Finite Group Databasebeta

← #13161967

Subgroups — #13161967

Order 1728.

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

normal subgroups only

This group has 734 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×#64149C211yesmaximal, characteristic
2C4⋊#6296C211yesmaximal, characteristic, complemented
3C3×#63659C212yesmaximal, complemented
4#11722075C212yesmaximal, complemented
5#11722070C211yesmaximal, characteristic, complemented
6#10591487—32nomaximal
7#10591402—31nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—21no—
10C2×C3×#638—21no—
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×#6296—21no—
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#88815—21no—
17#64149D311yescharacteristic
18C3×#940—32no—
19C3×#940—61no—
20#63919—31no—
21#63918—32no—
22#63659D312yescomplemented
23C3×#832—32no—
24C3×#832—32no—
25C3×#832—62no—
26#63564—32no—
27#63563—32no—
28#63562—32no—
29#63553—31no—
30Dic3×C3⋊C8—32no—
31C23×C33—21no—
32C22×C3×C3⋊D3—21no—
33C33×D4—22no—
34C2×C33×C4C2311yescharacteristic
35C3×#638—42no—
36C3×#638—42no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3D411yescharacteristic
39C2×C3×C3⋊Dic3D411yescharacteristic
40C3×C4⋊(C3⋊D3)—22no—
41#6296—42no—
42C3×C3⋊(C3⋊C8)—22no—
43#5070—91no—
44#4760—92no—
45C2×C32×D4—22no—
46C2×C32×D4—21no—
47C2×C32×D4—41no—
48C2×C32×D4C2×D312yes—
49C2×C32×D4C2×D311yescharacteristic
50C2×#638—21no—
51C2×#638—21no—
52C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
53C2×C3×#67—62no—
54C2×C3×#67—62no—
55C2×C3×#67—62no—
56C2×C3×#67—62no—
57C2×C3×D12—32no—
58C2×C3×D12—61no—
59C2×#627—62no—
60C2×#627—61no—
61C4⋊(C3⋊Dic3)C2×D312yes—
62C4×C3⋊Dic3C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D312yes—
64C3×C4⋊Dic3—32no—
65C3×C4⋊Dic3—32no—
66C3×C4⋊Dic3—62no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—62no—
72#3724—61no—
73Dic3⋊Dic3—62no—
74C22×C33—41no—
75C22×C33—41no—
76C22×C33—41no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C33×C4#4112yes—
82C3×C3⋊Dic3—42no—
83C3×C3⋊Dic3D812yes—
84C3×C3⋊Dic3SD1612yes—
85C3×#127—91no—
86#940—32no—
87#940—61no—
88#888—92no—
89#869—91no—
90C3×#102—92no—
91#832—32no—
92#832—32no—
93#832—62no—
94#832—92no—
95#821—92no—
96C8×Dic3—92no—
97#809—92no—
98C4×C3⋊C8—91no—
99C23×C32—21no—
100C23×C32—22no—
101C23×C32—22no—
102C23×C32—42no—
103C23×C32—42no—
104C22×C3⋊D3—21no—
105C22×C3×D3—62no—
106C22×C3×D3—62no—
107C32×D4—22no—
108C32×D4—24no—
109C32×D4—22no—
110C32×D4—22no—
111C32×D4—42no—
112C32×D4—42no—
113C32×D4—44no—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—41no—
117C2×C32×C4C22×D312yes—
118C2×C32×C4C22×D311yescharacteristic
119#638—42no—
120#638—42no—
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3#6711yescharacteristic
123C2×C3⋊Dic3#6711yescharacteristic
124C4⋊(C3⋊D3)—22no—
125C3×#67—124no—
126C3×#67—124no—
127C3×#67—124no—
128C3×#67—124no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—122no—
136C3×D12—62no—
137C3×D12—62no—
138C3×D12—122no—
139#627—124no—
140#627—122no—
141C3⋊(C3⋊C8)—22no—
142C3×C3⋊C8—62no—
143C3×C3⋊C8—62no—
144C3×C3⋊C8—122no—
145#486—271noSylow
146C2×C33—81no—
147C2×C33#12411yescharacteristic
148C2×C33C2×D811yescharacteristic
149C2×C33C2×SD1611yescharacteristic
150C3×C3⋊D3—81no—
151C2×C3×D4—22no—
152C2×C3×D4—21no—
153C2×C3×D4—41no—
154C2×C3×D4—91no—
155C2×C3×D4—181no—
156C2×C3×D4—181no—
157C2×C3×D4D3211yescharacteristic
158C2×C3×D4D3212yes—
159C2×#67—62no—
160C2×#67—62no—
161C2×#67—62no—
162C2×#67—62no—
163C2×D12—32no—
164C2×D12—61no—
165C2×Dic6—182no—
166C2×Dic6—181no—
167C2×C3×C8—92no—
168C3×C4⋊C4—92no—
169C3×C42—91no—
170C4⋊Dic3—32no—
171C4⋊Dic3—32no—
172C4⋊Dic3—62no—
173C4⋊Dic3—181no—
174Dic3⋊C4—182no—
175C4×Dic3—32no—
176C4×Dic3—61no—
177C2×C3⋊C8—32no—
178C2×C3⋊C8—32no—
179C2×C3⋊C8—62no—
180C22×C32—22no—
181C22×C32—21no—
182C22×C32—41no—
183C22×C32—44no—
184C22×C32—42no—
185C22×C32—44no—
186C22×C32—42no—
187C22×C32—41no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—42no—
191C22×C32—42no—
192C22×C32—42no—
193C22×C32—42no—
194C22×C32—82no—
195C22×C32—82no—
196C22×C32—82no—
197C22×C32—82no—
198C22×C32D3×D412yes—
199C22×C32C2×#6711yescharacteristic
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3×D3—122no—
204C2×C3×D3—122no—
205C2×C3×D3—122no—
206C2×C3×D3—122no—
