Finite Group Databasebeta

← #13156279

Subgroups — #13156279

Order 1728.

680 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C4⋊#6296C211yesmaximal, characteristic, complemented
2#11723494C211yesmaximal, characteristic
3#11722036C212yesmaximal
4C3×#63575C212yesmaximal, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6#10590764—32nomaximal
7#10590430—31nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×#6296—21no—
10C2×#6289—21no—
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×C4⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15#88815—21no—
16#88804—21no—
17#64102—31no—
18#63919—31no—
19#63918—32no—
20#63894—32no—
21#63894—61no—
22#63648—32no—
23C3×#818—32no—
24C3×Dic3⋊C8—32no—
25C3×#807—32no—
26#63575D312yescomplemented
27#63572D311yescharacteristic, complemented
28#63564—32no—
29#63564—32no—
30#63564—62no—
31C22×C3⋊(C3⋊D3)—21no—
32C2×C33×C4C2311yescharacteristic
33C4⋊(C3⋊(C3⋊D3))—21no—
34C4⋊(C3⋊(C3⋊D3))Q812yescomplemented
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C32×Dic3—21no—
39#6296—42no—
40#6289—42no—
41C32×C3⋊C8—22no—
42#4890—92no—
43#4846—91no—
44C2×C4⋊(C3⋊D3)—31no—
45C2×C4⋊(C3⋊D3)—32no—
46C2×C4⋊(C3⋊D3)—61no—
47C2×C4⋊(C3⋊D3)—62no—
48C2×C4⋊(C3⋊D3)—121no—
49C2×C4×C3⋊D3—61no—
50C2×#627—62no—
51C2×#627—61no—
52C2×#624—62no—
53C2×#624—62no—
54C2×C32×C8—32no—
55C32×C4⋊C4—31no—
56#3752—61no—
57C4⋊(C3⋊Dic3)C2×D312yes—
58C4×C3⋊Dic3C2×D311yescharacteristic
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3—32no—
61C3×C4⋊Dic3—32no—
62C3×C4⋊Dic3—62no—
63C3×C4⋊Dic3C2×D312yes—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—22no—
67C2×C3×C3⋊C8C2×D312yes—
68C2×C3×C3⋊C8C2×D312yes—
69#3724—61no—
70Dic3⋊Dic3—62no—
71#3722—62no—
72#3722—62no—
73C22×C33C2×D411yescharacteristic
74C2×C3⋊(C3⋊D3)—22no—
75C2×C3⋊(C3⋊D3)—41no—
76C2×C3⋊(C3⋊D3)—41no—
77C33×C4C2×Q811yescharacteristic
78C33×C4#4111yescharacteristic
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3—42no—
81C3×C3⋊Dic3D812yes—
82C32×Dic3—41no—
83#893—92no—
84#893—181no—
85#888—92no—
86C4×D12—91no—
87#869—91no—
88C3×#107—92no—
89C3×C4⋊C8—91no—
90#821—92no—
91#821—92no—
92#821—182no—
93#818—32no—
94Dic3⋊C8—32no—
95#809—92no—
96#807—32no—
97C22×C3⋊D3—61no—
98C22×C3⋊D3—62no—
99C22×C3⋊D3—62no—
100C22×C3⋊D3—122no—
101C22×C3⋊D3—122no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—41no—
105C2×C32×C4—61no—
106C2×C32×C4C22×D312yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3D1211yescharacteristic
110C2×C3⋊Dic3D1211yescharacteristic
111C4⋊(C3⋊D3)—32no—
112C4⋊(C3⋊D3)—34no—
113C4⋊(C3⋊D3)—62no—
114C4⋊(C3⋊D3)—64no—
115C4⋊(C3⋊D3)—61no—
116C4⋊(C3⋊D3)—62no—
117C4⋊(C3⋊D3)—62no—
118C4⋊(C3⋊D3)—122no—
119C4⋊(C3⋊D3)—122no—
120C4⋊(C3⋊D3)—122no—
