Finite Group Databasebeta

← #13159558

Subgroups — #13159558

Order 1728.

884 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 144 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#11723531C211yesmaximal, characteristic
2#11723503C211yesmaximal, characteristic, complemented
3#11722190C211yesmaximal, characteristic, complemented
4C3×#63645C211yesmaximal, characteristic, complemented
5C32×#821C211yesmaximal, characteristic, complemented
6#11722056C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592473—31nomaximal
9#10590650—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×#6291—21no—
13C2×#6289—21no—
14C2×C33×C8C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88806—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#64147—31no—
22C4⋊C4×C3⋊D3—31no—
23#63916—34no—
24#63900—34no—
25C32×#102—31no—
26#63659—31no—
27#63647—31no—
28#63647—34no—
29#63647—64no—
30#63645D311yescharacteristic, complemented
31#63634—31no—
32C3×#821D314yescomplemented
33C3×#818—34no—
34#63576—34no—
35#63562—34no—
36C22×C3⋊(C3⋊D3)D411yescharacteristic
37C22×C32×D3—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C4×C3⋊(C3⋊D3)—21no—
41C4×C3⋊(C3⋊D3)—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C32×Dic3—21no—
45C32×D12—21no—
46C32×D12—21no—
47#6291—42no—
48#6289—42no—
49C33×C8—21no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51#4882—91no—
52#4774—94no—
53C2×C32×D4—31no—
54C2×#638—61no—
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—34no—
57C2×C4×C3⋊D3—61no—
58C2×C4×C3⋊D3—64no—
59C2×C3×D12C2×D314yes—
60C2×#626—64no—
61C2×#624—64no—
62C2×C32×C8—24no—
63C2×C32×C8C2×D314yes—
64C2×C32×C8C2×D311yescharacteristic
65C32×C4⋊C4—31no—
66#3757—61no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69#3750—61no—
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—34no—
72C2×C3⋊(C3⋊C8)—64no—
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3C2×D314yes—
76#3724—64no—
77(C2×D3)⋊Dic3—64no—
78C22×C33C2×D411yescharacteristic
79C2×C3⋊(C3⋊D3)—21no—
80C2×C3⋊(C3⋊D3)—21no—
81C2×C3⋊(C3⋊D3)D812yes—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C2×C32×D3—41no—
85C33×C4#4111yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—21no—
88C3×C3⋊Dic3—41no—
89C3×C3⋊Dic3—41no—
90C32×Dic3—41no—
91#938—94no—
92#895—91no—
93D3×C4⋊C4—94no—
94D3×C4⋊C4—91no—
95C3×#107—91no—
96C3×#102—34no—
97#832—94no—
98#821C3⋊D311yescharacteristic, complemented
99(C2×D3)⋊C8—94no—
100(C2×D3)⋊C8—91no—
101(C2×D3)⋊C8—184no—
102#818—34no—
103#809—91no—
104#807—94no—
105#807—91no—
106C23×C32—61no—
107C22×C3⋊D3—31no—
108C22×C3⋊D3—34no—
109C22×C3⋊D3—64no—
110C22×C3×D3—24no—
111C32×D4—61no—
112C32×D4—61no—
113C2×C32×C4—24no—
114C2×C32×C4—61no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117#638—122no—
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—34no—
122C2×C3⋊Dic3—64no—
123C4×C3⋊D3—61no—
124C4×C3⋊D3—61no—
125C4×C3⋊D3—64no—
126C4×C3⋊D3—64no—
127C4×C3⋊D3—124no—
128C4×C3⋊D3—124no—
129C4×C3⋊D3—122no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—64no—
132C2×C3×Dic3—64no—
133C3×D12—24no—
134C3×D12—24no—
135#626—128no—
136#624—128no—
137C32×C8—21no—
138C32×C8—24no—
139C32×C8—44no—
140C3⋊(C3⋊C8)—61no—
141C3⋊(C3⋊C8)—64no—
142C3⋊(C3⋊C8)—124no—
143#480—271noSylow
144C2×C33#12311yescharacteristic
145C2×C33C2×D811yescharacteristic
146C2×C33#13711yescharacteristic
147C3⋊(C3⋊D3)—22no—
148C32×D3—81no—
149C2×C3×D4—34no—
150C2×#67—184no—
151C2×D12—181no—
152C2×D12C2×C3⋊D311yescharacteristic
153C2×C4×D3—94no—
154C2×C4×D3—91no—
155C2×C4×D3—184no—
156C2×C4×D3—181no—
157C2×C4×D3—184no—
158C2×C3×C8—24no—
159C2×C3×C8D3214yes—
160C2×C3×C8C2×C3⋊D311yescharacteristic
161C3×C4⋊C4—34no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164#217—184no—
165(C2×D3)⋊C4—181no—
166C4⋊Dic3—34no—
167C4⋊Dic3—34no—
168C4⋊Dic3C2×C3⋊D311yescharacteristic
169Dic3⋊C4—181no—
170Dic3⋊C4—184no—
171C2×C3⋊C8—94no—
172C2×C3⋊C8—91no—
173C2×C3⋊C8—184no—
174C22×C32—24no—
175C22×C32—121no—
176C22×C32—121no—
177C22×C32—121no—
178C22×C32D3×D414yes—
179C22×C32D3×D411yescharacteristic
180C2×C3⋊D3—32no—
181C2×C3⋊D3—38no—
182C2×C3⋊D3—68no—
183C2×C3⋊D3—61no—
