Finite Group Databasebeta

← #13138981

Subgroups — #13138981

Order 1728.

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

normal subgroups only

This group has 516 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×#88528C211yesmaximal, characteristic, complemented
2#11721158C211yesmaximal, characteristic
3#11721083C211yesmaximal, characteristic, complemented
4C2×#88438C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#11720324C211yesmaximal, characteristic, complemented
8#10590876—31nomaximal
9#10590799—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#6200—21no—
14C2×#6200—21no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#88528—22no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88449—21no—
21#88447—21no—
22#88438—22no—
23C2×C3×#204—31no—
24C2×C3×#204—31no—
25C2×C3×#204—61no—
26#63914—31no—
27#63911—31no—
28#63843—31no—
29Dic3×Dic6—31no—
30C2×#3717—31no—
31C2×#3715—31no—
32#63648D311yescharacteristic, complemented
33#63641D311yescharacteristic, complemented
34C3×#821—31no—
35C3×Dic3⋊C8—31no—
36#63563—31no—
37#63562—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C4⋊#272—21no—
43C4⋊#272—21no—
44#6229—21no—
45#6229D412yescomplemented
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#6200—42no—
52#6200—42no—
53C8×H(3,𝔽3)—21no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55#5163—91no—
56#4848—91no—
57#4762—91no—
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×C3×D12—31no—
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—61no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—61no—
66C2×#626—61no—
67C2×#626—61no—
68C2×#625—61no—
69C2×#625—61no—
70C2×C32×C8—21no—
71C2×C32×C8C2×D311yescharacteristic
72C2×C32×C8C2×D311yescharacteristic
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×C4⋊Dic3—31no—
77C3×C4×Dic3—31no—
78C2×C3×C3⋊C8—31no—
79C3×#204—62no—
80C3×#204—62no—
81C3×#204—64no—
82Dic3⋊Dic3—61no—
83Dic3⋊Dic3—61no—
84Dic32—61no—
85Dic32—61no—
86#3717—62no—
87#3715—62no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×#272—41no—
90C2×#272—41no—
91C2×#272—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97#1084—22no—
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3—41no—
100(C3⋊Dic3)⋊C3SD1612yescomplemented
101C2×C3×SD16—91no—
102Q8×Dic3—91no—
103C2×#215—91no—
104#940—91no—
105#940—91no—
106C2×#214—91no—
107C2×#204—31no—
108C2×#204—31no—
109C2×#204—61no—
110C2×#204—91no—
111#887—91no—
112#874—91no—
113C4×Dic6—91no—
114C3×C4⋊C8—91no—
115C3×#102—91no—
116#832—91no—
117#821—31no—
118#821—91no—
119#821—91no—
120Dic3⋊C8—31no—
121Dic3⋊C8—91no—
122#809—91no—
123C22×C3⋊D3—21no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C22×C3×D3—61no—
127C22×C3×D3—121no—
128C2×C32×C4—21no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3D1211yescharacteristic
133C2×C3⋊Dic3D1211yescharacteristic
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—21no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—121no—
142C3×D12—62no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—121no—
150C3×Dic6—32no—
151C3×Dic6—32no—
152C3×Dic6—62no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C3×Dic6—121no—
156#626—122no—
157#626—122no—
158#625—122no—
159#625—122no—
160C32×C8—22no—
161C32×C8—21no—
162C32×C8—21no—
163C3⋊(C3⋊C8)—21no—
164C3×C3⋊C8—61no—
165#460—271noSylow
166C2×H(3,𝔽3)#12111yescharacteristic
167C2×H(3,𝔽3)C2×SD1611yescharacteristic
168C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
169#272—81no—
170(C3⋊D3)⋊C3—81no—
171C2×C3×Q8—91no—
172C2×C3×D4—91no—
173C2×C3×D4—91no—
174C2×#67—181no—
175C2×#67—181no—
176C2×#67—181no—
177C2×#67—181no—
178C2×D12—31no—
179C2×D12—31no—
180C2×D12—31no—
181C2×D12—61no—
182C2×D12—91no—
183C2×D12—181no—
184C2×D12—181no—
185C2×Dic6—31no—
186C2×Dic6—31no—
187C2×Dic6—61no—
188C3×SD16—182no—
189C2×C3×C8—31no—
190C2×C3×C8—31no—
191C2×C3×C8—61no—
192C2×C3×C8—91no—
193C2×C3×C8D3211yescharacteristic
194C3×C4⋊C4—91no—
195C3×C42—91no—
196#215—182no—
197#214—182no—
198C4⋊Dic3—31no—
199C4⋊Dic3—91no—
200C4⋊Dic3—181no—
201Dic3⋊C4—181no—
202Dic3⋊C4—181no—
203C4×Dic3—31no—
204C4×Dic3—91no—
205C4×Dic3—181no—
206C4×Dic3—181no—
207C4×Dic3—181no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—91no—
210#204—62no—
211#204—62no—
212#204—64no—
213#204—182no—
214C22×C32—21no—
215C22×C32C2×D1211yescharacteristic
216C22×C32D3×D411yescharacteristic
217C2×C3⋊D3—41no—
218C2×C3⋊D3—41no—
219C2×C3⋊D3—41no—
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—121no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—241no—
