Finite Group Databasebeta

← #13161965

Subgroups — #13161965

Order 1728.

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

normal subgroups only

This group has 858 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, complemented
2C2×C3×#3753C212yesmaximal, complemented
3#11723697C211yesmaximal, characteristic, complemented
4C2×#88811C212yesmaximal, complemented
5#11722071C211yesmaximal, characteristic, complemented
6#10591485—31nomaximal
7#10591483—32nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C2×C3×#634C2211yescharacteristic
11C2×#6295—22no—
12C3×#3757—22no—
13C3×#3753—24no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#88811—24no—
17#64145D311yescharacteristic, complemented
18C2×#3753D312yescomplemented
19C3×#936—32no—
20C3×#936—61no—
21C2×C3×#213—32no—
22C2×C3×#213—32no—
23C2×C3×#213—62no—
24#63914—31no—
25#63911—32no—
26C2×#3717—32no—
27C2×#3716—32no—
28C2×#3715—32no—
29#63555—31no—
30#63554—32no—
31C23×C33—21no—
32C22×C3×C3⋊D3—22no—
33C33×D4—22no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D412yes—
37C3×C4⋊(C3⋊D3)—22no—
38C3×C4⋊(C3⋊D3)—22no—
39C3×#634—22no—
40#6295—44no—
41C3×C3⋊(C3⋊C8)D414yescomplemented
42#5154—92no—
43#5058—91no—
44C2×C32×D4—22no—
45C2×C32×D4—21no—
46C2×C32×D4—41no—
47C2×C32×D4C2×D311yescharacteristic
48C2×C32×D4C2×D312yes—
49C2×C4⋊(C3⋊D3)C2×D312yes—
50C2×#634C2×D311yescharacteristic
51C2×C3×D12—32no—
52C2×C3×D12—32no—
53C2×C3×D12—62no—
54C2×C3×Dic6—32no—
55C2×C3×Dic6—61no—
56C2×#626—64no—
57C2×#625—62no—
58#3757—22no—
59#3753—24no—
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D312yes—
62C3×#217—64no—
63C3×#217—64no—
64C3×#213—64no—
65C3×#213—64no—
66C3×#213—68no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—62no—
72#3717—64no—
73#3716—64no—
74#3715—64no—
75C22×C33—41no—
76C22×C33—42no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—42no—
79C2×C3×C3⋊D3—44no—
80C33×C4C2×D412yes—
81C3×C3⋊Dic3—22no—
82C3×C3⋊Dic3—41no—
83C2×C3×D8—92no—
84C3×#124—91no—
85C2×#215—92no—
86#940—92no—
87#936—32no—
88#936—61no—
89C2×#214—92no—
90C2×#213—32no—
91C2×#213—32no—
92C2×#213—62no—
93C2×#204—92no—
94#874—91no—
95#815—92no—
96#803—91no—
97C23×C32—21no—
98C23×C32—22no—
99C23×C32—22no—
100C23×C32—42no—
101C23×C32—42no—
102C22×C3⋊D3—22no—
103C22×C3×D3—62no—
104C22×C3×D3—62no—
105C22×C3×D3—122no—
106C32×D4—22no—
107C32×D4—24no—
108C32×D4—22no—
109C32×D4—22no—
110C32×D4—42no—
111C32×D4—42no—
112C32×D4—44no—
113C2×C32×C4—22no—
114C2×C32×C4—21no—
115C2×C32×C4—41no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C22×D312yes—
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3#6712yes—
120C4⋊(C3⋊D3)—22no—
121C4⋊(C3⋊D3)—22no—
122#634—22no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C3×D12—64no—
128C3×D12—62no—
129C3×D12—62no—
130C3×D12—62no—
131C3×D12—62no—
132C3×D12—122no—
133C3×Dic6—62no—
134C3×Dic6—62no—
135C3×Dic6—122no—
136#626—128no—
137#625—124no—
138C3⋊(C3⋊C8)#6714yescomplemented
139C3×C3⋊C8—34no—
140C3×C3⋊C8—34no—
141C3×C3⋊C8—64no—
142#496—271noSylow
143C2×C33—81no—
144C2×C33#12711yescharacteristic
145C2×C33AGL(1,ℤ/8)12yes—
146C3×C3⋊D3—82no—
147C2×C3×Q8—91no—
148C2×C3×D4—22no—
149C2×C3×D4—21no—
150C2×C3×D4—41no—
151C2×C3×D4—92no—
152C2×C3×D4D3211yescharacteristic
153C2×C3×D4D3212yes—
154C2×#67—184no—
155C2×D12—32no—
156C2×D12—32no—
157C2×D12—62no—
158C2×D12—182no—
159C2×Dic6—32no—
160C2×Dic6—61no—
161C3×D8—184no—
162C2×C3×C8—92no—
163C3×C22⋊C4—182no—
164C3×C42—91no—
165#217—64no—
166#217—64no—
167#215—184no—
168#214—184no—
169#213—64no—
170#213—64no—
171#213—68no—
172C4×Dic3—32no—
173C4×Dic3—61no—
174C2×C3⋊C8—32no—
175C2×C3⋊C8—32no—
176C2×C3⋊C8—62no—
177#204—184no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32—44no—
182C22×C32—42no—
183C22×C32—44no—
184C22×C32—42no—
185C22×C32—41no—
186C22×C32—42no—
187C22×C32—44no—
188C22×C32—44no—
189C22×C32—84no—
190C22×C32—84no—
191C22×C32D3×D412yes—
