Finite Group Databasebeta

← #13137926

Subgroups — #13137926

Order 1728.

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

normal subgroups only

This group has 505 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
1#64104⋊C3C211yesmaximal, characteristic, complemented
2#11721141C211yesmaximal, characteristic, complemented
3C2×#88432C211yesmaximal, characteristic, complemented
4C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
5#11720529C211yesmaximal, characteristic, complemented
6#63636⋊C3C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590576—31nomaximal
9#10590509—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#6198—21no—
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17C2×#6180C2211yescharacteristic
18#3752⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21#88432—22no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23#64104D311yescharacteristic, complemented
24C3×#895—31no—
25#63916—31no—
26#63909—31no—
27C2×#3714—31no—
28C2×#3714—31no—
29C2×#3710—31no—
30C2×D3×C3⋊C8—31no—
31#63636D311yescharacteristic, complemented
32C3×#809—31no—
33C3×#807—31no—
34C3×#807—31no—
35C3×#807—61no—
36#63564—31no—
37#63562—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—42no—
50C4×(C3⋊D3)⋊C3—21no—
51C4×(C3⋊D3)⋊C3—21no—
52#6198—42no—
53#6180D412yescomplemented
54(C3⋊(C3⋊C8))⋊C3—21no—
55#5038—91no—
56#4883—91no—
57#4775—91no—
58C2×C4⋊(C3⋊D3)—21no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×D12—61no—
65C2×C3×C4×D3—31no—
66C2×#626—61no—
67C2×#626—61no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4C2×D311yescharacteristic
70C32×C4⋊C4C2×D311yescharacteristic
71#3752—21no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×(C2×D3)⋊C4—61no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—61no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—61no—
81#3722—61no—
82(C2×D3)⋊Dic3—61no—
83#3714—62no—
84#3714—62no—
85#3710—62no—
86D3×C3⋊C8—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×(C3⋊D3)⋊C3—21no—
89C2×(C3⋊D3)⋊C3—21no—
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3D812yescomplemented
97C4×H(3,𝔽3)—41no—
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)#4111yescharacteristic
100#1084—41no—
101(C3⋊Dic3)⋊C3—21no—
102C3×#121—91no—
103#938—91no—
104C2×#213—91no—
105C2×#213—91no—
106#930—91no—
107C22×C3⋊C8—91no—
108C2×D24—91no—
109C2×D3×C8—91no—
110C2×D3×C8—91no—
111#895—31no—
112#895—91no—
113#895—91no—
114C3×#107—91no—
115C3×#102—91no—
116#832—91no—
117#821—91no—
118#809—31no—
119#809—91no—
120#809—91no—
121#807—31no—
122#807—31no—
123#807—61no—
124C22×C3⋊D3—21no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3#6711yescharacteristic
127C22×C3×D3—31no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—41no—
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C22×D311yescharacteristic
136C2×C3⋊Dic3#6711yescharacteristic
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)—42no—
142C4×C3⋊D3—21no—
143C4×C3⋊D3—21no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—121no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—122no—
153C3×C4×D3—61no—
154C3×C4×D3—61no—
155#626—122no—
156#626—122no—
157C3⋊(C3⋊C8)—21no—
158C3×C3⋊C8—32no—
159C3×C3⋊C8—32no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—61no—
162#466—271noSylow
163C2×H(3,𝔽3)#12111yescharacteristic
164C2×H(3,𝔽3)C2×D811yescharacteristic
165C2×H(3,𝔽3)#13511yescharacteristic
166(C3⋊D3)⋊C3—22no—
167(C3⋊D3)⋊C3—81no—
168(C3⋊D3)⋊C3—81no—
169C2×C3×D4—91no—
170C2×C3×D4—91no—
171C2×C3×D4—181no—
172C22×C3×C4—91no—
173C2×#67—181no—
174C2×#67—181no—
175C2×D12—31no—
176C2×D12—31no—
177C2×D12—61no—
178C2×D12—91no—
179C2×D12—91no—
180C2×D12—181no—
181C2×D12—181no—
182C2×C4×D3—31no—
183C2×C4×D3—91no—
184C2×C3×C8—91no—
185C2×C3×C8—91no—
186C3×C4⋊C4—31no—
187C3×C4⋊C4—31no—
188C3×C4⋊C4—61no—
189C3×C4⋊C4—91no—
190C3×C4⋊C4D3211yescharacteristic
191C3×C22⋊C4—181no—
192#217—181no—
193#213—182no—
194#213—182no—
195(C2×D3)⋊C4—61no—
196(C2×D3)⋊C4—181no—
197(C2×D3)⋊C4—181no—
198(C2×D3)⋊C4—181no—
199C4⋊Dic3—31no—
200C4⋊Dic3—31no—
201C4⋊Dic3—61no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—61no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—182no—
208D24—182no—
209D3×C8—182no—
210D3×C8—182no—
211C22×C32—21no—
212C22×C32D3×D411yescharacteristic
213C22×C32C2×#6711yescharacteristic
214C2×C3⋊D3—21no—
215C2×C3⋊D3—21no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—41no—
219C2×C3⋊D3—41no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3#21312yescomplemented
223C2×C3×D3—32no—
224C2×C3×D3—61no—
225C2×C3×D3—61no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—41no—
235C32×C4—41no—