184C2×C3⋊D3—61no—
185C2×C3⋊D3—64no—
186C2×C3⋊D3—64no—
187C2×C3⋊D3—124no—
188C2×C3⋊D3—124no—
189C2×C3×D3—44no—
190C2×C3×D3—44no—
191C2×C3×D3—44no—
192C32×C4—24no—
193C32×C4—24no—
194C32×C4—121no—
195C32×C4#23514yes—
196C32×C4#23511yescharacteristic
197C32×C4#23714yes—
198C32×C4#23911yescharacteristic
199C3⋊Dic3—41no—
200C3⋊Dic3—41no—
201C3⋊Dic3—61no—
202C3⋊Dic3—64no—
203C3⋊Dic3—124no—
204C3×Dic3—44no—
205C3×Dic3—124no—
206C3×Dic3—124no—
207#121—271no—
208C2×C4⋊C4—271no—
209#107—91no—
210#107—271no—
211#102—31no—
212#102—271no—
213C22⋊C8—271no—
214C33#48011yescharacteristic, complemented, Sylow
215C23×C3—64no—
216C22×D3—21no—
217C22×D3—94no—
218C22×D3—91no—
219C22×D3—184no—
220C3×D4—64no—
221C3×D4—64no—
222C2×C3×C4—24no—
223C2×C3×C4—64no—
224C2×C3×C4—181no—
225C2×C3×C4—181no—
226C2×C3×C4C2×D3214yes—
227C2×C3×C4C22×C3⋊D311yescharacteristic
228#67—368no—
229C2×Dic3—21no—
230C2×Dic3—64no—
231C2×Dic3—64no—
232C2×Dic3—94no—
233C2×Dic3—91no—
234C2×Dic3—184no—
235D12—21no—
236D12—21no—
237D12—362no—
238C4×D3—184no—
239C4×D3—184no—
240C4×D3—181no—
241C4×D3—181no—
242C4×D3—364no—
243C4×D3—364no—
244C4×D3—362no—
245C4×D3—368no—
246C3×C8—24no—
247C3×C8—21no—
248C3×C8—44no—
249C3⋊C8—184no—
250C3⋊C8—181no—
251C3⋊C8—364no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32—24no—
255C2×C32—241no—
256C2×C32#88314yes—
257C2×C32#89411yescharacteristic
258C2×C32D3×D814yes—
259C2×C32D3×D811yescharacteristic
260C2×C32#91514yes—
261C2×C32#91811yescharacteristic
262C3⋊D3—62no—
263C3⋊D3—68no—
264C3⋊D3—128no—
265C3×D3—84no—
266C2×D4—31no—
267C2×D4—541no—
268C22×C4—271no—
269C22×C4—541no—
270C2×C8—271no—
271C2×C8D3×C3⋊D311yescharacteristic
272C4⋊C4—31no—
273C4⋊C4—91no—
274C4⋊C4—91no—
275C4⋊C4—541no—
276C22⋊C4—541no—
277C22×C3—24no—
278C22×C3—124no—
279C22×C3—124no—
280C22×C3—124no—
281C22×C3#380214yes—
282C22×C3D4×C3⋊D311yescharacteristic
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—98no—
287C2×D3—92no—
288C2×D3—188no—
289C2×D3—184no—
290C2×D3—181no—
291C2×D3—184no—
292C2×D3—181no—
293C2×D3—364no—
294C2×D3—364no—
295C3×C4—24no—
296C3×C4—24no—
297C3×C4—124no—
298C3×C4—361no—
299C3×C4—361no—
300C3×C4#379914yes—
301C3×C4#380414yes—
302C3×C4#382811yescharacteristic
303C3×C4#383011yescharacteristic
304Dic3—41no—
305Dic3—124no—
306Dic3—124no—
307Dic3—181no—
308Dic3—184no—
309Dic3—364no—
310C32—24no—
311C32#477414yescomplemented
312C32#488211yescharacteristic, complemented
313C23—61no—
314C23—271no—
315D4—61no—
316D4—61no—
317D4—1082no—
318C2×C4—61no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4—271no—
322C2×C4—541no—
323C2×C4—541no—
324C2×C4—1082no—
325C2×C4C2×D3×C3⋊D311yescharacteristic
326C8—21no—
327C8—541no—
328C2×C3—24no—
329C2×C3—24no—
330C2×C3—24no—
331C2×C3—244no—
332C2×C3#6379714yes—
333C2×C3#6381114yes—
334C2×C3#6385414yes—
335C2×C3#6409211yescharacteristic
336C2×C3D8×C3⋊D311yescharacteristic
337C2×C3#6412411yescharacteristic
338D3—81no—
339D3—188no—
340D3—182no—
341D3—368no—
342C22—121no—
343C22—121no—
344C22—121no—
345C22—272no—
346C22—541no—
347C22—541no—
348C22#8902611yescharacteristic
349C4—121no—
350C4—361no—
351C4—361no—
352C4—541no—
353C4#8902211yescharacteristic
354C4#8902811yescharacteristic
355C3—24no—
356C3#1059065014yescomplemented
357C3#1059247311yescharacteristic, complemented
358C2—241no—
359C2—542no—
360C2#1172332211yescharacteristic
361C2#1172334411yescharacteristic
362C2#1172342211yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radical#11723531C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#480
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11723503C2semidirect
3. #11722190C2semidirect
4. C3×#63645C2semidirect
5. C32×#821C2semidirect
6. #11722056C2semidirect
7. #11722031C2semidirect
30. #63645D3semidirect
32. C3×#821D3semidirect
98. #821C3⋊D3semidirect
214. C33#480semidirect
311. C32#4774semidirect
312. C32#4882semidirect
356. C3#10590650semidirect
357. C3#10592473semidirect

Maximal subgroups

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

subgroupindex
#117235312
#117235032
#117221902
C3×#636452
C32×#8212
#117220562
#117220312
#105924733
#105906503