Finite Group Databasebeta

← #13161969

Subgroups — #13161969

Order 1728.

630 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 99 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
2C4⋊#6296C211yesmaximal, characteristic, complemented
3C3×#63659C212yesmaximal
4#11722076C212yesmaximal, complemented
5#11722071C211yesmaximal, characteristic, complemented
6#10591496—31nomaximal
7#10591488—32nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C2×#6296—21no—
11C3×#3757—21no—
12C3×#3757—21no—
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#88815—21no—
17#64145D311yescharacteristic
18C3×#936—32no—
19C3×#936—61no—
20#63919—31no—
21#63918—32no—
22#63659D312yes—
23C3×#832—32no—
24C3×#832—32no—
25C3×#832—62no—
26#63567—32no—
27#63566—32no—
28#63565—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⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×#634—22no—
39#6296—42no—
40C3×C3⋊(C3⋊C8)—22no—
41#5055—91no—
42#4765—92no—
43C2×C32×D4—22no—
44C2×C32×D4—21no—
45C2×C32×D4—41no—
46C2×C32×D4C2×D312yes—
47C2×C32×D4C2×D311yescharacteristic
48C2×#634C2×D311yescharacteristic
49C2×C3×Dic6—32no—
50C2×C3×Dic6—61no—
51C2×#627—62no—
52C2×#627—61no—
53#3757—21no—
54#3757—21no—
55C4⋊(C3⋊Dic3)C2×D312yes—
56C4×C3⋊Dic3C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)C2×D312yes—
58C3×#217—62no—
59C3×#217—62no—
60C3×#217—62no—
61C3×#217—62no—
62C3×C4⋊Dic3—32no—
63C3×C4⋊Dic3—32no—
64C3×C4⋊Dic3—62no—
65C3×C4×Dic3—32no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—62no—
70#3724—61no—
71Dic3⋊Dic3—62no—
72C22×C33—41no—
73C22×C33—41no—
74C22×C33—41no—
75C22×C33C2×D411yescharacteristic
76C33×C4#4112yes—
77C3×C3⋊Dic3—21no—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3—41no—
80C3×C3⋊Dic3—42no—
81C3×#124—91no—
82#936—32no—
83#936—61no—
84#888—92no—
85#869—91no—
86C3×#102—92no—
87#835—92no—
88#832—32no—
89#832—32no—
90#832—62no—
91#816—92no—
92#815—92no—
93#810—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×D312yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—22no—
114C2×C3⋊Dic3#6711yescharacteristic
115C2×C3⋊Dic3#6711yescharacteristic
116#634—22no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—122no—
126C3×Dic6—62no—
127C3×Dic6—62no—
128C3×Dic6—122no—
129#627—124no—
130#627—122no—
131C3⋊(C3⋊C8)—22no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—122no—
135#489—271noSylow
136C2×C33—81no—
137C2×C33#12411yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C2×C33#13711yescharacteristic
140C2×C3×Q8—91no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—41no—
144C2×C3×D4D3211yescharacteristic
145C2×C3×D4D3212yes—
146C2×Dic6—32no—
147C2×Dic6—61no—
148C2×Dic6—182no—
149C2×Dic6—181no—
150C2×C3×C8—92no—
151C3×C4⋊C4—92no—
152C3×C22⋊C4—181no—
153C3×C22⋊C4—181no—
154C3×C42—91no—
155#217—62no—
156#217—62no—
157#217—62no—
158#217—62no—
159C4⋊Dic3—32no—
160C4⋊Dic3—32no—
161C4⋊Dic3—62no—
162C4⋊Dic3—181no—
163Dic3⋊C4—182no—
164C4×Dic3—32no—
165C4×Dic3—61no—
166C2×C3⋊C8—32no—
167C2×C3⋊C8—32no—
168C2×C3⋊C8—62no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—41no—
172C22×C32—44no—
173C22×C32—42no—
174C22×C32—44no—
175C22×C32—42no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—42no—
180C22×C32—42no—
181C22×C32—42no—
182C22×C32—42no—
183C22×C32—82no—
184C22×C32—82no—
185C22×C32—82no—
186C22×C32—82no—
187C22×C32D3×D412yes—
188C22×C32C2×#6711yescharacteristic
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—42no—
193C32×C4#23512yes—
194C32×C4#23712yes—
195C32×C4#23712yes—
196C3⋊Dic3—21no—
197C3⋊Dic3—21no—
198C3⋊Dic3—41no—
