Finite Group Databasebeta

← #13140604

Subgroups — #13140604

Order 1728.

580 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 75 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×#88545C211yesmaximal, characteristic, complemented
2#64149⋊C3C211yesmaximal, characteristic, complemented
3C2×#3754⋊C3C211yesmaximal, characteristic, complemented
4#11721114C211yesmaximal, characteristic, complemented
5C2×#88436C211yesmaximal, characteristic, complemented
6C2×#88431C211yesmaximal, characteristic, complemented
7#11720312C211yesmaximal, characteristic, complemented
8#10591475—31nomaximal
9#10591469—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#638⋊C3—22no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×#634⋊C3C2211yescharacteristic
15#88545—22no—
16C2×#6180C2211yescharacteristic
17#3754⋊C3—22no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88451—22no—
21#88436—22no—
22#88431—22no—
23#64149D311yescharacteristic, complemented
24C2×#3754D311yescharacteristic, complemented
25C3×#940—31no—
26C2×C3×#214—31no—
27C2×C3×#213—31no—
28C2×C3×#213—31no—
29C2×C3×#213—61no—
30#63863—31no—
31#63860—31no—
32C2×#3717—31no—
33C2×#3716—31no—
34C2×#3714—31no—
35C2×#3713—31no—
36#63555—31no—
37#63554—31no—
38C23×H(3,𝔽3)—21no—
39C22×#272—21no—
40C22×(C3⋊D3)⋊C3—21no—
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C4⋊#272—21no—
45C4⋊#272—21no—
46#638⋊C3—44no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3D412yes—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#634⋊C3—21no—
52#634⋊C3—21no—
53#6180D412yescomplemented
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55#5176—91no—
56#5154—91no—
57#5074—91no—
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—22no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×#634C2×D311yescharacteristic
64C2×C3×#67—62no—
65C2×C3×D12—31no—
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×Dic6—31no—
70#3754—22no—
71C4×C3⋊Dic3C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×#214—62no—
74C3×#213—62no—
75C3×#213—62no—
76C3×#213—64no—
77C3×C4×Dic3—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—61no—
82(C2×D3)⋊Dic3—62no—
83(C2×D3)⋊Dic3—62no—
84#3717—62no—
85#3716—62no—
86#3714—62no—
87#3713—62no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—42no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×#272—41no—
92C2×#272—42no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—42no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)C2×D411yescharacteristic
97(C3⋊Dic3)⋊C3—22no—
98(C3⋊Dic3)⋊C3—41no—
99C2×C3×SD16—91no—
100C2×C3×D8—91no—
101C3×#127—91no—
102#947—91no—
103C2×#215—91no—
104#940—31no—
105#940—91no—
106#936—91no—
107C2×#214—31no—
108C2×#214—91no—
109C2×#214—91no—
110C2×#213—31no—
111C2×#213—31no—
112C2×#213—61no—
113C2×#213—91no—
114C2×#213—91no—
115C2×D24—91no—
116C2×#204—91no—
117#875—91no—
118#815—91no—
119#815—91no—
120#803—91no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—41no—
124C22×C3⋊D3—21no—
125C22×C3×D3—61no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C22×C3×D3—121no—
129C32×D4—22no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—21no—
134C32×D4—41no—
135C2×C32×C4—21no—
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138#638—44no—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3#6712yes—
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)—21no—
143#634—21no—
144#634—21no—
145C3×#67—124no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—61no—
148C3×D12—62no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—61no—
154C3×D12—61no—
155C3×D12—121no—
156C3×Dic6—61no—
157C3×Dic6—61no—
158C3⋊(C3⋊C8)#6712yescomplemented
159C3×C3⋊C8—32no—
160C3×C3⋊C8—32no—
161C3×C3⋊C8—32no—
162C3×C3⋊C8—62no—
163#496—271noSylow
164C2×H(3,𝔽3)—81no—
165C2×H(3,𝔽3)#12711yescharacteristic
166C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
167#272—81no—
168(C3⋊D3)⋊C3—81no—
169C2×C3×Q8—91no—
170C2×C3×D4—31no—
171C2×C3×D4—31no—
172C2×C3×D4—61no—
173C2×C3×D4—91no—
174C2×C3×D4—91no—
175C2×C3×D4—182no—
176C2×C3×D4D3211yescharacteristic
177C2×#67—62no—
178C2×#67—182no—
179C2×D12—31no—
180C2×D12—31no—
181C2×D12—31no—
182C2×D12—61no—
183C2×D12—91no—
184C2×Dic6—31no—
185C2×Dic6—91no—
186C3×SD16—182no—
187C3×D8—182no—
188C2×C3×C8—91no—
189C2×C3×C8—91no—
190C3×C42—91no—
191#217—182no—
192#215—182no—
193#214—62no—
194#214—182no—
195#214—182no—
196#213—62no—
197#213—62no—
198#213—64no—
199#213—182no—
200#213—182no—
201(C2×D3)⋊C4—182no—
202(C2×D3)⋊C4—182no—
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—
211#204—182no—
212C22×C32—21no—
213C22×C32—42no—
214C22×C32—42no—
215C22×C32—42no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—82no—
