Finite Group Databasebeta

← #13141076

Subgroups — #13141076

Order 1728.

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

normal subgroups only

This group has 432 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
2#64160⋊C3C211yesmaximal, characteristic, complemented
3#11721689C211yesmaximal, characteristic, complemented
4C2×#88437C211yesmaximal, characteristic, complemented
5#11720325C211yesmaximal, characteristic, complemented
6#11720321C211yesmaximal, characteristic, complemented
7#11720319C211yesmaximal, characteristic, complemented
8#10591681—31nomaximal
9#10591679—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13#88549—21no—
14#88545—21no—
15#88538C2211yescharacteristic
16#3756⋊C3—21no—
17#3754⋊C3—21no—
18#3748⋊C3—21no—
19#3748⋊C3C2211yescharacteristic
20#3748⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88437—21no—
24#88437—21no—
25#64160D311yescharacteristic, complemented
26#64157D311yescharacteristic, complemented
27C3×#951—31no—
28C3×#949—31no—
29C3×#949—31no—
30C3×#949—61no—
31C3×#948—31no—
32C2×#3716—31no—
33C2×#3715—31no—
34#63578—31no—
35#63577—31no—
36#63561—31no—
37#63560—31no—
38#63558—31no—
39#63558—31no—
40C22×#272—21no—
41Q8×H(3,𝔽3)D411yescharacteristic
42D4×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C4⋊#272—21no—
47C4⋊#272—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49#634⋊C3—21no—
50#634⋊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#5203—91no—
57#5200—91no—
58#5141—91no—
59C32×#41—21no—
60C32×#41C2×D311yescharacteristic
61C32×#41C2×D311yescharacteristic
62C2×#634C2×D311yescharacteristic
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—61no—
66C2×C3×Dic6—31no—
67#3756—21no—
68#3754—21no—
69#3748—21no—
70#3748C2×D311yescharacteristic
71#3748C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—21no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×#216—61no—
75C3×#215—62no—
76C3×#215—61no—
77C3×#215—61no—
78C3×#214—61no—
79C3×#213—62no—
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#3716—61no—
91#3716—61no—
92#3715—61no—
93#3715—61no—
94C22×H(3,𝔽3)—21no—
95C22×H(3,𝔽3)#4111yescharacteristic
96C2×#272—41no—
97C2×#272—81no—
98C4×H(3,𝔽3)—21no—
99C4×H(3,𝔽3)C2×D411yescharacteristic
100C4×H(3,𝔽3)C2×D411yescharacteristic
101(C3⋊Dic3)⋊C3—41no—
102C3×#137—91no—
103C3×AGL(1,ℤ/8)—91no—
104C3×#131—91no—
105#951—31no—
106#951—91no—
107#949—31no—
108#949—31no—
109#949—61no—
110#948—31no—
111#948—91no—
112C2×#215—91no—
113C2×#214—91no—
114C2×#204—91no—
115#836—91no—
116#833—91no—
117#824—91no—
118#824—91no—
119#823—91no—
120#823—91no—
121#822—91no—
122#822—91no—
123#819—91no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C22×C3×D3—121no—
127C32×Q8—21no—
128C32×Q8D1211yescharacteristic
129C32×Q8#6711yescharacteristic
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—41no—
134C32×D4D1211yescharacteristic
135C32×D4#6711yescharacteristic
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—41no—
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C3⋊Dic3—21no—
143#634—21no—
144#634—21no—
145C2×C3×Dic3—61no—
146C3×D12—62no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—121no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)#6711yescharacteristic
157C3⋊(C3⋊C8)#6711yescharacteristic
158C3×C3⋊C8—31no—
159C3×C3⋊C8—31no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—61no—
165#471—271noSylow
166C2×H(3,𝔽3)—21no—
167C2×H(3,𝔽3)—41no—
168C2×H(3,𝔽3)#12111yescharacteristic
169#272—81no—
170C3×#41—31no—
171C3×#41—31no—
172C3×#41—61no—
173C3×#41D3211yescharacteristic
174C2×C3×Q8—91no—
175C2×C3×D4—91no—
176C2×D12—31no—
177C2×D12—31no—
178C2×D12—61no—
179C2×Dic6—31no—
180C2×Dic6—91no—
181C3×Q16—181no—
182C3×SD16—181no—
183C3×SD16—181no—
184C3×D8—181no—
185C3×M16—91no—
186C3×M16—91no—
187C3×M16—91no—
188C3×M16—181no—
189C2×C3×C8—91no—
190C2×C3×C8—181no—
191#216—61no—
192#216—181no—
193#215—61no—
194#215—61no—
195#215—62no—
196#215—181no—
197#215—181no—
198#214—61no—
199#214—181no—
200#214—181no—
201#214—181no—
202#213—61no—
203#213—61no—
204#213—62no—
205#208—31no—
206#208—31no—
207#208—31no—
208#208—31no—
209#208—61no—
210#208—61no—
211#208—91no—
212#208—91no—
213#208—181no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—61no—