236C32×C4—81no—
237C32×C4#23511yescharacteristic
238C32×C4D3×D411yescharacteristic
239C32×C4D3×D411yescharacteristic
240C32×C4#23711yescharacteristic
241C3⋊Dic3—21no—
242C3×Dic3—61no—
243C3×Dic3—121no—
244C3×Dic3—121no—
245C3×Dic3—241no—
246C2×D8—271no—
247C22×C8—271no—
248#121—91no—
249#121—271no—
250#107—91no—
251#102—91no—
252#102—271no—
253H(3,𝔽3)#46611yescharacteristic, complemented, Sylow
254C23×C3—91no—
255C23×C3—181no—
256C23×C3—181no—
257C22×D3—31no—
258C22×D3—61no—
259C22×D3—61no—
260C22×D3—91no—
261C22×D3—181no—
262C22×D3—181no—
263C3×D4—181no—
264C3×D4—181no—
265C3×D4—181no—
266C3×D4—181no—
267C3×D4—362no—
268C2×C3×C4—21no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—91no—
275C2×C3×C4—121no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4C2×D3211yescharacteristic
280#67—362no—
281#67—362no—
282C2×Dic3—31no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—91no—
286C2×Dic3—121no—
287D12—61no—
288D12—61no—
289D12—61no—
290D12—61no—
291D12—122no—
292D12—181no—
293D12—181no—
294D12—181no—
295D12—181no—
296D12—362no—
297D12—362no—
298C4×D3—61no—
299C4×D3—61no—
300C4×D3—181no—
301C4×D3—181no—
302C3×C8—92no—
303C3×C8—181no—
304C3⋊C8—32no—
305C3⋊C8—32no—
306C3⋊C8—62no—
307C3⋊C8—61no—
308C3⋊C8—181no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32#88411yescharacteristic
313C2×C32D3×D811yescharacteristic
314C2×C32#91611yescharacteristic
315C2×C32C2×#21311yescharacteristic
316C2×C32#93911yescharacteristic
317C2×C32#95011yescharacteristic
318C3⋊D3—22no—
319C3⋊D3—81no—
320C3⋊D3—81no—
321C3×D3—62no—
322C3×D3—241no—
323C3×D3—241no—
324C2×D4—91no—
325C2×D4—91no—
326C2×D4—181no—
327C2×D4—541no—
328C22×C4—91no—
329D8—542no—
330C2×C8—91no—
331C2×C8—91no—
332C2×C8—542no—
333C4⋊C4—91no—
334C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C22⋊C4—181no—
336C22⋊C4—541no—
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—61no—
340C22×C3—92no—
341C22×C3—181no—
342C22×C3—181no—
343C22×C3—361no—
344C22×C3—361no—
345C22×C3—361no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3—361no—
349C22×C3D3×#6711yescharacteristic
350C2×D3—32no—
351C2×D3—61no—
352C2×D3—61no—
353C2×D3—92no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—121no—
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—181no—
361C2×D3—181no—
362C2×D3—361no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C2×D3—361no—
367C2×D3—361no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—31no—
372C3×C4—41no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—121no—
376C3×C4—121no—
377C3×C4—181no—
378C3×C4—241no—
379C3×C4—361no—
380C3×C4#379511yescharacteristic
381C3×C4#380211yescharacteristic
382Dic3—61no—
383Dic3—121no—
384Dic3—121no—
385Dic3—181no—
386Dic3—241no—
387C32—21no—
388C32#477511yescharacteristic, complemented
389C32#503811yescharacteristic, complemented
390C23—91no—
391C23—181no—
392C23—181no—
393D4—181no—
394D4—181no—
395D4—181no—
396D4—181no—
397D4—362no—
398D4—1082no—
399C2×C4—21no—
400C2×C4—91no—
401C2×C4—181no—
402C2×C4—181no—
403C2×C4—181no—
404C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
405C8—92no—
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—182no—
417C2×C3—721no—
418C2×C3—721no—
419C2×C3#6388511yescharacteristic
420C2×C3D3×#21311yescharacteristic
421C2×C3#6394711yescharacteristic
422D3—62no—
423D3—182no—
424D3—241no—
425D3—241no—
426D3—721no—
427D3—721no—
428C22—92no—
429C22—181no—
430C22—181no—
431C22—361no—
432C22—361no—
433C22—361no—
434C22—361no—
435C22—361no—
436C22—361no—
437C22#8867611yescharacteristic
438C4—41no—
439C4—181no—
440C4—361no—
441C4#8864811yescharacteristic
442C4#8865511yescharacteristic
443C3—31no—
444C3—31no—
445C3—61no—
446C3#1059049911yescharacteristic
447C2—182no—
448C2—721no—
449C2—721no—
450C2#1172112611yescharacteristic
451C2#1172119411yescharacteristic
452C2#1172123911yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
hypercenterC4⋊C4((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 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64104⋊C3C2semidirect
2. #11721141C2semidirect
3. C2×#88432C2semidirect
4. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
5. #11720529C2semidirect
6. #63636⋊C3C2semidirect
7. #11720308C2semidirect
23. #64104D3semidirect
31. #63636D3semidirect
53. #6180D4semidirect
96. C2×(C3⋊D3)⋊C3D8semidirect
222. C2×C3⋊D3#213semidirect
253. H(3,𝔽3)#466semidirect
388. C32#4775semidirect
389. C32#5038semidirect

Maximal subgroups

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

subgroupindex
#64104⋊C32
#117211412
C2×#884322
C2×((C3⋊D3)⋊C3)⋊C82
#117205292
#63636⋊C32
#117203082
#105905763
#105905093