Finite Group Databasebeta

← #13140926

Subgroups — #13140926

Order 1728.

563 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 97 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×#88722C211yesmaximal, characteristic
2#11721053C212yesmaximal, complemented
3#11720584C211yesmaximal, characteristic, complemented
4#63663⋊C3C211yesmaximal, characteristic, complemented
5#11720320C212yesmaximal, complemented
6#10591640—31nomaximal
7#10591637—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9#88722—22no—
10#88722—22no—
11C2×C4⋊((C3⋊D3)⋊C3)C412yes—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C412yes—
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15#88538C2212yes—
16#3748⋊C3C2212yes—
17#88430—22no—
18((C3⋊D3)⋊C3)⋊C8—22no—
19C2×#3832D311yescharacteristic
20C2×C3×#239—31no—
21#63818—32no—
22D3×#208—32no—
23#63663D311yescharacteristic, complemented
24C3×#836—31no—
25C3×#836—31no—
26C3×#836—31no—
27C3×#836—61no—
28#63559—32no—
29#63558—32no—
30C22×(C3⋊D3)⋊C3C2×C412yes—
31C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
32Q8×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)C2×C412yes—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
36C4⋊((C3⋊D3)⋊C3)—24no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38C4×(C3⋊D3)⋊C3—22no—
39C4×(C3⋊D3)⋊C3—22no—
40C4×(C3⋊D3)⋊C3D412yes—
41C4×(C3⋊D3)⋊C3D412yes—
42#6180—22no—
43(C3⋊(C3⋊C8))⋊C3—22no—
44#5237—91no—
45#4751—91no—
46#4751—91no—
47C2×C32×Q8—21no—
48C2×C32×Q8C2×D311yescharacteristic
49C2×C32×Q8C2×D311yescharacteristic
50#3832—22no—
51#3832—22no—
52C2×C4⋊(C3⋊D3)Dic312yes—
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C2×C4×C3⋊D3Dic312yes—
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C3×#239—62no—
57C3×#239—62no—
58C2×C3×D12—32no—
59C2×C3×D12—31no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—32no—
62#3748C2×D312yes—
63C3×#208—32no—
64C3×#208—32no—
65C3×#208—32no—
66C3×#208—62no—
67#3712—62no—
68#3711—62no—
69#3710—62no—
70D3×C3⋊C8—62no—
71C22×H(3,𝔽3)C22×C411yescharacteristic
72C2×(C3⋊D3)⋊C3—22no—
73C2×(C3⋊D3)⋊C3—41no—
74C2×(C3⋊D3)⋊C3—42no—
75C2×(C3⋊D3)⋊C3C22⋊C412yes—
76C4×H(3,𝔽3)—22no—
77C4×H(3,𝔽3)C2×D412yes—
78(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
79(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
80C2×C3×#41—91no—
81C2×#239—31no—
82C2×#239—91no—
83C2×#208—92no—
84D3×M16—92no—
85D3×M16—92no—
86C3×#101—91no—
87C3×#101—91no—
88#836—31no—
89#836—31no—
90#836—31no—
91#836—61no—
92#836—91no—
93#833—92no—
94#823—92no—
95#823—92no—
96C22×C3⋊D3C2×Dic312yes—
97C22×C3⋊D3C2×Dic311yescharacteristic
98C22×C3×D3—31no—
99C22×C3×D3—32no—
100C32×Q8—22no—
101C32×Q8—22no—
102C32×Q8—42no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4C4×D312yes—
106C2×C32×C4C2×Dic312yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3C2×Dic311yescharacteristic
110C4⋊(C3⋊D3)—24no—
111C4⋊(C3⋊D3)—22no—
112C4×C3⋊D3—22no—
113C4×C3⋊D3—22no—
114C4×C3⋊D3#6712yes—
115C4×C3⋊D3#6712yes—
116C2×C3×Dic3—31no—
117C3×D12—64no—
118C3×D12—62no—
119C3×C4×D3—32no—
120C3×C4×D3—32no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3⋊(C3⋊C8)—22no—
124C3×C3⋊C8—62no—
125C3×C3⋊C8—62no—
126C3×C3⋊C8—62no—
127C3×C3⋊C8—122no—
128#413—271noSylow
129C2×H(3,𝔽3)—21no—
130C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
131(C3⋊D3)⋊C3—22no—
132(C3⋊D3)⋊C3—42no—
133C3×#41—182no—
134C3×#41—182no—
135C2×C3×Q8—31no—
136C2×C3×Q8—31no—
137C2×C3×Q8—61no—
138C2×C3×Q8D3211yescharacteristic
139C2×C3×D4—91no—
140C2×C3×D4—92no—
141C22×C3×C4—91no—
142C22×C3×C4—92no—
143#239—62no—
144#239—62no—
145#239—182no—
146#239—182no—
147C2×D12—32no—
148C2×D12—31no—
149C2×D12—91no—
150C2×D12—92no—
151C2×C4×D3—31no—
152C2×C4×D3—32no—
153C2×C4×D3—91no—
154C2×C4×D3—92no—
155C3×M16—92no—
156C3×M16—92no—
157#208—32no—
158#208—32no—
159#208—32no—
160#208—62no—
161#208—92no—
162#208—182no—
