Finite Group Databasebeta

← #13140608

Subgroups — #13140608

Order 1728.

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

normal subgroups only

This group has 545 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#64147⋊C3C211yesmaximal, characteristic, complemented
3#11721156C211yesmaximal, characteristic, complemented
4C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720315C211yesmaximal, characteristic, complemented
8#10591498—31nomaximal
9#10591455—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#638⋊C3—21no—
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#6200—21no—
15#88545—22no—
16C2×#6180C2211yescharacteristic
17#3757⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88451—21no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23#64147D311yescharacteristic, complemented
24C3×#938—31no—
25C2×C3×#213—31no—
26C2×C3×#213—31no—
27C2×C3×#213—61no—
28#63916—31no—
29#63908—31no—
30C2×#3710—31no—
31C2×D3×C3⋊C8—31no—
32#63659D311yescharacteristic, complemented
33C3×#832—31no—
34#63575—31no—
35#63575—31no—
36#63563—31no—
37#63562—31no—
38C23×H(3,𝔽3)—21no—
39C22×#272—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
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—42no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3D411yescharacteristic
50C4×(C3⋊D3)⋊C3—21no—
51C4×(C3⋊D3)⋊C3—21no—
52#6200—42no—
53#6180D412yescomplemented
54(C3⋊(C3⋊C8))⋊C3—21no—
55#5157—91no—
56#5038—91no—
57#4775—91no—
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—21no—
62C2×C4×C3⋊D3C2×D311yescharacteristic
63C2×C3×#67—61no—
64C2×C3×D12—31no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67C2×C3×C4×D3—31no—
68C2×#626—61no—
69C2×#625—61no—
70#3757—21no—
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×#217—61no—
75C3×#213—62no—
76C3×#213—62no—
77C3×#213—64no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—61no—
84(C2×D3)⋊Dic3—61no—
85(C2×D3)⋊Dic3—61no—
86#3710—62no—
87D3×C3⋊C8—62no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×#272—41no—
93C2×#272—41no—
94C2×#272—41no—
95C2×(C3⋊D3)⋊C3—21no—
96C2×(C3⋊D3)⋊C3—21no—
97C2×(C3⋊D3)⋊C3D812yescomplemented
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)#4111yescharacteristic
100(C3⋊Dic3)⋊C3—21no—
101(C3⋊Dic3)⋊C3—41no—
102(C3⋊Dic3)⋊C3—41no—
103C2×C3×D8—91no—
104C3×#121—91no—
105#938—31no—
106#938—91no—
107#938—91no—
108C2×#213—31no—
109C2×#213—31no—
110C2×#213—61no—
111#930—91no—
112C22×C3⋊C8—91no—
113C2×D3×C8—91no—
114C2×D3×C8—91no—
115#895—91no—
116C3×#102—91no—
117#832—31no—
118#832—91no—
119#832—91no—
120#821—91no—
121#818—91no—
122#809—91no—
123#807—91no—
124#807—91no—
125C23×C32—21no—
126C23×C32—21no—
127C23×C32—41no—
128C22×C3⋊D3#6711yescharacteristic
129C22×C3×D3—31no—
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—42no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3#6711yescharacteristic
146C4×C3⋊D3—21no—
147C4×C3⋊D3—21no—
148C3×#67—122no—
149C2×C3×Dic3—31no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C3×D12—62no—
153C3×D12—61no—
154C3×D12—61no—
155C3×D12—61no—
156C3×D12—61no—
157C3×D12—121no—
158C3×C4×D3—61no—
159C3×C4×D3—61no—
160#626—122no—
161#625—122no—
162C3⋊(C3⋊C8)—21no—
