Finite Group Databasebeta

← #13141074

Subgroups — #13141074

Order 1728.

446 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 69 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#11721870C211yesmaximal, characteristic, complemented
2#64158⋊C3C211yesmaximal, characteristic, complemented
3#11721689C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#11720325C211yesmaximal, characteristic, complemented
6#11720320C211yesmaximal, characteristic, complemented
7#11720319C211yesmaximal, characteristic, complemented
8#10591678—31nomaximal
9#10591677—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13#88549—21no—
14#88545—21no—
15#88538C2211yescharacteristic
16#3755⋊C3—21no—
17#3753⋊C3—21no—
18#3748⋊C3—21no—
19#3748⋊C3C2211yescharacteristic
20#3748⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88436—21no—
24#88436—21no—
25#64158D311yescharacteristic, complemented
26#64157D311yescharacteristic, complemented
27C3×#949—31no—
28C3×#949—31no—
29C3×#949—31no—
30C3×#949—61no—
31C3×#948—31no—
32C2×#3714—31no—
33C2×#3713—31no—
34#63578—31no—
35#63577—31no—
36#63559—31no—
37#63558—31no—
38#63558—31no—
39#63558—31no—
40C22×#272—21no—
41C22×(C3⋊D3)⋊C3—21no—
42Q8×H(3,𝔽3)D411yescharacteristic
43D4×H(3,𝔽3)—21no—
44D4×H(3,𝔽3)D411yescharacteristic
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C4⋊#272—21no—
48C4⋊#272—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#6180—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
56#5199—91no—
57#5199—91no—
58#5142—91no—
59C32×#41—21no—
60C32×#41C2×D311yescharacteristic
61C32×#41C2×D311yescharacteristic
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67#3755—21no—
68#3753—21no—
69#3748—21no—
70#3748C2×D311yescharacteristic
71#3748C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—21no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×#215—62no—
75C3×#215—61no—
76C3×#215—61no—
77C3×#215—61no—
78C3×#213—62no—
79C3×#213—61no—
80C3×#213—61no—
81C3×#213—61no—
82C3×#208—31no—
83C3×#208—31no—
84C3×#208—31no—
85C3×#208—31no—
86C3×#208—61no—
87C3×#208—61no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—61no—
90#3714—61no—
91#3714—61no—
92#3713—61no—
93#3713—61no—
94C22×H(3,𝔽3)—21no—
95C22×H(3,𝔽3)#4111yescharacteristic
96C2×#272—41no—
97C2×#272—81no—
98C2×(C3⋊D3)⋊C3—41no—
99C2×(C3⋊D3)⋊C3—81no—
100C4×H(3,𝔽3)—21no—
101C4×H(3,𝔽3)C2×D411yescharacteristic
102C4×H(3,𝔽3)C2×D411yescharacteristic
103C3×AGL(1,ℤ/8)—91no—
104C3×AGL(1,ℤ/8)—91no—
105C3×#131—91no—
106#949—31no—
107#949—31no—
108#949—31no—
109#949—61no—
110#949—91no—
111#948—31no—
112#948—91no—
113C2×#213—91no—
114C2×#213—91no—
115C2×D24—91no—
116#833—91no—
117#833—91no—
118#823—91no—
119#823—91no—
120#823—91no—
121#823—91no—
122#822—91no—
123#822—91no—
124#819—91no—
125C22×C3⋊D3—21no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C22×C3×D3—121no—
130C32×Q8—21no—
131C32×Q8D1211yescharacteristic
132C32×Q8#6711yescharacteristic
133C32×D4—21no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—41no—
137C32×D4D1211yescharacteristic
138C32×D4#6711yescharacteristic
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—41no—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)—21no—
147C3×D12—62no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—61no—
154C3×D12—121no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)—21no—
157C3⋊(C3⋊C8)#6711yescharacteristic
158C3⋊(C3⋊C8)#6711yescharacteristic
159C3×C3⋊C8—31no—
160C3×C3⋊C8—31no—
161C3×C3⋊C8—62no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—61no—
165C3×C3⋊C8—61no—
166#472—271noSylow
167C2×H(3,𝔽3)—21no—
168C2×H(3,𝔽3)—41no—
169C2×H(3,𝔽3)#12111yescharacteristic
170#272—81no—
171(C3⋊D3)⋊C3—81no—
172C3×#41—31no—
173C3×#41—31no—
174C3×#41—61no—
175C3×#41D3211yescharacteristic
176C2×C3×D4—91no—
177C2×C3×D4—91no—
178C2×D12—31no—
179C2×D12—31no—
180C2×D12—31no—
181C2×D12—61no—
182C2×D12—91no—
183C3×SD16—181no—
184C3×SD16—181no—
185C3×D8—181no—
186C3×D8—181no—
187C3×M16—91no—
188C3×M16—91no—
189C3×M16—91no—
190C3×M16—181no—
191C2×C3×C8—91no—
192C2×C3×C8—181no—
193#215—61no—
194#215—61no—
195#215—62no—
196#215—61no—
197#215—181no—
198#213—61no—
199#213—61no—
200#213—62no—
201#213—61no—
202#213—181no—
203#213—181no—
204#213—181no—
205#213—181no—
206#213—181no—
207#208—31no—
208#208—31no—
209#208—31no—
210#208—31no—
211#208—61no—
212#208—61no—
213#208—91no—
214#208—91no—
215#208—181no—
216C2×C3⋊C8—31no—
217C2×C3⋊C8—61no—
218C2×C3⋊C8—91no—