219C22×C32D3×D411yescharacteristic
220C22×C32C2×#6711yescharacteristic
221C2×C3⋊D3—42no—
222C2×C3⋊D3—41no—
223C2×C3×D3—122no—
224C2×C3×D3—122no—
225C2×C3×D3—122no—
226C2×C3×D3—122no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—242no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4D3×D411yescharacteristic
234C32×C4D3×D411yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4C2×#6711yescharacteristic
237C3⋊Dic3—22no—
238C3⋊Dic3—41no—
239C3×Dic3—62no—
240C3×Dic3—121no—
241C2×SD16—91no—
242C2×SD16—271no—
243C2×D8—91no—
244C2×D8—271no—
245#127—91no—
246#124—271no—
247#97—271no—
248H(3,𝔽3)#49611yescharacteristic, complemented, Sylow
249C23×C3—21no—
250C23×C3—61no—
251C23×C3—61no—
252C23×C3—121no—
253C23×C3—181no—
254C23×C3—181no—
255C22×D3—61no—
256C22×D3—61no—
257C22×D3—61no—
258C22×D3—121no—
259C22×D3—181no—
260C3×Q8—181no—
261C3×Q8—181no—
262C3×D4—21no—
263C3×D4—21no—
264C3×D4—61no—
265C3×D4—61no—
266C3×D4—61no—
267C3×D4—61no—
268C3×D4—62no—
269C3×D4—121no—
270C3×D4—181no—
271C3×D4—181no—
272C3×D4—181no—
273C3×D4—181no—
274C3×D4—364no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—61no—
278C2×C3×C4—92no—
279C2×C3×C4—181no—
280C2×C3×C4C2×D3211yescharacteristic
281#67—124no—
282#67—364no—
283C2×Dic3—32no—
284C2×Dic3—61no—
285C2×Dic3—92no—
286C2×Dic3—181no—
287D12—61no—
288D12—61no—
289D12—61no—
290D12—61no—
291D12—62no—
292D12—61no—
293D12—61no—
294D12—121no—
295D12—181no—
296D12—181no—
297Dic6—61no—
298Dic6—61no—
299Dic6—181no—
300Dic6—181no—
301C3×C8—92no—
302C3×C8—92no—
303C3⋊C8—32no—
304C3⋊C8—32no—
305C3⋊C8—32no—
306C3⋊C8—62no—
307C3⋊C8—92no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—82no—
311C2×C32—81no—
312C2×C32—81no—
313C2×C32#91412yes—
314C2×C32#93212yes—
315C2×C32#94011yescharacteristic
316C2×C32#94011yescharacteristic
317C3⋊D3—81no—
318C3×D3—242no—
319C3×D3—241no—
320C3×D3—241no—
321C3×D3—241no—
322C2×Q8—91no—
323C2×D4—91no—
324C2×D4—91no—
325C2×D4—182no—
326C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
327SD16—182no—
328SD16—542no—
329D8—182no—
330D8—542no—
331C2×C8—91no—
332C2×C8—91no—
333C22⋊C4—542no—
334C42—91no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—42no—
338C22×C3—41no—
339C22×C3—61no—
340C22×C3—122no—
341C22×C3—121no—
342C22×C3—121no—
343C22×C3—122no—
344C22×C3—122no—
345C22×C3—242no—
346C22×C3—361no—
347C22×C3—362no—
348C22×C3—362no—
349C22×C3—361no—
350C22×C3D3×#6711yescharacteristic
351C2×D3—121no—
352C2×D3—121no—
353C2×D3—122no—
354C2×D3—122no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—121no—
358C2×D3—242no—
359C2×D3—361no—
360C2×D3—362no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—31no—
365C3×C4—61no—
366C3×C4—61no—
367C3×C4—182no—
368C3×C4—361no—
369C3×C4D3×#6711yescharacteristic
370C3×C4#381111yescharacteristic
371Dic3—62no—
372Dic3—121no—
373Dic3—182no—
374Dic3—361no—
375C32—21no—
376C32#507411yescharacteristic, complemented
377C32#517611yescharacteristic, complemented
378C23—21no—
379C23—181no—
380C23—181no—
381Q8—181no—
382Q8—181no—
383D4—21no—
384D4—21no—
385D4—181no—
386D4—181no—
387D4—181no—
388D4—181no—
389D4—364no—
390C2×C4—92no—
391C2×C4—181no—
392C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
393C8—92no—
394C8—92no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—32no—
398C2×C3—32no—
399C2×C3—61no—
400C2×C3—62no—
401C2×C3—81no—
402C2×C3—241no—
403C2×C3—241no—
404C2×C3—242no—
405C2×C3—721no—
406C2×C3—721no—
407C2×C3#6393412yes—
408C2×C3#6397911yescharacteristic
409D3—241no—
410D3—241no—
411D3—242no—
412D3—241no—
413D3—721no—
414C22—41no—
415C22—42no—
416C22—361no—
417C22—361no—
418C22—362no—
419C22—362no—
420C22#8867611yescharacteristic
421C4—182no—
422C4—361no—
423C4#8867611yescharacteristic
424C4#8867711yescharacteristic
425C3—31no—
426C3—31no—
427C3—61no—
428C3#1059147211yescharacteristic
429C2—81no—
430C2—721no—
431C2—721no—
432C2#1172119612yes—
433C2#1172131711yescharacteristic
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)#496
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×#88545C2semidirect
2. #64149⋊C3C2semidirect
3. C2×#3754⋊C3C2semidirect
4. #11721114C2semidirect
5. C2×#88436C2semidirect
6. C2×#88431C2semidirect
7. #11720312C2semidirect
23. #64149D3semidirect
24. C2×#3754D3semidirect
53. #6180D4semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
158. C3⋊(C3⋊C8)#67semidirect
248. H(3,𝔽3)#496semidirect
376. C32#5074semidirect
377. C32#5176semidirect

Maximal subgroups

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

subgroupindex
C2×#885452
#64149⋊C32
C2×#3754⋊C32
#117211142
C2×#884362
C2×#884312
#117203122
#105914753
#105914693