163C3×C3⋊C8—32no—
164C3×C3⋊C8—32no—
165C3×C3⋊C8—62no—
166C3×C3⋊C8—61no—
167#466—271noSylow
168C2×H(3,𝔽3)—81no—
169C2×H(3,𝔽3)#12111yescharacteristic
170C2×H(3,𝔽3)C2×D811yescharacteristic
171C2×H(3,𝔽3)#13511yescharacteristic
172#272—81no—
173(C3⋊D3)⋊C3—22no—
174C2×C3×D4—31no—
175C2×C3×D4—31no—
176C2×C3×D4—61no—
177C2×C3×D4—91no—
178C2×C3×D4—181no—
179C2×C3×D4D3211yescharacteristic
180C22×C3×C4—91no—
181C2×#67—61no—
182C2×#67—181no—
183C2×#67—181no—
184C2×#67—181no—
185C2×D12—31no—
186C2×D12—31no—
187C2×D12—61no—
188C2×D12—181no—
189C2×C4×D3—31no—
190C2×C4×D3—91no—
191C3×D8—182no—
192C2×C3×C8—91no—
193C2×C3×C8—91no—
194C3×C4⋊C4—91no—
195C3×C4⋊C4—91no—
196C3×C22⋊C4—181no—
197#217—61no—
198#217—181no—
199#217—181no—
200#213—62no—
201#213—62no—
202#213—64no—
203(C2×D3)⋊C4—181no—
204(C2×D3)⋊C4—181no—
205C4⋊Dic3—31no—
206C4⋊Dic3—31no—
207C4⋊Dic3—91no—
208C4⋊Dic3—91no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—61no—
213C2×C3⋊C8—91no—
214C2×C3⋊C8—182no—
215D3×C8—182no—
216D3×C8—182no—
217C22×C32—21no—
218C22×C32—42no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—81no—
226C22×C32—81no—
227C22×C32D3×D411yescharacteristic
228C22×C32C2×#6711yescharacteristic
229C2×C3⋊D3—21no—
230C2×C3⋊D3—21no—
231C2×C3⋊D3#21312yescomplemented
232C2×C3×D3—32no—
233C2×C3×D3—61no—
234C2×C3×D3—61no—
235C2×C3×D3—122no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C2×C3×D3—121no—
241C2×C3×D3—121no—
242C2×C3×D3—241no—
243C2×C3×D3—241no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4#23511yescharacteristic
247C32×C4D3×D411yescharacteristic
248C32×C4D3×D411yescharacteristic
249C32×C4#23711yescharacteristic
250C3⋊Dic3—21no—
251C3⋊Dic3—41no—
252C3⋊Dic3—41no—
253C3×Dic3—61no—
254C3×Dic3—121no—
255C3×Dic3—121no—
256C2×D8—91no—
257C22×C8—271no—
258#121—91no—
259#121—271no—
260#107—271no—
261#102—91no—
262#102—271no—
263H(3,𝔽3)#46611yescharacteristic, complemented, Sylow
264C23×C3—21no—
265C23×C3—61no—
266C23×C3—61no—
267C23×C3—91no—
268C23×C3—121no—
269C23×C3—181no—
270C22×D3—31no—
271C22×D3—61no—
272C22×D3—61no—
273C22×D3—91no—
274C22×D3—121no—
275C3×D4—21no—
276C3×D4—21no—
277C3×D4—61no—
278C3×D4—61no—
279C3×D4—61no—
280C3×D4—61no—
281C3×D4—62no—
282C3×D4—121no—
283C3×D4—181no—
284C3×D4—181no—
285C3×D4—362no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—61no—
289C2×C3×C4—91no—
290C2×C3×C4—181no—
291C2×C3×C4—181no—
292C2×C3×C4—181no—
293C2×C3×C4—181no—
294C2×C3×C4C2×D3211yescharacteristic
295#67—122no—
296#67—362no—
297#67—362no—
298#67—362no—
299C2×Dic3—31no—
300C2×Dic3—61no—
301C2×Dic3—61no—
302C2×Dic3—91no—
303C2×Dic3—181no—
304C2×Dic3—181no—
305D12—61no—
306D12—61no—
307D12—61no—
308D12—61no—
309D12—62no—
310D12—121no—
311D12—362no—
312C4×D3—61no—
313C4×D3—61no—
314C4×D3—181no—
315C4×D3—181no—
316C3×C8—92no—
317C3×C8—181no—
318C3⋊C8—32no—
319C3⋊C8—32no—
320C3⋊C8—62no—
321C3⋊C8—61no—
322C3⋊C8—181no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—82no—
327C2×C32—81no—
328C2×C32—81no—
329C2×C32#88411yescharacteristic
330C2×C32D3×D811yescharacteristic
331C2×C32#91611yescharacteristic
332C2×C32C2×#21311yescharacteristic
333C2×C32#93911yescharacteristic
334C2×C32#95011yescharacteristic
