Finite Group Databasebeta

← #13140612

Subgroups — #13140612

Order 1728.

643 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 643 conjugacy classes of subgroups — too many to draw. Only its 83 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×#3753⋊C3C211yesmaximal, characteristic, complemented
4#11721158C211yesmaximal, characteristic, complemented
5C2×#88436C211yesmaximal, characteristic, complemented
6C2×#88432C211yesmaximal, characteristic, complemented
7#11720310C211yesmaximal, characteristic, complemented
8#10591484—31nomaximal
9#10591482—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#638⋊C3—22no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×#6200—22no—
16#88545—22no—
17C2×#6180C2211yescharacteristic
18#3753⋊C3—22no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88436—22no—
22#88432—22no—
23#64149D311yescharacteristic, complemented
24C2×#3753D311yescharacteristic, complemented
25C3×#940—31no—
26C2×C3×#213—31no—
27C2×C3×#213—31no—
28C2×C3×#213—31no—
29C2×C3×#213—61no—
30#63914—31no—
31#63911—31no—
32C2×#3714—31no—
33C2×#3714—31no—
34C2×#3714—31no—
35C2×#3713—31no—
36#63553—31no—
37Dic3×C3⋊C8—31no—
38C23×H(3,𝔽3)—21no—
39C22×#272—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3—21no—
42D4×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C4⋊#272—21no—
46C4⋊#272—21no—
47#638⋊C3—44no—
48C2×(C3⋊Dic3)⋊C3D412yes—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53#6200—44no—
54#6180D412yescomplemented
55(C3⋊(C3⋊C8))⋊C3D412yescomplemented
56#5151—91no—
57#5151—91no—
58#5073—91no—
59C2×C32×D4—21no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C2×D311yescharacteristic
62C2×#638—22no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C3×#67—62no—
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—31no—
70C2×C3×D12—61no—
71C2×#626—62no—
72C2×#625—62no—
73#3753—22no—
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×#213—62no—
77C3×#213—62no—
78C3×#213—62no—
79C3×#213—64no—
80C3×C4×Dic3—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—61no—
85#3714—62no—
86#3714—62no—
87#3714—62no—
88#3713—62no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—42no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×#272—41no—
93C2×#272—42no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3—42no—
97C2×(C3⋊D3)⋊C3—42no—
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)C2×D411yescharacteristic
100(C3⋊Dic3)⋊C3D814yescomplemented
101C2×C3×D8—91no—
102C2×C3×D8—91no—
103C3×#127—91no—
104#940—31no—
105#940—91no—
106#940—91no—
107#940—91no—
108C2×#213—31no—
109C2×#213—31no—
110C2×#213—31no—
111C2×#213—61no—
112C2×#213—91no—
113C2×#213—91no—
114C2×#213—91no—
115C2×#213—91no—
116C2×#213—91no—
117C2×D24—91no—
118C2×D24—91no—
119#874—91no—
120C8×Dic3—91no—
121C8×Dic3—91no—
122C4×C3⋊C8—91no—
123C23×C32—21no—
124C23×C32—21no—
125C23×C32—41no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3—21no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—121no—
133C32×D4—22no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—21no—
138C32×D4—41no—
139C2×C32×C4—21no—
