Finite Group Databasebeta

← #13162022

Subgroups — #13162022

Order 1728.

730 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#64156C211yesmaximal, characteristic
2C2×C3×#3756C212yesmaximal, complemented
3#11723678C211yesmaximal, characteristic, complemented
4C2×#88811C212yesmaximal
5#11722071C211yesmaximal, characteristic, complemented
6#10591629—31nomaximal
7#10591627—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×#634C2212yes—
11C3×#3756—24no—
12C3×#3752—22no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2212yes—
15#88815—22no—
16#88811—24no—
17#64156D311yescharacteristic
18C2×#3756D312yescomplemented
19C3×#947—32no—
20C3×#947—61no—
21C2×C3×#216—32no—
22C2×C3×#216—32no—
23C2×C3×#216—62no—
24#63873—31no—
25#63866—32no—
26C2×#3717—32no—
27C2×#3716—32no—
28C2×#3715—32no—
29#63555—31no—
30#63554—32no—
31C22×C3×C3⋊D3—21no—
32C33×Q8—22no—
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3D412yes—
37C3×C4⋊(C3⋊D3)—22no—
38C3×#634—22no—
39C3×#634—22no—
40C3×C3⋊(C3⋊C8)D414yes—
41#5190—92no—
42#5060—91no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—41no—
46C2×C32×Q8C2×D311yescharacteristic
47C2×C32×Q8C2×D312yes—
48C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
49C2×#634C2×D312yes—
50C2×C3×D12—32no—
51C2×C3×D12—61no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—32no—
54C2×C3×Dic6—62no—
55#3756—24no—
56#3752—22no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D312yes—
59C3×#216—64no—
60C3×#216—64no—
61C3×#216—68no—
62C3×(C2×D3)⋊C4—64no—
63C3×(C2×D3)⋊C4—64no—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—62no—
69#3724—62no—
70Dic3⋊Dic3—64no—
71#3717—64no—
72#3716—64no—
73#3715—64no—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—41no—
76C2×C3×C3⋊D3—42no—
77C33×C4—41no—
78C33×C4C2×D412yes—
79C3×C3⋊Dic3—22no—
80C3×C3⋊Dic3—42no—
81C2×C3×Q16—92no—
82C3×#124—91no—
83#947—32no—
84#947—61no—
85#944—92no—
86C2×#216—32no—
87C2×#216—32no—
88C2×#216—62no—
89C2×#215—92no—
90C2×#214—92no—
91C2×#204—92no—
92#869—91no—
93#815—92no—
94#803—91no—
95C22×C3⋊D3—21no—
96C22×C3×D3—62no—
97C22×C3×D3—62no—
98C32×Q8—22no—
99C32×Q8—24no—
100C32×Q8—22no—
101C32×Q8—22no—
102C32×Q8—42no—
103C32×Q8—42no—
104C32×Q8—44no—
105C2×C32×C4—21no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4—41no—
111C2×C32×C4—42no—
112C2×C32×C4—42no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D312yes—
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3#6712yes—
117C4⋊(C3⋊D3)—22no—
118#634—22no—
119#634—22no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—122no—
125C3×D12—62no—
126C3×D12—62no—
127C3×D12—122no—
128C3×Dic6—64no—
129C3×Dic6—62no—
130C3×Dic6—62no—
131C3×Dic6—62no—
132C3×Dic6—62no—
133C3×Dic6—122no—
134C3⋊(C3⋊C8)#6714yes—
135C3×C3⋊C8—34no—
136C3×C3⋊C8—34no—
137C3×C3⋊C8—64no—
138#497—271noSylow
139C2×C33#12711yescharacteristic
140C2×C33#13712yes—
141C3×C3⋊D3—81no—
142C2×C3×Q8—21no—
143C2×C3×Q8—22no—
144C2×C3×Q8—41no—
145C2×C3×Q8—92no—
146C2×C3×Q8D3211yescharacteristic
147C2×C3×Q8D3212yes—
148C2×C3×D4—91no—
149C2×D12—32no—
150C2×D12—61no—
151C2×Dic6—32no—
152C2×Dic6—32no—
153C2×Dic6—62no—
154C3×Q16—184no—
155C2×C3×C8—92no—
156C3×C22⋊C4—182no—
157C3×C42—91no—
158#216—64no—
159#216—64no—
160#216—68no—
161#215—184no—
162#214—184no—
163(C2×D3)⋊C4—64no—
164(C2×D3)⋊C4—64no—
165C4⋊Dic3—182no—
166Dic3⋊C4—184no—
167C4×Dic3—32no—
168C4×Dic3—61no—
169C2×C3⋊C8—32no—
170C2×C3⋊C8—32no—
171C2×C3⋊C8—62no—
172#204—184no—
173C22×C32—22no—
174C22×C32—21no—