216C2×C3⋊C8—91no—
217C2×C3⋊C8—181no—
218#204—181no—
219#204—181no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—41no—
224C22×C32#23511yescharacteristic
225C22×C32#23711yescharacteristic
226C2×C3×D3—122no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—242no—
230C2×C3×D3—241no—
231C2×C3×D3—241no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—41no—
237C32×C4C2×D1211yescharacteristic
238C32×C4D3×D411yescharacteristic
239C32×C4C2×#6711yescharacteristic
240C32×C4C2×#6711yescharacteristic
241C3⋊Dic3—41no—
242C3×Dic3—121no—
243#137—91no—
244AGL(1,ℤ/8)—91no—
245C2×SD16—271no—
246#131—91no—
247#108—271no—
248#101—271no—
249#100—271no—
250H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
251C23×C3—181no—
252C22×D3—61no—
253C22×D3—61no—
254C22×D3—121no—
255C3×Q8—31no—
256C3×Q8—31no—
257C3×Q8—61no—
258C3×Q8—181no—
259C3×Q8—181no—
260C3×Q8#62611yescharacteristic
261C3×D4—21no—
262C3×D4—31no—
263C3×D4—31no—
264C3×D4—61no—
265C3×D4—61no—
266C3×D4—61no—
267C3×D4—121no—
268C3×D4—181no—
269C3×D4—181no—
270C3×D4#62611yescharacteristic
271C2×C3×C4—21no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—121no—
278C2×C3×C4—181no—
279C2×C3×C4C2×D3211yescharacteristic
280C2×Dic3—61no—
281C2×Dic3—181no—
282D12—61no—
283D12—61no—
284D12—61no—
285D12—61no—
286D12—62no—
287D12—121no—
288Dic6—61no—
289Dic6—61no—
290Dic6—181no—
291Dic6—181no—
292C3×C8—91no—
293C3×C8—91no—
294C3×C8—181no—
295C3×C8—181no—
296C3×C8—181no—
297C3⋊C8—31no—
298C3⋊C8—31no—
299C3⋊C8—61no—
300C3⋊C8—61no—
301C3⋊C8—62no—
302C3⋊C8—61no—
303C3⋊C8—61no—
304C3⋊C8—91no—
305C3⋊C8—91no—
306C3⋊C8—181no—
307C3⋊C8—181no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—41no—
314C2×C32—81no—
315C2×C32#93011yescharacteristic
316C2×C32#93911yescharacteristic
317C3×D3—242no—
318C3×D3—241no—
319C3×D3—241no—
320#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
321C2×Q8—91no—
322C2×D4—91no—
323Q16—181no—
324SD16—181no—
325SD16—181no—
326SD16—541no—
327SD16—541no—
328D8—181no—
329M16—91no—
330M16—91no—
331M16—91no—
332M16—181no—
333C2×C8—91no—
334C2×C8—181no—
335C22×C3—21no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—61no—
339C22×C3—61no—
340C22×C3—61no—
341C22×C3—121no—
342C22×C3—361no—
343C22×C3—721no—
344C22×C3#380411yescharacteristic
345C2×D3—121no—
346C2×D3—121no—
347C2×D3—122no—
348C2×D3—242no—
349C2×D3—241no—
350C2×D3—241no—
351C3×C4—21no—
352C3×C4—31no—
353C3×C4—31no—
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—61no—
357C3×C4—61no—
358C3×C4—61no—
359C3×C4—61no—
360C3×C4—121no—
361C3×C4—361no—
362C3×C4C2×#62611yescharacteristic
363C3×C4D3×#6711yescharacteristic
364Dic3—121no—
365Dic3—361no—
366C32—21no—
367C32#514111yescharacteristic, complemented
368C32#520311yescharacteristic, complemented
369C23—181no—
370Q8—181no—
371Q8—181no—
372Q8#620011yescharacteristic
373D4—21no—
374D4—181no—
375D4—181no—
376D4#620011yescharacteristic
377C2×C4—21no—
378C2×C4—181no—
379C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
380C8—91no—
381C8—91no—
382C8—181no—
383C8—181no—
384C8—181no—
385C2×C3—21no—
386C2×C3—31no—
387C2×C3—31no—
388C2×C3—41no—
389C2×C3—61no—
390C2×C3—61no—
391C2×C3—61no—
392C2×C3—121no—
393C2×C3—121no—
394C2×C3—121no—
395C2×C3—241no—
396C2×C3—721no—
397C2×C3#6397811yescharacteristic
398D3—241no—
399D3—241no—
400D3—242no—
401C22—21no—
402C22—361no—
403C22—721no—
404C22#8867111yescharacteristic
405C4—21no—
406C4—361no—
407C4C2×#620011yescharacteristic
408C4#8867611yescharacteristic
409C3—31no—
410C3—31no—
411C3—61no—
412C3#1059168011yescharacteristic
413C2—21no—
414C2—41no—
415C2—721no—
416C2#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
prime radical#11721870C2
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#471
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
2. #64160⋊C3C2semidirect
3. #11721689C2semidirect
4. C2×#88437C2semidirect
5. #11720325C2semidirect
6. #11720321C2semidirect
7. #11720319C2semidirect
25. #64160D3semidirect
26. #64157D3semidirect
250. H(3,𝔽3)#471semidirect
367. C32#5141semidirect
368. C32#5203semidirect

Maximal subgroups

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

subgroupindex
#117218702
#64160⋊C32
#117216892
C2×#884372
#117203252
#117203212
#117203192
#105916813
#105916793