231C2×C3×D3—241no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4#23511yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4D3×D411yescharacteristic
237C32×C4#23911yescharacteristic
238C3⋊Dic3—21no—
239C3⋊Dic3—41no—
240C3⋊Dic3#20412yescomplemented
241C3×Dic3—32no—
242C3×Dic3—62no—
243C3×Dic3—62no—
244C3×Dic3—61no—
245C3×Dic3—122no—
246C3×Dic3—121no—
247C2×SD16—91no—
248C2×SD16—271no—
249#127—271no—
250C4×Q8—271no—
251C4⋊C8—91no—
252#102—91no—
253#102—271no—
254H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
255C23×C3—181no—
256C23×C3—181no—
257C22×D3—61no—
258C22×D3—61no—
259C22×D3—61no—
260C22×D3—121no—
261C22×D3—181no—
262C3×Q8—92no—
263C3×Q8—181no—
264C3×D4—181no—
265C3×D4—181no—
266C3×D4—181no—
267C3×D4—181no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—61no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—181no—
274C2×C3×C4—181no—
275C2×C3×C4C2×D3211yescharacteristic
276#67—362no—
277#67—362no—
278#67—362no—
279#67—362no—
280C2×Dic3—31no—
281C2×Dic3—31no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—91no—
286C2×Dic3—91no—
287C2×Dic3—121no—
288C2×Dic3—181no—
289D12—61no—
290D12—61no—
291D12—61no—
292D12—61no—
293D12—62no—
294D12—61no—
295D12—61no—
296D12—121no—
297D12—181no—
298D12—181no—
299D12—362no—
300D12—362no—
301Dic6—32no—
302Dic6—32no—
303Dic6—62no—
304Dic6—61no—
305Dic6—61no—
306Dic6—121no—
307C3×C8—21no—
308C3×C8—61no—
309C3×C8—61no—
310C3×C8—62no—
311C3×C8—181no—
312C3⋊C8—61no—
313C3⋊C8—181no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32#88411yescharacteristic
318C2×C32#89511yescharacteristic
319C2×C32C2×#20411yescharacteristic
320C2×C32#90811yescharacteristic
321C2×C32#91111yescharacteristic
322C2×C32D3×SD1611yescharacteristic
323C3⋊D3—81no—
324C3×D3—242no—
325C3×D3—241no—
326C3×D3—241no—
327C3×D3—241no—
328C2×Q8—91no—
329C2×D4—91no—
330C2×D4—91no—
331C2×D4—541no—
332C2×D4—541no—
333SD16—182no—
334SD16—542no—
335C2×C8—91no—
336C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
337C4⋊C4—91no—
338C4⋊C4—541no—
339C42—91no—
340C42—541no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—61no—
344C22×C3—361no—
345C22×C3—361no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3D3×D1211yescharacteristic
351C2×D3—121no—
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—122no—
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—121no—
361C2×D3—241no—
362C2×D3—241no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—61no—
371C3×C4—61no—
372C3×C4—92no—
373C3×C4—181no—
374C3×C4—182no—
375C3×C4—361no—
376C3×C4#379911yescharacteristic
377C3×C4#381111yescharacteristic
378Dic3—32no—
379Dic3—62no—
380Dic3—62no—
381Dic3—61no—
382Dic3—92no—
383Dic3—122no—
384Dic3—121no—
385Dic3—181no—
386Dic3—361no—
387C32—21no—
388C32#476211yescharacteristic, complemented
389C32#484811yescharacteristic, complemented
390C23—181no—
391C23—181no—
392Q8—92no—
393Q8—181no—
394D4—181no—
395D4—181no—
396D4—181no—
397D4—181no—
398D4—1082no—
399D4—1082no—
400C2×C4—91no—
401C2×C4—91no—
402C2×C4—181no—
403C2×C4—181no—
404C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
405C8—21no—
406C8—181no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—61no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—721no—
417C2×C3—721no—
418C2×C3D3×#20411yescharacteristic
419C2×C3#6379511yescharacteristic
420C2×C3#6391011yescharacteristic
421D3—241no—
422D3—241no—
423D3—242no—
424D3—241no—
425D3—721no—
426C22—361no—
427C22—361no—
428C22—361no—
429C22—361no—
430C22—361no—
431C22—361no—
432C22#8865411yescharacteristic
433C4—92no—
434C4—181no—
435C4—182no—
436C4—361no—
437C4#8865211yescharacteristic
438C4#8867711yescharacteristic
439C3—31no—
440C3—31no—
441C3—61no—
442C3#1059087911yescharacteristic
443C2—721no—
444C2—721no—
445C2#1172084611yescharacteristic
446C2#1172084911yescharacteristic
447C2#1172115711yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radical#11721158C2
hypercenterC2×C8((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×C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#88528C2semidirect
3. #11721083C2semidirect
4. C2×#88438C2semidirect
5. #63648⋊C3C2semidirect
6. #63641⋊C3C2semidirect
7. #11720324C2semidirect
32. #63648D3semidirect
33. #63641D3semidirect
45. #6229D4semidirect
100. (C3⋊Dic3)⋊C3SD16semidirect
240. C3⋊Dic3#204semidirect
254. H(3,𝔽3)#460semidirect
388. C32#4762semidirect
389. C32#4848semidirect

Maximal subgroups

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

subgroupindex
C2×#885282
#117211582
#117210832
C2×#884382
#63648⋊C32
#63641⋊C32
#117203242
#105908763
#105907993