175C22×C32—41no—
176C22×C32D3×D412yes—
177C22×C32C2×#6711yescharacteristic
178C2×C3⋊D3—41no—
179C2×C3⋊D3—42no—
180C2×C3×D3—122no—
181C2×C3×D3—124no—
182C2×C3×D3—124no—
183C2×C3×D3—122no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—42no—
188C32×C4—42no—
189C32×C4—44no—
190C32×C4—44no—
191C32×C4—41no—
192C32×C4—42no—
193C32×C4D3×D412yes—
194C32×C4D3×D412yes—
195C32×C4C2×#6712yes—
196C3⋊Dic3—22no—
197C3⋊Dic3—42no—
198C3×Dic3—64no—
199C3×Dic3—122no—
200C3×Dic3—124no—
201C3×Dic3—122no—
202C3×Dic3—122no—
203C2×Q16—92no—
204C2×SD16—272no—
205C4⋊Q8—271no—
206#124—91no—
207#97—271no—
208C33#49711yescharacteristic, complemented, Sylow
209C23×C3—181no—
210C22×D3—62no—
211C22×D3—62no—
212C3×Q8—24no—
213C3×Q8—22no—
214C3×Q8—22no—
215C3×Q8—22no—
216C3×Q8—42no—
217C3×Q8—42no—
218C3×Q8—44no—
219C3×Q8—182no—
220C3×Q8—182no—
221C3×D4—182no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—21no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—92no—
231C2×C3×C4—182no—
232C2×C3×C4C2×D3211yescharacteristic
233C2×C3×C4C2×D3212yes—
234C2×Dic3—34no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—122no—
239D12—62no—
240D12—62no—
241D12—122no—
242Dic6—64no—
243Dic6—62no—
244Dic6—62no—
245Dic6—62no—
246Dic6—62no—
247Dic6—122no—
248C3×C8—94no—
249C3⋊C8—34no—
250C3⋊C8—34no—
251C3⋊C8—64no—
252C2×C32—24no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—21no—
256C2×C32—41no—
257C2×C32—42no—
258C2×C32#91814yes—
259C2×C32#94012yes—
260C2×C32#94011yescharacteristic
261C2×C32#95112yes—
262C3⋊D3—81no—
263C3×D3—242no—
264C3×D3—242no—
265C2×Q8—92no—
266C2×Q8#111311yescharacteristic
267C2×D4—91no—
268Q16—184no—
269SD16—544no—
270C2×C8—92no—
271C4⋊C4—542no—
272C22⋊C4—182no—
273C42—91no—
274C22×C3—21no—
275C22×C3—22no—
276C22×C3—41no—
277C22×C3—362no—
278C22×C3—361no—
279C22×C3D3×#6712yes—
280C22×C3#381111yescharacteristic
281C2×D3—122no—
282C2×D3—124no—
283C2×D3—124no—
284C2×D3—122no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—42no—
289C3×C4—44no—
290C3×C4—44no—
291C3×C4—42no—
292C3×C4—41no—
293C3×C4—42no—
294C3×C4—182no—
295C3×C4—362no—
296C3×C4D3×#6712yes—
297C3×C4D3×#6712yes—
298C3×C4#381112yes—
299Dic3—64no—
300Dic3—122no—
301Dic3—124no—
302Dic3—122no—
303Dic3—122no—
304C32—22no—
305C32—21no—
306C32—41no—
307C32#506011yescharacteristic, complemented
308C32#519012yescomplemented
309C23—181no—
310Q8—22no—
311Q8—182no—
312Q8—182no—
313D4—182no—
314C2×C4—21no—
315C2×C4—92no—
316C2×C4—182no—
317C2×C4C2×#111311yescharacteristic
318C8—94no—
319C2×C3—22no—
320C2×C3—24no—
321C2×C3—22no—
322C2×C3—21no—
323C2×C3—42no—
324C2×C3—41no—
325C2×C3—721no—
326C2×C3#6394314yes—
327C2×C3#6394512yes—
328C2×C3#6397912yes—
329C2×C3#6397911yescharacteristic
330D3—242no—
331D3—242no—
332C22—362no—
333C22—361no—
334C22#8904911yescharacteristic
335C4—41no—
336C4—182no—
337C4—362no—
338C4#8904912yes—
339C3—21no—
340C3—22no—
341C3—41no—
342C3#1059162712yescomplemented
343C3#1059162911yescharacteristic, complemented
344C2—721no—
345C2#1172371112yes—
346C2#1172372811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×Q8#1113
hypocenterC33#497
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3756C2semidirect
3. #11723678C2semidirect
5. #11722071C2semidirect
18. C2×#3756D3semidirect
208. C33#497semidirect
307. C32#5060semidirect
308. C32#5190semidirect
342. C3#10591627semidirect
343. C3#10591629semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C2×C3×#37562
#117236782
C2×#888112
#117220712
#105916293
#105916273