121C4×C3⋊D3—122no—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—32no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—122no—
131#627—124no—
132#627—122no—
133#624—124no—
134#624—124no—
135C32×C8—62no—
136C3×C3⋊C8—22no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—42no—
139#474—271noSylow
140C2×C33C22⋊Q811yescharacteristic
141C2×C33C2×D811yescharacteristic
142C2×C33#13711yescharacteristic
143C3⋊(C3⋊D3)—42no—
144C2×D12—92no—
145C2×D12—91no—
146C2×D12—181no—
147C2×D12—182no—
148C2×D12—361no—
149C2×C4×D3—181no—
150C2×C4×D3—182no—
151C2×C4×D3—182no—
152C2×Dic6—182no—
153C2×Dic6—181no—
154C2×C3×C8—32no—
155C2×C3×C8—32no—
156C2×C3×C8—62no—
157C3×C4⋊C4—32no—
158C3×C4⋊C4—61no—
159C3×C4⋊C4—92no—
160C3×C42—91no—
161(C2×D3)⋊C4—181no—
162(C2×D3)⋊C4—182no—
163(C2×D3)⋊C4—182no—
164C4⋊Dic3—32no—
165C4⋊Dic3—32no—
166C4⋊Dic3—62no—
167C4⋊Dic3—181no—
168C4⋊Dic3D3211yescharacteristic
169Dic3⋊C4—182no—
170C4×Dic3—32no—
171C4×Dic3—61no—
172C2×C3⋊C8D3212yes—
173C22×C32—22no—
174C22×C32—21no—
175C22×C32—41no—
176C22×C32C2×D1211yescharacteristic
177C22×C32D3×D412yes—
178C2×C3⋊D3—62no—
179C2×C3⋊D3—64no—
180C2×C3⋊D3—64no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3—124no—
183C2×C3⋊D3—121no—
184C2×C3⋊D3—121no—
185C2×C3⋊D3—122no—
186C2×C3⋊D3—122no—
187C2×C3⋊D3—122no—
188C2×C3⋊D3—122no—
189C2×C3⋊D3—242no—
190C2×C3⋊D3—242no—
191C2×C3⋊D3—242no—
192C2×C3⋊D3—242no—
193C32×C4—21no—
194C32×C4—22no—
195C32×C4—21no—
196C32×C4—22no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4—121no—
200C32×C4#23512yes—
201C32×C4#23711yescharacteristic
202C32×C4D3×Q812yes—
203C32×C4D3×Q811yescharacteristic
204C3⋊Dic3—21no—
205C3⋊Dic3—42no—
206C3⋊Dic3D2412yes—
207C3×Dic3—34no—
208C3×Dic3—42no—
209C3×Dic3—42no—
210C3×Dic3—62no—
211C3×Dic3—62no—
212C3×Dic3—121no—
213C3×Dic3—122no—
214C3×Dic3—122no—
215C3×Dic3—242no—
216C4⋊Q8—271no—
217C4×D4—271no—
218#107—92no—
219C4⋊C8—91no—
220#102—272no—
221C33#47411yescharacteristic, complemented, Sylow
222C22×D3—182no—
223C22×D3—182no—
224C22×D3—181no—
225C22×D3—362no—
226C22×D3—362no—
227C2×C3×C4—21no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—62no—
231C2×C3×C4—62no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—182no—
235C2×C3×C4C2×D3211yescharacteristic
236C2×C3×C4C2×D3212yes—
237C2×Dic3—21no—
238C2×Dic3—32no—
239C2×Dic3—32no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—62no—
243C2×Dic3—62no—
244C2×Dic3—122no—
245D12—92no—
246D12—94no—
247D12—182no—
248D12—184no—
249D12—182no—
250D12—181no—
251D12—182no—
252D12—362no—
253D12—362no—
254D12—362no—
255C4×D3—364no—
256C4×D3—362no—
257C4×D3—364no—
258Dic6—364no—
259Dic6—362no—
260C3×C8—62no—
261C3×C8—62no—
262C3×C8—122no—
263C3⋊C8—22no—
264C2×C32—22no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—22no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32#89612yes—