207C2×C3×D3—122no—
208C2×C3×D3—122no—
209C32×C4—22no—
210C32×C4—22no—
211C32×C4—22no—
212C32×C4—42no—
213C32×C4#23512yes—
214C32×C4#23712yes—
215C32×C4#23712yes—
216C3⋊Dic3—42no—
217C3⋊Dic3#21312yes—
218C3⋊Dic3#21412yes—
219C3×Dic3—34no—
220C3×Dic3—34no—
221C3×Dic3—62no—
222C3×Dic3—62no—
223C3×Dic3—122no—
224C3×Dic3—122no—
225C3×Dic3—242no—
226C4⋊Q8—271no—
227#127—91no—
228#102—92no—
229#102—272no—
230C4×C8—271no—
231C33#48611yescharacteristic, complemented, Sylow
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—21no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—181no—
238C22×D3—62no—
239C22×D3—62no—
240C3×D4—24no—
241C3×D4—22no—
242C3×D4—22no—
243C3×D4—22no—
244C3×D4—42no—
245C3×D4—42no—
246C3×D4—44no—
247C3×D4—182no—
248C3×D4—362no—
249C3×D4—362no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—41no—
253C2×C3×C4—91no—
254C2×C3×C4—91no—
255C2×C3×C4—182no—
256C2×C3×C4C2×D3211yescharacteristic
257C2×C3×C4C2×D3212yes—
258#67—124no—
259#67—124no—
260#67—124no—
261#67—124no—
262C2×Dic3—32no—
263C2×Dic3—32no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—122no—
269D12—62no—
270D12—62no—
271D12—122no—
272Dic6—364no—
273Dic6—362no—
274C3×C8—182no—
275C3⋊C8—62no—
276C3⋊C8—62no—
277C3⋊C8—122no—
278C2×C32—22no—
279C2×C32—21no—
280C2×C32—22no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—22no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32—82no—
288C2×C32—84no—
289C2×C32—84no—
290C2×C32—82no—
291C2×C32—81no—
292C2×C32#88512yes—
293C2×C32D3×D812yes—
294C2×C32D3×SD1612yes—
295C2×C32C2×#21311yescharacteristic
296C2×C32C2×#21411yescharacteristic
297C2×C32#93611yescharacteristic
298C3⋊D3—81no—
299C3×D3—242no—
300C3×D3—242no—
301C2×Q8—541no—
302C2×D4—91no—
303C2×D4—181no—
304C2×D4—181no—
305C2×D4#111311yescharacteristic
306C2×C8—92no—
307C4⋊C4—92no—
308C4⋊C4—541no—
309C42—91no—
310C22×C3—21no—
311C22×C3—22no—
312C22×C3—42no—
313C22×C3—42no—
314C22×C3—44no—
315C22×C3—42no—
316C22×C3—44no—
317C22×C3—42no—
318C22×C3—41no—
319C22×C3—42no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—42no—
323C22×C3—41no—
324C22×C3—82no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3—82no—
328C22×C3—361no—
329C22×C3—361no—
330C22×C3—361no—
331C22×C3D3×#6712yes—
332C22×C3#381111yescharacteristic
333C2×D3—122no—
334C2×D3—122no—
335C2×D3—122no—
336C2×D3—122no—
337C2×D3—122no—
338C2×D3—122no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—42no—
343C3×C4—92no—
344C3×C4—92no—
345C3×C4—362no—
346C3×C4#380412yes—
347C3×C4#380412yes—
348C3×C4#380512yes—
349Dic3—34no—
350Dic3—34no—
351Dic3—62no—
352Dic3—62no—
353Dic3—122no—
354Dic3—122no—
355Dic3—242no—
356C32—21no—
357C32—22no—
358C32—41no—
359C32#476012yescomplemented
360C32#507011yescharacteristic, complemented
361C23—21no—
362C23—181no—
363Q8—1082no—
364D4—22no—
365D4—182no—
366D4—362no—
367D4—362no—
368C2×C4—91no—
369C2×C4—91no—
370C2×C4—182no—
371C2×C4C2×#111311yescharacteristic
372C8—182no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—22no—
377C2×C3—21no—
378C2×C3—22no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3—82no—
383C2×C3—84no—
384C2×C3—84no—
385C2×C3—81no—
386C2×C3—82no—
387C2×C3—721no—
388C2×C3D3×#21312yes—
389C2×C3#6392611yescharacteristic
390C2×C3D3×#21412yes—
391C2×C3#6393011yescharacteristic
392C2×C3#6395311yescharacteristic
393C2×C3#6395912yes—
394D3—242no—
395D3—242no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—361no—
400C22—361no—
401C22—361no—
402C22#8904911yescharacteristic
403C4—92no—
404C4—92no—
405C4—362no—
406C4#8904612yes—
407C3—21no—
408C3—22no—
409C3—41no—
410C3#1059140211yescharacteristic, complemented
411C3#1059148712yescomplemented
412C2—81no—
413C2—721no—
414C2#1172370211yescharacteristic
415C2#1172370411yescharacteristic
416C2#1172371811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64149C2
hypercenterC2×D4#1113
hypocenterC33#486
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
3. C3×#63659C2semidirect
4. #11722075C2semidirect
5. #11722070C2semidirect
22. #63659D3semidirect
231. C33#486semidirect
359. C32#4760semidirect
360. C32#5070semidirect
410. C3#10591402semidirect
411. C3#10591487semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
C4⋊#62962
C3×#636592
#117220752
#117220702
#105914873
#105914023