no—
388D4—92no—
389D4—181no—
390D4—181no—
391D4—181no—
392C2×C4—21no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—181no—
396C2×C4—1082no—
397C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
398C8—181no—
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—362no—
410C2×C3—721no—
411C2×C3#6393911yescharacteristic
412C2×C3#6394911yescharacteristic
413C2×C3#6395811yescharacteristic
414D3—122no—
415D3—122no—
416D3—242no—
417D3—241no—
418D3—721no—
419C22—182no—
420C22—361no—
421C22—361no—
422C22—361no—
423C22—361no—
424C22—361no—
425C22#8867611yescharacteristic
426C4—41no—
427C4—181no—
428C4—181no—
429C4—361no—
430C4#8867111yescharacteristic
431C4#8867711yescharacteristic
432C3—31no—
433C3—31no—
434C3—61no—
435C3#1059156811yescharacteristic
436C2—362no—
437C2—721no—
438C2#1172123311yescharacteristic
439C2#1172123511yescharacteristic
440C2#1172130311yescharacteristic
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 radicalC2×#88436C2
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#463
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. #64156⋊C3C2semidirect
3. #11721149C2semidirect
5. #63662⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720311C2semidirect
23. #64156D3semidirect
32. #63662D3semidirect
251. H(3,𝔽3)#463semidirect
382. C32#4819semidirect
383. C32#5057semidirect

Maximal subgroups

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

subgroupindex
C2×#885492
#64156⋊C32
#117211492
C2×#884362
#63662⋊C32
#117203182
#117203112
#105915703
#105915673