Finite Group Databasebeta

← #13156644

Subgroups — #13156644

Order 1728.

730 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 114 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#11723678C211yesmaximal, characteristic, complemented
2#11723429C211yesmaximal, characteristic
3C2×#88800C212yesmaximal, complemented
4C2×C3×#3716C212yesmaximal
5C3×#63555C211yesmaximal, characteristic, complemented
6#10591493—31nomaximal
7#10590897—32nomaximal
8C2×#6308C2211yescharacteristic
9C2×C3×#634C2212yes—
10C2×C32×D12C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C32×C3⋊C8C2212yes—
13#88815—22no—
14(C2×D3)⋊(C3⋊Dic3)—22no—
15#88800—24no—
16C3×#3716—24no—
17#64145—31no—
18C2×#3746—32no—
19C2×C3×#214—32no—
20C2×C3×#204—32no—
21#63873—31no—
22#63866—32no—
23#63860—32no—
24#63860—61no—
25C2×#3719—32no—
26C2×#3719—32no—
27C2×#3719—62no—
28C2×#3716D312yes—
29C3×#815—32no—
30#63555D311yescharacteristic, complemented
31C22×C32×D3—21no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34#6308—22no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3D412yes—
37C3×#634—22no—
38C3×#634—22no—
39C32×D12—22no—
40C32×C3⋊C8D414yes—
41#5167—92no—
42#4715—91no—
43C2×C32×D4—31no—
44C2×#634—31no—
45C2×#634—32no—
46C2×#634—62no—
47C2×#634—61no—
48C2×#634—121no—
49C2×#634C2×D312yes—
50C2×C3×D12—21no—
51C2×C3×D12C2×D312yes—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—32no—
54C2×C3×Dic6—62no—
55C2×C32×C8—32no—
56#3757—62no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58#3746—64no—
59C3×#214—64no—
60C3×C4×Dic3—32no—
61C3×C4×Dic3—61no—
62C2×C3×C3⋊C8—22no—
63C2×C3×C3⋊C8C2×D312yes—
64C2×C3×C3⋊C8C2×D312yes—
65C3×#204—64no—
66#3724—62no—
67Dic3⋊Dic3—64no—
68(C2×D3)⋊Dic3—64no—
69(C2×D3)⋊Dic3—64no—
70#3719—68no—
71#3719—64no—
72#3719—64no—
73#3716—24no—
74C22×C33C2×D411yescharacteristic
75C2×C32×D3—41no—
76C2×C32×D3—42no—
77C33×C4C2×D412yes—
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3—22no—
80C3×C3⋊Dic3—42no—
81C2×C3×SD16—92no—
82#944—92no—
83C2×#216—92no—
84#936—92no—
85#936—181no—
86C2×#214—32no—
87C2×Dic12—92no—
88C2×Dic12—92no—
89C2×Dic12—182no—
90C2×#204—32no—
91#875—91no—
92#869—91no—
93C3×#97—91no—
94#815—32no—
95C23×C32—61no—
96C22×C3×D3—22no—
97C22×C3×D3—22no—
98C32×D4—62no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—41no—
102C2×C32×C4C22×D311yescharacteristic
103C2×C32×C4C22×D312yes—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—61no—
106C2×C3⋊Dic3—62no—
107C2×C3⋊Dic3—62no—
108C2×C3⋊Dic3—122no—
109C2×C3⋊Dic3—122no—
110C2×C3⋊Dic3D1212yes—
111#634—22no—
112#634—22no—
113#634—62no—
114#634—64no—
115#634—62no—
116#634—62no—
117#634—122no—
118#634—122no—
119#634—124no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—122no—
125C3×D12—22no—
126C3×D12—22no—
127C3×D12—42no—
128C3×Dic6—64no—
129C3×Dic6—62no—
130C3×Dic6—62no—
131C3×Dic6—62no—
132C3×Dic6—62no—
133C3×Dic6—122no—
134C32×C8—34no—
135C3×C3⋊C8—24no—
136C3×C3⋊C8D1214yes—
137C3×C3⋊C8#6714yes—
138#497—271noSylow
139C2×C33#12711yescharacteristic
140C2×C33#13712yes—
141C32×D3—81no—
142C2×C3×Q8—92no—
143C2×C3×D4—32no—
144C2×C3×D4—61no—
145C2×D12D3211yescharacteristic
146C2×Dic6—32no—
147C2×Dic6—32no—
148C2×Dic6—62no—
149C2×Dic6—91no—
150C2×Dic6—92no—
151C2×Dic6—182no—
152C2×Dic6—181no—
153C2×Dic6—361no—
154C3×SD16—184no—
155C2×C3×C8—32no—
156C2×C3×C8—32no—
157C2×C3×C8—62no—
158C3×C42—91no—
159#217—184no—
160#217—184no—
161#216—184no—
162#214—64no—
163(C2×D3)⋊C4—182no—
164C4⋊Dic3—182no—
165Dic3⋊C4—184no—
166C4×Dic3—32no—
167C4×Dic3—61no—
168C2×C3⋊C8D3212yes—
169Dic12—184no—
170Dic12—184no—
171Dic12—188no—
172#204—64no—