199C3⋊Dic3—42no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204C3×Dic3—122no—
205C3×Dic3—122no—
206C3×Dic3—122no—
207C3×Dic3—122no—
208C3×Dic3—242no—
209C4⋊Q8—271no—
210#124—91no—
211#103—272no—
212#102—92no—
213#97—271no—
214C33#48911yescharacteristic, complemented, Sylow
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—21no—
218C23×C3—42no—
219C23×C3—42no—
220C3×Q8—182no—
221C3×D4—24no—
222C3×D4—22no—
223C3×D4—22no—
224C3×D4—22no—
225C3×D4—42no—
226C3×D4—42no—
227C3×D4—44no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—182no—
234C2×C3×C4—181no—
235C2×C3×C4C2×D3211yescharacteristic
236C2×C3×C4C2×D3212yes—
237C2×Dic3—32no—
238C2×Dic3—32no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—62no—
242C2×Dic3—62no—
243C2×Dic3—62no—
244C2×Dic3—62no—
245C2×Dic3—122no—
246Dic6—62no—
247Dic6—62no—
248Dic6—122no—
249Dic6—364no—
250Dic6—362no—
251C3×C8—182no—
252C3⋊C8—62no—
253C3⋊C8—62no—
254C3⋊C8—122no—
255C2×C32—22no—
256C2×C32—21no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—22no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—82no—
265C2×C32—84no—
266C2×C32—84no—
267C2×C32—82no—
268C2×C32—81no—
269C2×C32#88512yes—
270C2×C32#91112yes—
271C2×C32#91512yes—
272C2×C32#93611yescharacteristic
273C2×C32#94911yescharacteristic
274C2×C32#95111yescharacteristic
275C2×Q8—91no—
276C2×Q8—541no—
277C2×D4#111311yescharacteristic
278C2×C8—92no—
279C4⋊C4—92no—
280C4⋊C4—541no—
281C22⋊C4—181no—
282C22⋊C4—181no—
283C42—91no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3—44no—
287C22×C3—42no—
288C22×C3—42no—
289C22×C3—42no—
290C22×C3—44no—
291C22×C3—42no—
292C22×C3—41no—
293C22×C3—42no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—82no—
299C22×C3—82no—
300C22×C3—82no—
301C22×C3—82no—
302C22×C3D3×#6712yes—
303C22×C3#381111yescharacteristic
304C3×C4—22no—
305C3×C4—22no—
306C3×C4—22no—
307C3×C4—42no—
308C3×C4—181no—
309C3×C4—181no—
310C3×C4—362no—
311C3×C4—361no—
312C3×C4#380412yes—
313C3×C4#380412yes—
314C3×C4#380512yes—
315Dic3—62no—
316Dic3—62no—
317Dic3—121no—
318Dic3—121no—
319Dic3—122no—
320Dic3—122no—
321Dic3—122no—
322Dic3—122no—
323Dic3—242no—
324C32—22no—
325C32—21no—
326C32—41no—
327C32#476512yescomplemented
328C32#505511yescharacteristic, complemented
329C23—21no—
330Q8—182no—
331Q8—1082no—
332D4—22no—
333C2×C4—91no—
334C2×C4—91no—
335C2×C4—181no—
336C2×C4—182no—
337C2×C4C2×#111311yescharacteristic
338C8—182no—
339C2×C3—21no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—22no—
343C2×C3—21no—
344C2×C3—22no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—82no—
349C2×C3—84no—
350C2×C3—84no—
351C2×C3—82no—
352C2×C3—81no—
353C2×C3#6392512yes—
354C2×C3#6392711yescharacteristic
355C2×C3#6392912yes—
356C2×C3#6393111yescharacteristic
357C2×C3#6395311yescharacteristic
358C2×C3#6395912yes—
359C22—41no—
360C22—41no—
361C22—41no—
362C22#8904911yescharacteristic
363C4—181no—
364C4—181no—
365C4—362no—
366C4—361no—
367C4#8904612yes—
368C3—22no—
369C3—21no—
370C3—41no—
371C3#1059148812yescomplemented
372C3#1059149611yescharacteristic, complemented
373C2—81no—
374C2#1172370311yescharacteristic
375C2#1172370511yescharacteristic
376C2#1172371811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33#489
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
2. C4⋊#6296C2semidirect
4. #11722076C2semidirect
5. #11722071C2semidirect
214. C33#489semidirect
327. C32#4765semidirect
328. C32#5055semidirect
371. C3#10591488semidirect
372. C3#10591496semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C4⋊#62962
C3×#636592
#117220762
#117220712
#105914963
#105914883