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142#638—44no—
143C2×C3⋊Dic3#6712yes—
144C4⋊(C3⋊D3)—21no—
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)—21no—
148C3×#67—124no—
149C2×C3×Dic3—32no—
150C3×D12—62no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—61no—
154C3×D12—61no—
155C3×D12—61no—
156C3×D12—61no—
157C3×D12—61no—
158C3×D12—61no—
159C3×D12—121no—
160#626—124no—
161#625—124no—
162C3⋊(C3⋊C8)#6712yescomplemented
163C3×C3⋊C8—32no—
164C3×C3⋊C8—32no—
165C3×C3⋊C8—32no—
166C3×C3⋊C8—62no—
167#493—271noSylow
168C2×H(3,𝔽3)—81no—
169C2×H(3,𝔽3)#12711yescharacteristic
170C2×H(3,𝔽3)C2×D812yes—
171#272—81no—
172(C3⋊D3)⋊C3—81no—
173(C3⋊D3)⋊C3—81no—
174C2×C3×D4—31no—
175C2×C3×D4—31no—
176C2×C3×D4—61no—
177C2×C3×D4—91no—
178C2×C3×D4—91no—
179C2×C3×D4—91no—
180C2×C3×D4—182no—
181C2×C3×D4D3211yescharacteristic
182C2×#67—62no—
183C2×#67—182no—
184C2×#67—182no—
185C2×#67—182no—
186C2×D12—31no—
187C2×D12—31no—
188C2×D12—31no—
189C2×D12—31no—
190C2×D12—61no—
191C2×D12—91no—
192C2×D12—91no—
193C2×D12—182no—
194C3×D8—182no—
195C3×D8—182no—
196C2×C3×C8—91no—
197C2×C3×C8—91no—
198C3×C42—91no—
199#213—62no—
200#213—62no—
201#213—62no—
202#213—64no—
203#213—182no—
204#213—182no—
205#213—182no—
206#213—182no—
207#213—182no—
208C4×Dic3—31no—
209C4×Dic3—91no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8—61no—
214C2×C3⋊C8—91no—
215D24—182no—
216D24—182no—
217C22×C32—21no—
218C22×C32—42no—
219C22×C32—42no—
220C22×C32—42no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—82no—
224C22×C32D3×D411yescharacteristic
225C22×C32C2×#6711yescharacteristic
226C2×C3⋊D3—42no—
227C2×C3⋊D3—42no—
228C2×C3⋊D3—41no—
229C2×C3⋊D3—41no—
230C2×C3×D3—122no—
231C2×C3×D3—122no—
232C2×C3×D3—122no—
233C2×C3×D3—122no—
234C2×C3×D3—122no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3—242no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4D3×D411yescharacteristic
243C32×C4D3×D411yescharacteristic
244C32×C4D3×D411yescharacteristic
245C32×C4C2×#6711yescharacteristic
246C3⋊Dic3#21314yescomplemented
247C3×Dic3—34no—
248C2×D8—91no—
249C2×D8—91no—
250C2×D8—271no—
251C2×D8—271no—
252#127—91no—
253#127—271no—
254C4×C8—271no—
255H(3,𝔽3)#49311yescharacteristic, complemented, Sylow
256C23×C3—21no—
257C23×C3—61no—
258C23×C3—61no—
259C23×C3—121no—
260C23×C3—181no—
261C23×C3—181no—
262C23×C3—181no—
263C22×D3—61no—
264C22×D3—61no—
265C22×D3—61no—
266C22×D3—61no—
267C22×D3—121no—
268C22×D3—181no—
269C22×D3—181no—
270C3×D4—21no—
271C3×D4—21no—
272C3×D4—61no—
273C3×D4—61no—
274C3×D4—61no—
275C3×D4—61no—
276C3×D4—62no—
277C3×D4—121no—
278C3×D4—181no—
279C3×D4—181no—
280C3×D4—181no—
281C3×D4—181no—
282C3×D4—181no—
283C3×D4—181no—
284C3×D4—364no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—61no—
288C2×C3×C4—92no—
289C2×C3×C4C2×D3211yescharacteristic
290#67—124no—
291#67—364no—
292#67—364no—
293#67—364no—
294C2×Dic3—32no—
295C2×Dic3—92no—
296D12—61no—
297D12—61no—
298D12—61no—
299D12—61no—
300D12—62no—
301D12—61no—
302D12—61no—
303D12—61no—
304D12—61no—
305D12—121no—
306D12—181no—
307D12—181no—
308D12—181no—
309D12—181no—
310D12—364no—
311C3×C8—92no—
312C3×C8—92no—
