Finite Group Databasebeta

← #13161972

Subgroups — #13161972

Order 1728.

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

normal subgroups only

This group has 762 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×#64145C211yesmaximal, characteristic
2C2×C3×#3754C212yesmaximal
3#11723678C211yesmaximal, characteristic, complemented
4C2×#88812C212yesmaximal, complemented
5#11722071C211yesmaximal, characteristic, complemented
6#10591493—31nomaximal
7#10591491—32nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#634C2212yes—
10C2×C3×#634C2211yescharacteristic
11C3×#3757—22no—
12C3×#3754—24no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2212yes—
15#88815—22no—
16#88812—24no—
17#64145D311yescharacteristic
18C2×#3754D312yes—
19C3×#936—32no—
20C3×#936—61no—
21C2×C3×#214—32no—
22C2×C3×#214—32no—
23C2×C3×#214—62no—
24#63873—31no—
25#63866—32no—
26C2×#3719—32no—
27C2×#3719—32no—
28C2×#3718—32no—
29#63555—31no—
30#63554—32no—
31C23×C33—21no—
32C33×D4—22no—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3D412yes—
37C3×#634—22no—
38C3×#634—22no—
39C3×#634—22no—
40C3×C3⋊(C3⋊C8)D414yes—
41#5167—92no—
42#5060—91no—
43C2×C32×D4—22no—
44C2×C32×D4—21no—
45C2×C32×D4—41no—
46C2×C32×D4C2×D311yescharacteristic
47C2×C32×D4C2×D312yes—
48C2×#634C2×D311yescharacteristic
49C2×#634C2×D312yes—
50C2×C3×Dic6—32no—
51C2×C3×Dic6—32no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—62no—
54C2×C3×Dic6—61no—
55#3757—22no—
56#3754—24no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D312yes—
59C3×#217—64no—
60C3×#217—64no—
61C3×#214—64no—
62C3×#214—64no—
63C3×#214—68no—
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#3719—64no—
72#3719—64no—
73#3718—64no—
74C22×C33—41no—
75C22×C33—42no—
76C22×C33C2×D411yescharacteristic
77C33×C4C2×D412yes—
78C3×C3⋊Dic3—22no—
79C3×C3⋊Dic3—42no—
80C3×C3⋊Dic3—41no—
81C2×C3×SD16—92no—
82C3×#124—91no—
83#944—92no—
84C2×#216—92no—
85C2×#216—92no—
86#936—32no—
87#936—61no—
88C2×#214—32no—
89C2×#214—32no—
90C2×#214—62no—
91C2×Dic12—92no—
92#869—91no—
93#815—92no—
94#803—91no—
95C23×C32—21no—
96C23×C32—22no—
97C23×C32—22no—
98C23×C32—42no—
99C23×C32—42no—
100C32×D4—22no—
101C32×D4—24no—
102C32×D4—22no—
103C32×D4—22no—
104C32×D4—42no—
105C32×D4—42no—
106C32×D4—44no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C32×C4C22×D312yes—
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—22no—
114C2×C3⋊Dic3#6712yes—
115#634—22no—
116#634—22no—
117#634—22no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—122no—
125C3×Dic6—64no—
126C3×Dic6—62no—
127C3×Dic6—62no—
128C3×Dic6—62no—
129C3×Dic6—62no—
130C3×Dic6—62no—
131C3×Dic6—62no—
132C3×Dic6—122no—
133C3×Dic6—122no—
134C3⋊(C3⋊C8)#6714yes—
135C3×C3⋊C8—34no—
136C3×C3⋊C8—34no—
137C3×C3⋊C8—64no—
138#497—271noSylow
139C2×C33—81no—
140C2×C33#12711yescharacteristic
141C2×C33#13712yes—
142C2×C3×Q8—91no—
143C2×C3×Q8—92no—
144C2×C3×D4—22no—
145C2×C3×D4—21no—
146C2×C3×D4—41no—
147C2×C3×D4D3211yescharacteristic
148C2×C3×D4D3212yes—
149C2×Dic6—32no—
150C2×Dic6—32no—
151C2×Dic6—32no—
152C2×Dic6—62no—
153C2×Dic6—61no—
154C3×SD16—184no—
155C2×C3×C8—92no—
156C3×C22⋊C4—182no—
157C3×C42—91no—
158#217—64no—
159#217—64no—
160#216—184no—
161#216—184no—
162#214—64no—
163#214—64no—
164#214—68no—
165C4⋊Dic3—182no—
166Dic3⋊C4—184no—
167C4×Dic3—32no—
168C4×Dic3—61no—
169C2×C3⋊C8—32no—
170C2×C3⋊C8—32no—
171C2×C3⋊C8—62no—
172Dic12—184no—
173C22×C32—22no—
174C22×C32—21no—
175C22×C32—41no—
176C22×C32—44no—
177C22×C32—42no—
178C22×C32—44no—
179C22×C32—42no—