173C22×C32—22no—
174C22×C32—21no—
175C22×C32—41no—
176C22×C32—121no—
177C22×C32—122no—
178C22×C32C2×D1211yescharacteristic
179C22×C32D3×D412yes—
180C2×C3×D3—42no—
181C2×C3×D3—44no—
182C2×C3×D3—44no—
183C2×C3×D3—42no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—42no—
188C32×C4C2×D1212yes—
189C32×C4D3×D412yes—
190C32×C4C2×#6712yes—
191C3⋊Dic3—22no—
192C3⋊Dic3—42no—
193C3⋊Dic3—124no—
194C3⋊Dic3—122no—
195C3⋊Dic3—124no—
196C3⋊Dic3—121no—
197C3⋊Dic3—122no—
198C3×Dic3—64no—
199C3×Dic3—122no—
200C3×Dic3—124no—
201C3×Dic3—122no—
202C3×Dic3—122no—
203C2×Q16—272no—
204C2×SD16—92no—
205C4⋊Q8—271no—
206#124—271no—
207#97—91no—
208C33#49711yescharacteristic, complemented, Sylow
209C23×C3—62no—
210C23×C3—62no—
211C22×D3—21no—
212C3×Q8—182no—
213C3×Q8—182no—
214C3×D4—62no—
215C3×D4—62no—
216C3×D4—122no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—92no—
221C2×C3×C4—182no—
222C2×C3×C4C2×D3212yes—
223C2×C3×C4C2×D3211yescharacteristic
224C2×Dic3—34no—
225C2×Dic3—62no—
226C2×Dic3—62no—
227C2×Dic3—62no—
228C2×Dic3—122no—
229C2×Dic3—182no—
230C2×Dic3—182no—
231C2×Dic3—181no—
232C2×Dic3—362no—
233C2×Dic3—362no—
234D12—22no—
235Dic6—64no—
236Dic6—62no—
237Dic6—62no—
238Dic6—62no—
239Dic6—62no—
240Dic6—122no—
241Dic6—184no—
242Dic6—182no—
243Dic6—182no—
244Dic6—182no—
245Dic6—362no—
246Dic6—362no—
247Dic6—364no—
248C3×C8—34no—
249C3×C8—34no—
250C3×C8—64no—
251C3⋊C8#62614yes—
252C2×C32—24no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—21no—
256C2×C32—41no—
257C2×C32—42no—
258C2×C32—241no—
259C2×C32#87411yescharacteristic
260C2×C32#90912yes—
261C2×C32#91514yes—
262C2×C32#94012yes—
263C3×D3—82no—
264C3×D3—82no—
265C2×Q8—92no—
266C2×Q8—271no—
267C2×D4—31no—
268Q16—544no—
269SD16—184no—
270C2×C8—32no—
271C4⋊C4—542no—
272C22⋊C4—542no—
273C42—91no—
274C22×C3—21no—
275C22×C3—22no—
276C22×C3—41no—
277C22×C3—122no—
278C22×C3—124no—
279C22×C3—124no—
280C22×C3—122no—
281C22×C3D3×D1212yes—
282C22×C3#381111yescharacteristic
283C2×D3—42no—
284C2×D3—41no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—42no—
289C3×C4—182no—
290C3×C4—362no—
291C3×C4D3×D1212yes—
292C3×C4C2×#62612yes—
293C3×C4D3×#6712yes—
294Dic3—64no—
295Dic3—122no—
296Dic3—124no—
297Dic3—122no—
298Dic3—122no—
299Dic3—364no—
300Dic3—362no—
301Dic3—364no—
302Dic3—362no—
303Dic3—361no—
304C32—21no—
305C32—22no—
306C32—41no—
307C32#471511yescharacteristic, complemented
308C32#516712yescomplemented
309C23—61no—
310Q8—182no—
311Q8—182no—
312Q8—542no—
313D4—62no—
314C2×C4—92no—
315C2×C4—182no—
316C2×C4—541no—
317C2×C4D3311yescharacteristic
318C8—34no—
319C2×C3—22no—
320C2×C3—21no—
321C2×C3—22no—
322C2×C3—24no—
323C2×C3—41no—
324C2×C3—42no—
325C2×C3—242no—
326C2×C3—242no—
327C2×C3#6380014yes—
328C2×C3#6391112yes—
329C2×C3#6393112yes—
330C2×C3#6397911yescharacteristic
331D3—81no—
332C22—121no—
333C22—122no—
334C22#8895511yescharacteristic
335C4—182no—
336C4—362no—
337C4—1081no—
338C4D3×#62612yes—
339C3—21no—
340C3—22no—
341C3—41no—
342C3#1059089712yescomplemented
343C3#1059149311yescharacteristic, complemented
344C2—241no—
345C2#1172289512yes—
346C2#1172303011yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#497
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723678C2semidirect
3. C2×#88800C2semidirect
5. C3×#63555C2semidirect
30. #63555D3semidirect
208. C33#497semidirect
307. C32#4715semidirect
308. C32#5167semidirect
342. C3#10590897semidirect
343. C3#10591493semidirect

Maximal subgroups

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

subgroupindex
#117236782
#117234292
C2×#888002
C2×C3×#37162
C3×#635552
#105914933
#105908973