313C3⋊C8—32no—
314C3⋊C8—32no—
315C3⋊C8—32no—
316C3⋊C8—62no—
317C3⋊C8—92no—
318C2×C32—22no—
319C2×C32—21no—
320C2×C32—82no—
321C2×C32—81no—
322C2×C32—81no—
323C2×C32D3×D812yes—
324C2×C32C2×#21312yes—
325C2×C32#94011yescharacteristic
326C2×C32#94011yescharacteristic
327C3⋊D3—81no—
328C3⋊D3—81no—
329C3×D3—242no—
330C3×D3—241no—
331C3×D3—241no—
332C3×D3—241no—
333C3×D3—241no—
334C2×D4—91no—
335C2×D4—91no—
336C2×D4—91no—
337C2×D4—182no—
338C2×D4—542no—
339C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
340D8—182no—
341D8—182no—
342D8—542no—
343D8—542no—
344C2×C8—91no—
345C2×C8—91no—
346C42—91no—
347C22×C3—31no—
348C22×C3—31no—
349C22×C3—42no—
350C22×C3—41no—
351C22×C3—61no—
352C22×C3—122no—
353C22×C3—121no—
354C22×C3—121no—
355C22×C3—122no—
356C22×C3—122no—
357C22×C3—242no—
358C22×C3—362no—
359C22×C3—361no—
360C22×C3—361no—
361C22×C3—362no—
362C22×C3—362no—
363C22×C3—361no—
364C22×C3D3×#6711yescharacteristic
365C2×D3—121no—
366C2×D3—121no—
367C2×D3—122no—
368C2×D3—122no—
369C2×D3—122no—
370C2×D3—122no—
371C2×D3—122no—
372C2×D3—121no—
373C2×D3—121no—
374C2×D3—242no—
375C2×D3—361no—
376C2×D3—361no—
377C2×D3—362no—
378C2×D3—362no—
379C3×C4—31no—
380C3×C4—31no—
381C3×C4—31no—
382C3×C4—31no—
383C3×C4—61no—
384C3×C4—61no—
385C3×C4—94no—
386C3×C4D3×#6711yescharacteristic
387C3×C4#381111yescharacteristic
388Dic3—34no—
389Dic3—94no—
390C32—21no—
391C32#507311yescharacteristic, complemented
392C32#515111yescharacteristic, complemented
393C23—21no—
394C23—181no—
395C23—181no—
396C23—181no—
397D4—21no—
398D4—21no—
399D4—181no—
400D4—181no—
401D4—181no—
402D4—181no—
403D4—181no—
404D4—181no—
405D4—364no—
406D4—1084no—
407C2×C4—92no—
408C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
409C8—92no—
410C8—92no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—32no—
414C2×C3—32no—
415C2×C3—61no—
416C2×C3—62no—
417C2×C3—81no—
418C2×C3—241no—
419C2×C3—241no—
420C2×C3—242no—
421C2×C3—721no—
422C2×C3—721no—
423C2×C3—721no—
424C2×C3D3×#21312yes—
425C2×C3#6397911yescharacteristic
426D3—241no—
427D3—241no—
428D3—242no—
429D3—241no—
430D3—241no—
431D3—721no—
432D3—721no—
433C22—41no—
434C22—42no—
435C22—361no—
436C22—361no—
437C22—361no—
438C22—362no—
439C22—362no—
440C22—362no—
441C22#8867611yescharacteristic
442C4—94no—
443C4#8867611yescharacteristic
444C4#8867711yescharacteristic
445C3—31no—
446C3—31no—
447C3—61no—
448C3#1059148211yescharacteristic
449C2—81no—
450C2—721no—
451C2—721no—
452C2—721no—
453C2#1172119412yes—
454C2#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)#493
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×#3753⋊C3C2semidirect
4. #11721158C2semidirect
5. C2×#88436C2semidirect
6. C2×#88432C2semidirect
7. #11720310C2semidirect
23. #64149D3semidirect
24. C2×#3753D3semidirect
54. #6180D4semidirect
55. (C3⋊(C3⋊C8))⋊C3D4semidirect
100. (C3⋊Dic3)⋊C3D8semidirect
162. C3⋊(C3⋊C8)#67semidirect
246. C3⋊Dic3#213semidirect
255. H(3,𝔽3)#493semidirect
391. C32#5073semidirect
392. C32#5151semidirect

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×#3753⋊C32
#117211582
C2×#884362
C2×#884322
#117203102
#105914843
#105914823