192C22×C32C2×#6711yescharacteristic
193C2×C3⋊D3—42no—
194C2×C3⋊D3—44no—
195C2×C3×D3—124no—
196C2×C3×D3—124no—
197C2×C3×D3—122no—
198C2×C3×D3—122no—
199C2×C3×D3—124no—
200C2×C3×D3—244no—
201C32×C4—22no—
202C32×C4—22no—
203C32×C4—22no—
204C32×C4—42no—
205C32×C4D3×D412yes—
206C32×C4D3×D412yes—
207C32×C4C2×#6712yes—
208C3⋊Dic3—22no—
209C3⋊Dic3—41no—
210C3×Dic3—64no—
211C3×Dic3—122no—
212C3×Dic3—122no—
213C3×Dic3—122no—
214C2×SD16—272no—
215C2×D8—92no—
216#127—271no—
217#124—91no—
218#97—271no—
219C33#49611yescharacteristic, complemented, Sylow
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—21no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—182no—
226C22×D3—62no—
227C22×D3—62no—
228C22×D3—122no—
229C3×Q8—182no—
230C3×D4—24no—
231C3×D4—22no—
232C3×D4—22no—
233C3×D4—22no—
234C3×D4—42no—
235C3×D4—42no—
236C3×D4—44no—
237C3×D4—182no—
238C3×D4—182no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—92no—
243C2×C3×C4—181no—
244C2×C3×C4C2×D3211yescharacteristic
245C2×C3×C4C2×D3212yes—
246#67—368no—
247C2×Dic3—34no—
248C2×Dic3—62no—
249C2×Dic3—62no—
250C2×Dic3—62no—
251D12—64no—
252D12—62no—
253D12—62no—
254D12—62no—
255D12—62no—
256D12—122no—
257D12—364no—
258Dic6—62no—
259Dic6—62no—
260Dic6—122no—
261C3×C8—94no—
262C3⋊C8—34no—
263C3⋊C8—34no—
264C3⋊C8—64no—
265C2×C32—24no—
266C2×C32—22no—
267C2×C32—21no—
268C2×C32—22no—
269C2×C32—41no—
270C2×C32—42no—
271C2×C32—82no—
272C2×C32—84no—
273C2×C32—84no—
274C2×C32—82no—
275C2×C32—81no—
276C2×C32#91114yes—
277C2×C32#94012yes—
278C2×C32#94011yescharacteristic
279C2×C32#94912yes—
280C3⋊D3—82no—
281C3×D3—244no—
282C3×D3—242no—
283C3×D3—242no—
284C2×Q8—91no—
285C2×D4—92no—
286C2×D4—542no—
287C2×D4#111311yescharacteristic
288SD16—544no—
289D8—184no—
290C2×C8—92no—
291C22⋊C4—182no—
292C42—91no—
293C22×C3—21no—
294C22×C3—22no—
295C22×C3—44no—
296C22×C3—42no—
297C22×C3—44no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—84no—
305C22×C3—84no—
306C22×C3—364no—
307C22×C3—362no—
308C22×C3D3×#6712yes—
309C22×C3#381111yescharacteristic
310C2×D3—124no—
311C2×D3—124no—
312C2×D3—124no—
313C2×D3—122no—
314C2×D3—122no—
315C2×D3—244no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—22no—
319C3×C4—42no—
320C3×C4—182no—
321C3×C4—361no—
322C3×C4D3×#6712yes—
323C3×C4D3×#6712yes—
324C3×C4#381112yes—
325Dic3—64no—
326Dic3—122no—
327Dic3—122no—
328Dic3—122no—
329C32—22no—
330C32—21no—
331C32—41no—
332C32#505811yescharacteristic, complemented
333C32#515412yescomplemented
334C23—21no—
335C23—182no—
336Q8—182no—
337D4—22no—
338D4—182no—
339D4—182no—
340D4—1084no—
341C2×C4—92no—
342C2×C4—181no—
343C2×C4C2×#111311yescharacteristic
344C8—94no—
345C2×C3—22no—
346C2×C3—24no—
347C2×C3—21no—
348C2×C3—22no—
349C2×C3—42no—
350C2×C3—41no—
351C2×C3—82no—
352C2×C3—84no—
353C2×C3—84no—
354C2×C3—82no—
355C2×C3—81no—
356C2×C3—722no—
357C2×C3#6392514yes—
358C2×C3#6392712yes—
359C2×C3#6397912yes—
360C2×C3#6397911yescharacteristic
361D3—244no—
362D3—242no—
363D3—242no—
364C22—42no—
365C22—41no—
366C22—364no—
367C22—362no—
368C22#8904911yescharacteristic
369C4—182no—
370C4—361no—
371C4#8904912yes—
372C3—22no—
373C3—21no—
374C3—41no—
375C3#1059148312yescomplemented
376C3#1059148511yescharacteristic, complemented
377C2—81no—
378C2—722no—
379C2#1172370312yes—
380C2#1172372811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33#496
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
1. C3×#64145C2semidirect
2. C2×C3×#3753C2semidirect
3. #11723697C2semidirect
4. C2×#88811C2semidirect
5. #11722071C2semidirect
17. #64145D3semidirect
18. C2×#3753D3semidirect
41. C3×C3⋊(C3⋊C8)D4semidirect
138. C3⋊(C3⋊C8)#67semidirect
219. C33#496semidirect
332. C32#5058semidirect
333. C32#5154semidirect
375. C3#10591483semidirect
376. C3#10591485semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C2×C3×#37532
#117236972
C2×#888112
#117220712
#105914853
#105914833