163C2×C3⋊C8—182no—
164ΓL(1,𝔽25)—182no—
165ΓL(1,𝔽25)—182no—
166D3×C8—182no—
167D3×C8—182no—
168C22×C32—21no—
169C22×C32C2×C4×D311yescharacteristic
170C22×C32C22×Dic311yescharacteristic
171C2×C3⋊D3—22no—
172C2×C3⋊D3—41no—
173C2×C3⋊D3—42no—
174C2×C3⋊D3#21712yes—
175C2×C3×D3—32no—
176C2×C3×D3—62no—
177C2×C3×D3—121no—
178C2×C3×D3—122no—
179C32×C4—22no—
180C32×C4—22no—
181C32×C4—22no—
182C32×C4—42no—
183C32×C4D3×D412yes—
184C32×C4C2×#6712yes—
185C3⋊Dic3#21711yescharacteristic
186C3⋊Dic3#21711yescharacteristic
187C3×Dic3—31no—
188C3×Dic3—31no—
189C2×#41—91no—
190C2×M16—272no—
191#101—91no—
192#101—91no—
193#100—272no—
194H(3,𝔽3)#41311yescharacteristic, complemented, Sylow
195C23×C3—91no—
196C23×C3—92no—
197C22×D3—31no—
198C22×D3—32no—
199C22×D3—91no—
200C22×D3—92no—
201C3×Q8—22no—
202C3×Q8—62no—
203C3×Q8—62no—
204C3×Q8—122no—
205C3×D4—182no—
206C3×D4—184no—
207C2×C3×C4—32no—
208C2×C3×C4—32no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—62no—
212C2×C3×C4—61no—
213C2×C3×C4—92no—
214C2×C3×C4—92no—
215C2×C3×C4—91no—
216C2×C3×C4—182no—
217C2×C3×C4—182no—
218C2×C3×C4D3×Dic312yes—
219C2×C3×C4C2×D3211yescharacteristic
220C2×Dic3—31no—
221C2×Dic3—91no—
222D12—64no—
223D12—62no—
224D12—182no—
225D12—184no—
226C4×D3—32no—
227C4×D3—32no—
228C4×D3—62no—
229C4×D3—62no—
230C4×D3—92no—
231C4×D3—92no—
232C4×D3—182no—
233C4×D3—182no—
234C3×C8—182no—
235C3×C8—182no—
236C3⋊C8—62no—
237C3⋊C8—62no—
238C3⋊C8—62no—
239C3⋊C8—122no—
240C3⋊C8—182no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—41no—
245C2×C32D3×C22⋊C411yescharacteristic
246C2×C32C2×#21711yescharacteristic
247C3⋊D3—22no—
248C3⋊D3—42no—
249C3×D3—62no—
250C3×D3—122no—
251#41—182no—
252#41—182no—
253C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
254C2×D4—92no—
255C2×D4—91no—
256C22×C4—91no—
257C22×C4—92no—
258M16—92no—
259M16—92no—
260M16—542no—
261C2×C8—542no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—61no—
265C22×C3—92no—
266C22×C3—182no—
267C22×C3—361no—
268C22×C3—362no—
269C22×C3C2×D3×Dic311yescharacteristic
270C2×D3—32no—
271C2×D3—62no—
272C2×D3—92no—
273C2×D3—121no—
274C2×D3—122no—
275C2×D3—182no—
276C2×D3—361no—
277C2×D3—362no—
278C3×C4—22no—
279C3×C4—32no—
280C3×C4—32no—
281C3×C4—62no—
282C3×C4—62no—
283C3×C4—62no—
284C3×C4—91no—
285C3×C4—91no—
286C3×C4—122no—
287C3×C4D3×#6712yes—
288Dic3—31no—
289Dic3—31no—
290Dic3—91no—
291Dic3—91no—
292C32—21no—
293C32#475111yescharacteristic, complemented
294C32#523711yescharacteristic, complemented
295C23—91no—
296C23—92no—
297Q8—22no—
298D4—184no—
299D4—182no—
300C2×C4—91no—
301C2×C4—92no—
302C2×C4—92no—
303C2×C4—182no—
304C2×C4—182no—
305C2×C4((C3⋊D3)⋊C3)⋊C412yes—
306C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
307C8—182no—
308C8—182no—
309C2×C3—21no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—61no—
313C2×C3—61no—
314C2×C3—61no—
315C2×C3—121no—
316C2×C3—182no—
317C2×C3—362no—
318C2×C3D3×#21711yescharacteristic
319D3—62no—
320D3—122no—
321D3—182no—
322D3—362no—
323C22—92no—
324C22—182no—
325C22—361no—
326C22—362no—
327C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
328C4—22no—
329C4—91no—
330C4—91no—
331C4#8867612yes—
332C3—31no—
333C3—31no—
334C3—61no—
335C3#1059163711yescharacteristic
336C2—21no—
337C2—182no—
338C2—362no—
339C2#1172131011yescharacteristic
subgroupG/H
centerC2#11721310
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC2×C3D3×#217
prime radicalC2×#88722C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#413
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11721053C2semidirect
3. #11720584C2semidirect
4. #63663⋊C3C2semidirect
5. #11720320C2semidirect
23. #63663D3semidirect
194. H(3,𝔽3)#413semidirect
293. C32#4751semidirect
294. C32#5237semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
#117210532
#117205842
#63663⋊C32
#117203202
#105916403
#105916373