Finite Group Databasebeta

← #13139019

Subgroups — #13139019

Order 1728.

440 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721776C211yesmaximal, characteristic, complemented
2C2×#3745⋊C3C211yesmaximal, characteristic, complemented
3#11721071C212yesmaximal, complemented
4#63646⋊C3C211yesmaximal, characteristic, complemented
5#11720320C212yesmaximal, complemented
6#10590927—31nomaximal
7#10590926—31nomaximal
8#88740C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88538C2211yescharacteristic
12#88527—21no—
13C8×#272—21no—
14#3748⋊C3C2212yes—
15#3745⋊C3—21no—
16#3745⋊C3—21no—
17#88438—22no—
18#88436—22no—
19C2×#3745D311yescharacteristic, complemented
20C2×C3×D24—31no—
21C3×#901—31no—
22C3×#901—31no—
23C3×#901—61no—
24#63825—32no—
25#63823—32no—
26#63646D311yescharacteristic, complemented
27C3×#819—31no—
28#63559—32no—
29#63558—32no—
30C22×(C3⋊D3)⋊C3—22no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32#6235—21no—
33C4⋊#272D411yescharacteristic
34C4×#272—21no—
35#6229D411yescharacteristic
36C4⋊((C3⋊D3)⋊C3)—22no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38C8×H(3,𝔽3)D411yescharacteristic
39C8×H(3,𝔽3)D411yescharacteristic
40#6180—21no—
41(C3⋊(C3⋊C8))⋊C3—22no—
42#5160—91no—
43#5142—91no—
44#4897—91no—
45C2×C4⋊(C3⋊D3)C2×D312yes—
46C3×#235—31no—
47C3×#235—31no—
48C3×#235—61no—
49C2×C3×D12—32no—
50C2×C32×C8—21no—
51C2×C32×C8C2×D311yescharacteristic
52C2×C32×C8C2×D311yescharacteristic
53#3748C2×D312yes—
54#3745—21no—
55#3745—21no—
56C3×#208—31no—
57C3×#208—31no—
58C3×#208—32no—
59C3×#208—61no—
60C3×D24—61no—
61C3×D24—61no—
62C3×ΓL(1,𝔽25)—62no—
63C3×ΓL(1,𝔽25)—61no—
64C3×ΓL(1,𝔽25)—61no—
65C3×D3×C8—62no—
66C3×D3×C8—61no—
67C3×D3×C8—61no—
68#3717—62no—
69#3715—62no—
70#3714—62no—
71#3713—62no—
72C22×H(3,𝔽3)#4111yescharacteristic
73C2×#272—21no—
74C2×(C3⋊D3)⋊C3—42no—
75C2×(C3⋊D3)⋊C3—82no—
76C4×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)C2×D411yescharacteristic
78#1084—21no—
79C2×C3×D8—91no—
80C3×#131—91no—
81#949—92no—
82#932—92no—
83#908—92no—
84C2×D24—31no—
85C2×D24—91no—
86#901—31no—
87#901—31no—
88#901—61no—
89C3×#108—91no—
90#833—92no—
91#823—92no—
92#823—92no—
93#819—31no—
94#819—91no—
95C22×C3⋊D3—22no—
96C22×C3×D3—62no—
97C2×C32×C4—21no—
98C2×C32×C4C22×D311yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C4⋊(C3⋊D3)—22no—
101C4⋊(C3⋊D3)—22no—
102C3×#67—62no—
103C3×#67—61no—
104C3×#67—61no—
105C3×D12—31no—
106C3×D12—31no—
107C3×D12—61no—
108C3×D12—62no—
109C3×D12—62no—
110C3×C4×D3—62no—
111C3×C4×D3—61no—
112C3×C4×D3—61no—
113C3×Dic6—31no—
114C3×Dic6—31no—
115C3×Dic6—61no—
116C32×C8—21no—
117C32×C8—21no—
118C32×C8D1211yescharacteristic
119C32×C8D1211yescharacteristic
120C32×C8#6711yescharacteristic
121C32×C8#6711yescharacteristic
122C3⋊(C3⋊C8)—22no—
123C3×C3⋊C8—62no—
124C3×C3⋊C8—61no—
125C3×C3⋊C8—62no—
126C3×C3⋊C8—61no—
127#472—271noSylow
128C2×H(3,𝔽3)—21no—
129C2×H(3,𝔽3)#12111yescharacteristic
130#272—41no—
131(C3⋊D3)⋊C3—82no—
132C3×#41—91no—
133C2×C3×D4—92no—
134#235—31no—
135#235—31no—
136#235—61no—
137C2×D12—32no—
138C2×D12—92no—
139C3×D8—181no—
140C3×D8—181no—
141C3×M16—92no—
142C3×M16—91no—
143C3×M16—181no—
144C2×C3×C8—31no—
145C2×C3×C8—31no—
146C2×C3×C8—61no—
147C2×C3×C8—181no—
148C2×C3×C8D3211yescharacteristic
149#215—182no—
150#214—182no—
151#213—182no—
152#213—182no—
153#208—31no—
154#208—31no—
155#208—32no—
156#208—61no—
157#208—92no—
158D24—61no—
159D24—61no—
160D24—181no—
161D24—181no—
162D24—182no—
163#204—182no—
164ΓL(1,𝔽25)—61no—
165ΓL(1,𝔽25)—61no—
166ΓL(1,𝔽25)—62no—
167D3×C8—61no—
168D3×C8—61no—
169D3×C8—62no—
170C22×C32—21no—
171C22×C32#23711yescharacteristic
172C22×C32#23911yescharacteristic