274C2×C32#89711yescharacteristic
275C2×C32C2×D2411yescharacteristic
276C2×C32#90911yescharacteristic
277C2×C32D3×D812yes—
278C2×C32#91812yes—
279C3⋊D3—122no—
280C3⋊D3—124no—
281C3⋊D3—124no—
282C3⋊D3—244no—
283C3⋊D3—244no—
284C2×Q8—541no—
285C2×D4—271no—
286C22×C4—541no—
287C2×C8—32no—
288C4⋊C4—31no—
289C4⋊C4—92no—
290C4⋊C4—541no—
291C22⋊C4—541no—
292C42—91no—
293C22×C3—21no—
294C22×C3—22no—
295C22×C3—41no—
296C22×C3D3×D1212yes—
297C22×C3#381111yescharacteristic
298C2×D3—184no—
299C2×D3—182no—
300C2×D3—184no—
301C2×D3—364no—
302C2×D3—364no—
303C2×D3—362no—
304C2×D3—362no—
305C2×D3—362no—
306C2×D3—362no—
307C2×D3—361no—
308C2×D3—361no—
309C2×D3—722no—
310C2×D3—722no—
311C2×D3—722no—
312C2×D3—722no—
313C3×C4—21no—
314C3×C4—22no—
315C3×C4—22no—
316C3×C4—21no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—92no—
320C3×C4—122no—
321C3×C4—122no—
322C3×C4—181no—
323C3×C4—362no—
324C3×C4#379711yescharacteristic
325C3×C4#379712yes—
326C3×C4#379811yescharacteristic
327C3×C4#380412yes—
328Dic3—34no—
329Dic3—41no—
330Dic3—62no—
331Dic3—62no—
332Dic3—121no—
333Dic3—122no—
334Dic3—122no—
335Dic3—242no—
336C32—22no—
337C32—21no—
338C32—41no—
339C32#484611yescharacteristic, complemented
340C32#489012yescomplemented
341C23—541no—
342Q8—1082no—
343D4—272no—
344D4—541no—
345C2×C4—61no—
346C2×C4—91no—
347C2×C4—91no—
348C2×C4—182no—
349C2×C4—1082no—
350C2×C4D3311yescharacteristic
351C8—62no—
352C2×C3—21no—
353C2×C3—22no—
354C2×C3—21no—
355C2×C3—22no—
356C2×C3—21no—
357C2×C3—22no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3D3×D2412yes—
362C2×C3#6380112yes—
363C2×C3#6386511yescharacteristic
364C2×C3#6389512yes—
365C2×C3#6392611yescharacteristic
366C2×C3#6394511yescharacteristic
367D3—364no—
368D3—364no—
369D3—362no—
370D3—724no—
371D3—724no—
372C22—542no—
373C22—1081no—
374C22—1081no—
375C22#8895511yescharacteristic
376C4—92no—
377C4—121no—
378C4—181no—
379C4—362no—
380C4#8895811yescharacteristic
381C4#8895911yescharacteristic
382C3—21no—
383C3—22no—
384C3—41no—
385C3#1059043011yescharacteristic, complemented
386C3#1059076412yescomplemented
387C2—1082no—
388C2#1172283311yescharacteristic
389C2#1172289711yescharacteristic
390C2#1172298211yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#474
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. C4⋊#6296C2semidirect
4. C3×#63575C2semidirect
5. C3×#63572C2semidirect
26. #63575D3semidirect
27. #63572D3semidirect
34. C4⋊(C3⋊(C3⋊D3))Q8semidirect
221. C33#474semidirect
339. C32#4846semidirect
340. C32#4890semidirect
385. C3#10590430semidirect
386. C3#10590764semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
#117234942
#117220362
C3×#635752
C3×#635722
#105907643
#105904303