219C2×C3⋊C8—181no—
220D24—181no—
221D24—181no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—41no—
226C22×C32#23511yescharacteristic
227C22×C32#23711yescharacteristic
228C2×C3⋊D3—41no—
229C2×C3⋊D3—81no—
230C2×C3×D3—122no—
231C2×C3×D3—121no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C2×C3×D3—242no—
235C2×C3×D3—241no—
236C2×C3×D3—241no—
237C2×C3×D3—241no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—41no—
243C32×C4C2×D1211yescharacteristic
244C32×C4D3×D411yescharacteristic
245C32×C4C2×#6711yescharacteristic
246C32×C4C2×#6711yescharacteristic
247AGL(1,ℤ/8)—91no—
248AGL(1,ℤ/8)—91no—
249C2×D8—271no—
250#131—91no—
251#108—271no—
252#100—271no—
253#100—271no—
254H(3,𝔽3)#47211yescharacteristic, 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—31no—
263C3×Q8—31no—
264C3×Q8—61no—
265C3×Q8#62611yescharacteristic
266C3×D4—21no—
267C3×D4—31no—
268C3×D4—31no—
269C3×D4—61no—
270C3×D4—61no—
271C3×D4—61no—
272C3×D4—121no—
273C3×D4—181no—
274C3×D4—181no—
275C3×D4—181no—
276C3×D4—181no—
277C3×D4#62611yescharacteristic
278C2×C3×C4—21no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—121no—
285C2×C3×C4C2×D3211yescharacteristic
286D12—61no—
287D12—61no—
288D12—61no—
289D12—61no—
290D12—62no—
291D12—61no—
292D12—61no—
293D12—121no—
294D12—181no—
295D12—181no—
296C3×C8—91no—
297C3×C8—91no—
298C3×C8—181no—
299C3×C8—181no—
300C3×C8—181no—
301C3⋊C8—31no—
302C3⋊C8—31no—
303C3⋊C8—61no—
304C3⋊C8—61no—
305C3⋊C8—62no—
306C3⋊C8—61no—
307C3⋊C8—61no—
308C3⋊C8—91no—
309C3⋊C8—91no—
310C3⋊C8—181no—
311C3⋊C8—181no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—41no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—81no—
319C2×C32#93011yescharacteristic
320C2×C32#93911yescharacteristic
321C3⋊D3—81no—
322C3×D3—242no—
323C3×D3—241no—
324C3×D3—241no—
325C3×D3—241no—
326#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
327C2×D4—91no—
328C2×D4—91no—
329SD16—181no—
330SD16—181no—
331D8—181no—
332D8—181no—
333D8—541no—
334D8—541no—
335M16—91no—
336M16—91no—
337M16—91no—
338M16—181no—
339C2×C8—91no—
340C2×C8—181no—
341C22×C3—21no—
342C22×C3—31no—
343C22×C3—31no—
344C22×C3—61no—
345C22×C3—61no—
346C22×C3—61no—
347C22×C3—121no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3—721no—
351C22×C3—721no—
352C22×C3#380411yescharacteristic
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—122no—
356C2×D3—121no—
357C2×D3—242no—
358C2×D3—241no—
359C2×D3—241no—
360C2×D3—241no—
361C2×D3—361no—
362C2×D3—721no—
363C3×C4—21no—
364C3×C4—31no—
365C3×C4—31no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—61no—
369C3×C4—61no—
370C3×C4—61no—
371C3×C4—61no—
372C3×C4—121no—
373C3×C4C2×#62611yescharacteristic
374C3×C4D3×#6711yescharacteristic
375C32—21no—
376C32#514211yescharacteristic, complemented
377C32#519911yescharacteristic, complemented
378C23—181no—
379C23—181no—
380Q8#620011yescharacteristic
381D4—21no—
382D4—181no—
383D4—181no—
384D4—181no—
385D4—181no—
386D4#620011yescharacteristic
387C2×C4—21no—
388C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
389C8—91no—
390C8—91no—
391C8—181no—
392C8—181no—
393C8—181no—
394C2×C3—21no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—41no—
398C2×C3—61no—
399C2×C3—61no—
400C2×C3—61no—
401C2×C3—121no—
402C2×C3—121no—
403C2×C3—121no—
404C2×C3—241no—
405C2×C3—721no—
406C2×C3—721no—
407C2×C3#6397811yescharacteristic
408D3—241no—
409D3—241no—
410D3—242no—
411D3—241no—
412D3—721no—
413C22—21no—
414C22—361no—
415C22—361no—
416C22—721no—
417C22—721no—
418C22#8867111yescharacteristic
419C4—21no—
420C4C2×#620011yescharacteristic
421C4#8867611yescharacteristic
422C3—31no—
423C3—31no—
424C3—61no—
425C3#1059167711yescharacteristic
426C2—21no—
427C2—41no—
428C2—721no—
429C2—721no—
430C2#1172132211yescharacteristic
subgroupG/H
centerC2#11721322
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3#63978
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#472
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11721870C2semidirect
2. #64158⋊C3C2semidirect
3. #11721689C2semidirect
4. C2×#88436C2semidirect
5. #11720325C2semidirect
6. #11720320C2semidirect
7. #11720319C2semidirect
25. #64158D3semidirect
26. #64157D3semidirect
254. H(3,𝔽3)#472semidirect
376. C32#5142semidirect
377. C32#5199semidirect

Maximal subgroups

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

subgroupindex
#117218702
#64158⋊C32
#117216892
C2×#884362
#117203252
#117203202
#117203192
#105916783
#105916773