173C2×C3⋊D3—42no—
174C2×C3⋊D3—82no—
175C2×C3×D3—62no—
176C2×C3×D3—61no—
177C2×C3×D3—61no—
178C2×C3×D3—122no—
179C2×C3×D3—242no—
180C32×C4—21no—
181C32×C4—21no—
182C32×C4C2×D1211yescharacteristic
183C32×C4D3×D411yescharacteristic
184C32×C4D3×D411yescharacteristic
185C32×C4C2×#6711yescharacteristic
186C3×Dic3—62no—
187C3×Dic3—61no—
188C3×Dic3—61no—
189AGL(1,ℤ/8)—272no—
190C2×D8—91no—
191#131—91no—
192#108—91no—
193#100—272no—
194H(3,𝔽3)#47211yescharacteristic, complemented, Sylow
195C23×C3—182no—
196C22×D3—62no—
197C22×D3—182no—
198C3×Q8—91no—
199C3×D4—91no—
200C3×D4—182no—
201C3×D4—182no—
202C3×D4—181no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—61no—
206C2×C3×C4—181no—
207C2×C3×C4C2×D3211yescharacteristic
208#67—61no—
209#67—61no—
210#67—62no—
211D12—31no—
212D12—31no—
213D12—61no—
214D12—62no—
215D12—62no—
216D12—182no—
217D12—182no—
218C4×D3—61no—
219C4×D3—61no—
220C4×D3—62no—
221Dic6—31no—
222Dic6—31no—
223Dic6—61no—
224C3×C8—31no—
225C3×C8—31no—
226C3×C8—31no—
227C3×C8—31no—
228C3×C8—61no—
229C3×C8—61no—
230C3×C8—182no—
231C3×C8—181no—
232C3×C8#62611yescharacteristic
233C3×C8#62611yescharacteristic
234C3⋊C8—61no—
235C3⋊C8—61no—
236C3⋊C8—62no—
237C3⋊C8—62no—
238C3⋊C8—182no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—41no—
243C2×C32#89511yescharacteristic
244C2×C32#93811yescharacteristic
245C3⋊D3—82no—
246C3×D3—122no—
247C3×D3—121no—
248C3×D3—121no—
249C3×D3—242no—
250#41—91no—
251C2×D4—92no—
252SD16—542no—
253D8—181no—
254D8—181no—
255D8—542no—
256M16—91no—
257M16—92no—
258M16—181no—
259C2×C8—181no—
260C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
261C22×C3—31no—
262C22×C3—31no—
263C22×C3—61no—
264C22×C3—181no—
265C22×C3—362no—
266C22×C3—722no—
267C22×C3#379611yescharacteristic
268C2×D3—61no—
269C2×D3—61no—
270C2×D3—62no—
271C2×D3—122no—
272C2×D3—242no—
273C2×D3—362no—
274C2×D3—722no—
275C3×C4—31no—
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—61no—
280C3×C4—61no—
281C3×C4—181no—
282C3×C4#380211yescharacteristic
283C3×C4C2×#62611yescharacteristic
284Dic3—61no—
285Dic3—61no—
286Dic3—62no—
287C32—21no—
288C32#489711yescharacteristic, complemented
289C32#516011yescharacteristic, complemented
290C23—182no—
291Q8—91no—
292D4—91no—
293D4—181no—
294D4—182no—
295D4—182no—
296C2×C4—181no—
297C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
298C8—182no—
299C8—181no—
300C8#620011yescharacteristic
301C8#620011yescharacteristic
302C2×C3—21no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—61no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3—121no—
309C2×C3—361no—
310C2×C3—722no—
311C2×C3#6391611yescharacteristic
312D3—121no—
313D3—121no—
314D3—122no—
315D3—242no—
316D3—722no—
317C22—181no—
318C22—362no—
319C22—722no—
320C22#8864911yescharacteristic
321C4—181no—
322C4#8865511yescharacteristic
323C4C2×#620011yescharacteristic
324C3—31no—
325C3—31no—
326C3—61no—
327C3#1059092911yescharacteristic
328C2—21no—
329C2—361no—
330C2—722no—
331C2#1172115911yescharacteristic
subgroupG/H
centerC2#11721159
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3#63916
hypercenterC2×C8((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 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11721776C2semidirect
2. C2×#3745⋊C3C2semidirect
3. #11721071C2semidirect
4. #63646⋊C3C2semidirect
5. #11720320C2semidirect
19. C2×#3745D3semidirect
26. #63646D3semidirect
194. H(3,𝔽3)#472semidirect
288. C32#4897semidirect
289. C32#5160semidirect

Maximal subgroups

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

subgroupindex
#117217762
C2×#3745⋊C32
#117210712
#63646⋊C32
#117203202
#105909273
#105909263