180C22×C32—41no—
181C22×C32—42no—
182C22×C32—44no—
183C22×C32—44no—
184C22×C32—84no—
185C22×C32—84no—
186C22×C32D3×D412yes—
187C22×C32C2×#6711yescharacteristic
188C32×C4—22no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—42no—
192C32×C4D3×D412yes—
193C32×C4D3×D412yes—
194C32×C4C2×#6712yes—
195C3⋊Dic3—22no—
196C3⋊Dic3—42no—
197C3⋊Dic3—41no—
198C3×Dic3—64no—
199C3×Dic3—122no—
200C3×Dic3—124no—
201C3×Dic3—122no—
202C3×Dic3—122no—
203C3×Dic3—122no—
204C3×Dic3—122no—
205C2×Q16—272no—
206C2×SD16—92no—
207C4⋊Q8—271no—
208#124—91no—
209#97—271no—
210C33#49711yescharacteristic, complemented, Sylow
211C23×C3—22no—
212C23×C3—22no—
213C23×C3—21no—
214C23×C3—42no—
215C23×C3—42no—
216C3×Q8—182no—
217C3×Q8—182no—
218C3×Q8—182no—
219C3×D4—24no—
220C3×D4—22no—
221C3×D4—22no—
222C3×D4—22no—
223C3×D4—42no—
224C3×D4—42no—
225C3×D4—44no—
226C2×C3×C4—22no—
227C2×C3×C4—21no—
228C2×C3×C4—41no—
229C2×C3×C4—92no—
230C2×C3×C4—182no—
231C2×C3×C4—181no—
232C2×C3×C4C2×D3211yescharacteristic
233C2×C3×C4C2×D3212yes—
234C2×Dic3—34no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—122no—
241Dic6—64no—
242Dic6—62no—
243Dic6—62no—
244Dic6—62no—
245Dic6—62no—
246Dic6—62no—
247Dic6—62no—
248Dic6—122no—
249Dic6—122no—
250C3×C8—94no—
251C3⋊C8—34no—
252C3⋊C8—34no—
253C3⋊C8—64no—
254C2×C32—24no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—42no—
260C2×C32—82no—
261C2×C32—84no—
262C2×C32—84no—
263C2×C32—82no—
264C2×C32—81no—
265C2×C32#91514yes—
266C2×C32#94012yes—
267C2×C32#94011yescharacteristic
268C2×C32#95112yes—
269C2×Q8—91no—
270C2×Q8—92no—
271C2×D4#111311yescharacteristic
272Q16—544no—
273SD16—184no—
274C2×C8—92no—
275C4⋊C4—542no—
276C22⋊C4—182no—
277C42—91no—
278C22×C3—21no—
279C22×C3—22no—
280C22×C3—44no—
281C22×C3—42no—
282C22×C3—44no—
283C22×C3—44no—
284C22×C3—44no—
285C22×C3—42no—
286C22×C3—42no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—84no—
290C22×C3—84no—
291C22×C3D3×#6712yes—
292C22×C3#381111yescharacteristic
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—42no—
297C3×C4—182no—
298C3×C4—362no—
299C3×C4—361no—
300C3×C4D3×#6712yes—
301C3×C4D3×#6712yes—
302C3×C4#381112yes—
303Dic3—64no—
304Dic3—122no—
305Dic3—124no—
306Dic3—122no—
307Dic3—122no—
308Dic3—122no—
309Dic3—122no—
310C32—22no—
311C32—21no—
312C32—41no—
313C32#506011yescharacteristic, complemented
314C32#516712yescomplemented
315C23—21no—
316Q8—182no—
317Q8—182no—
318Q8—182no—
319D4—22no—
320C2×C4—92no—
321C2×C4—181no—
322C2×C4—182no—
323C2×C4C2×#111311yescharacteristic
324C8—94no—
325C2×C3—22no—
326C2×C3—24no—
327C2×C3—22no—
328C2×C3—21no—
329C2×C3—42no—
330C2×C3—41no—
331C2×C3—82no—
332C2×C3—84no—
333C2×C3—84no—
334C2×C3—82no—
335C2×C3—81no—
336C2×C3#6392914yes—
337C2×C3#6393112yes—
338C2×C3#6397912yes—
339C2×C3#6397911yescharacteristic
340C22—42no—
341C22—41no—
342C22#8904911yescharacteristic
343C4—182no—
344C4—361no—
345C4—362no—
346C4#8904912yes—
347C3—22no—
348C3—21no—
349C3—41no—
350C3#1059149112yescomplemented
351C3#1059149311yescharacteristic, complemented
352C2—81no—
353C2#1172370512yes—
354C2#1172372811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33#497
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #11723678C2semidirect
4. C2×#88812C2semidirect
5. #11722071C2semidirect
210. C33#497semidirect
313. C32#5060semidirect
314. C32#5167semidirect
350. C3#10591491semidirect
351. C3#10591493semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C2×C3×#37542
#117236782
C2×#888122
#117220712
#105914933
#105914913