335C3⋊D3—22no—
336C3×D3—62no—
337C3×D3—242no—
338C3×D3—241no—
339C3×D3—241no—
340C2×D4—91no—
341C2×D4—181no—
342C2×D4—541no—
343C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
344C22×C4—91no—
345D8—182no—
346C2×C8—91no—
347C2×C8—91no—
348C2×C8—542no—
349C4⋊C4—91no—
350C4⋊C4—91no—
351C22⋊C4—181no—
352C22⋊C4—541no—
353C22×C3—31no—
354C22×C3—31no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—61no—
359C22×C3—92no—
360C22×C3—121no—
361C22×C3—121no—
362C22×C3—121no—
363C22×C3—121no—
364C22×C3—121no—
365C22×C3—121no—
366C22×C3—122no—
367C22×C3—181no—
368C22×C3—181no—
369C22×C3—241no—
370C22×C3—241no—
371C22×C3—361no—
372C22×C3—361no—
373C22×C3—361no—
374C22×C3D3×#6711yescharacteristic
375C2×D3—32no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—92no—
379C2×D3—121no—
380C2×D3—121no—
381C2×D3—121no—
382C2×D3—121no—
383C2×D3—121no—
384C2×D3—121no—
385C2×D3—122no—
386C2×D3—181no—
387C2×D3—181no—
388C2×D3—241no—
389C2×D3—241no—
390C3×C4—31no—
391C3×C4—31no—
392C3×C4—31no—
393C3×C4—31no—
394C3×C4—61no—
395C3×C4—61no—
396C3×C4—181no—
397C3×C4—361no—
398C3×C4—361no—
399C3×C4#380411yescharacteristic
400C3×C4#381111yescharacteristic
401Dic3—61no—
402Dic3—121no—
403Dic3—121no—
404Dic3—181no—
405Dic3—361no—
406Dic3—361no—
407C32—21no—
408C32#477511yescharacteristic, complemented
409C32#503811yescharacteristic, complemented
410C23—21no—
411C23—91no—
412C23—181no—
413D4—21no—
414D4—21no—
415D4—181no—
416D4—181no—
417D4—362no—
418D4—1082no—
419C2×C4—91no—
420C2×C4—181no—
421C2×C4—181no—
422C2×C4—181no—
423C2×C4—181no—
424C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
425C8—92no—
426C8—181no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—61no—
434C2×C3—61no—
435C2×C3—61no—
436C2×C3—81no—
437C2×C3—182no—
438C2×C3—241no—
439C2×C3—241no—
440C2×C3—242no—
441C2×C3—721no—
442C2×C3D3×#21311yescharacteristic
443C2×C3#6393511yescharacteristic
444C2×C3#6397111yescharacteristic
445D3—62no—
446D3—182no—
447D3—241no—
448D3—241no—
449D3—242no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—92no—
454C22—181no—
455C22—181no—
456C22—361no—
457C22—361no—
458C22—361no—
459C22#8867611yescharacteristic
460C4—181no—
461C4—361no—
462C4—361no—
463C4#8867111yescharacteristic
464C4#8867711yescharacteristic
465C3—31no—
466C3—31no—
467C3—61no—
468C3#1059144911yescharacteristic
469C2—81no—
470C2—182no—
471C2—721no—
472C2#1172119411yescharacteristic
473C2#1172119711yescharacteristic
474C2#1172131211yescharacteristic
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)#466
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. #64147⋊C3C2semidirect
3. #11721156C2semidirect
4. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
5. #63659⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720315C2semidirect
23. #64147D3semidirect
32. #63659D3semidirect
53. #6180D4semidirect
97. C2×(C3⋊D3)⋊C3D8semidirect
231. C2×C3⋊D3#213semidirect
263. H(3,𝔽3)#466semidirect
408. C32#4775semidirect
409. C32#5038semidirect

Maximal subgroups

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

subgroupindex
C2×#885452
#64147⋊C32
#117211562
C2×((C3⋊D3)⋊C3)⋊C82
#63659⋊C32
#117203242
#117203152
#105914983
#105914553