Finite Group Databasebeta

← #13155934

Subgroups — #13155934

Order 1728.

802 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 98 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×#64308C211yesmaximal, characteristic
2#11723436C211yesmaximal, characteristic, complemented
3C3×#63827C212yesmaximal, complemented
4#11722073C211yesmaximal, characteristic, complemented
5#11722043C212yesmaximal, complemented
6#10591664—32nomaximal
7#10591658—31nomaximal
8C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
9C32×#235C2212yes—
10C2×C32×Dic6C2211yescharacteristic, complemented
11C3×#3811—22no—
12C3×D3×D12—22no—
13C3×#3799—22no—
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
15C3×#3748C2212yescomplemented
16#88803—22no—
17C3×#3718—22no—
18C3×#3715—22no—
19C3×#1009—32no—
20#64308D311yescharacteristic
21#64156—31no—
22C3×#951—32no—
23C3×#942—32no—
24#63863—32no—
25#63863—61no—
26#63827D312yescomplemented
27#63664—32no—
28#63568—32no—
29#63568—32no—
30#63568—62no—
31#63559—31no—
32#63558—32no—
33C22×C3×C3⋊D3D412yes—
34C2×C3×D32—22no—
35C2×C33×C4C2311yescharacteristic
36C2×C3⋊(C3⋊Dic3)—21no—
37C3×C4⋊(C3⋊D3)—22no—
38C32×#67—22no—
39C2×C32×Dic3—21no—
40C32×D12D412yes—
41C32×C4×D3—22no—
42C32×Dic6—22no—
43C32×Dic6D412yes—
44C3×#626—24no—
45C3×#624—22no—
46C3×C3⋊(C3⋊C8)—22no—
47#5242—91no—
48#5202—92no—
49C32×#41—32no—
50C2×C32×Q8—31no—
51C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
52C3×#239—62no—
53C3×D3×D4—62no—
54C3×D3×D4—64no—
55C3×#235—22no—
56C3×#235—62no—
57C3×#235C2×D312yes—
58C3×#235C2×D312yes—
59C2×C3×D12—32no—
60C2×C3×D12—62no—
61C2×C3×D12—61no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6C2×D312yescomplemented
64#3811—22no—
65D3×D12—22no—
66#3799—22no—
67#3752—62no—
68C4×C3⋊Dic3—31no—
69C4×C3⋊Dic3—32no—
70C4×C3⋊Dic3—61no—
71C4×C3⋊Dic3—62no—
72C4×C3⋊Dic3—121no—
73#3748C2×D312yescomplemented
74C3×#216—62no—
75C3×#216—62no—
76C3×#215—62no—
77C3×#214—62no—
78C3×#208—32no—
79C3×#208—32no—
80C3×#208—62no—
81(C2×D3)⋊Dic3—64no—
82(C2×D3)⋊Dic3—64no—
83#3718—22no—
84#3715—22no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—22no—
87C2×C3×C3⋊D3—42no—
88C2×C32×D3—22no—
89C3×D32—82no—
90C33×C4C2×D412yes—
91C3⋊(C3⋊Dic3)—41no—
92C32×Dic3—22no—
93C32×Dic3—41no—
94C3×H(5,𝔽2)—91no—
95#1009—32no—
96C3×#137—92no—
97#951—32no—
98#947—92no—
99#947—181no—
100#942—32no—
101#936—91no—
102#837—92no—
103#837—92no—
104#837—182no—
105#833—91no—
106#825—92no—
107#823—92no—
108C22×C3⋊D3#6712yes—
109C22×C3×D3—34no—
110C22×C3×D3—62no—
111C22×C3×D3—64no—
112C2×D32—22no—
113C32×Q8—32no—
114C32×Q8—62no—
115C32×D4—32no—
116C32×D4—62no—
117C2×C32×C4—22no—
118C2×C32×C4—21no—
119C2×C32×C4—41no—
120C2×C32×C4—61no—
121C2×C32×C4—62no—
122C2×C32×C4C22×D312yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—61no—
125C2×C3⋊Dic3—62no—
126C2×C3⋊Dic3—62no—
127C2×C3⋊Dic3—122no—
128C2×C3⋊Dic3—122no—
129C4⋊(C3⋊D3)—22no—
130C3×#67—22no—
131C3×#67—22no—
132C3×#67—42no—
133C3×#67—64no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—22no—
136C3×D12—22no—
137C3×D12—64no—
138C3×D12—62no—
139C3×D12—62no—
140C3×D12—122no—
141C3×D12#6712yes—
142C3×D12#6712yes—
143C3×C4×D3—22no—
144C3×C4×D3—22no—
145C3×C4×D3—42no—
146C3×C4×D3—64no—
147C3×Dic6—22no—
148C3×Dic6—22no—
149C3×Dic6—22no—
150C3×Dic6—42no—
151C3×Dic6#6712yes—
152C3×Dic6#6712yes—
153#626—24no—
154#624—22no—
155C3⋊(C3⋊C8)—22no—
156C3×C3⋊C8—62no—
157C3×C3⋊C8—62no—
158C3×C3⋊C8—122no—
159#455—271noSylow
160C2×C33—21no—
161C2×C33#12011yescharacteristic
162C3×C3⋊D3—42no—
163C32×D3—42no—
164C3×#41—32no—
165C3×#41—32no—
166C3×#41—62no—
167C3×#41—182no—
168C2×C3×Q8—32no—
169C2×C3×Q8—61no—
170C2×C3×D4—91no—
171C2×C3×D4—182no—
172C2×C3×D4—182no—
173#239—62no—
174D3×D4—62no—
175D3×D4—64no—
176#235—62no—
177#235D3212yes—
178C2×D12—32no—
179C2×D12—61no—
180C2×D12—62no—
181C2×Dic6D3211yescharacteristic, complemented
182C3×Q16—182no—
183C3×SD16—182no—
184C3×M16—92no—
185#217—182no—
186#216—62no—
187#216—62no—
188#215—62no—
189#214—62no—
190(C2×D3)⋊C4—184no—
191(C2×D3)⋊C4—184no—
192C4×Dic3—92no—
193C4×Dic3—91no—
194C4×Dic3—181no—
195C4×Dic3—182no—
196C4×Dic3—361no—
197#208—32no—
198#208—32no—
199#208—62no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32—41no—
203C22×C32—62no—
204C22×C32D3×D412yes—
205C22×C32C2×#6711yescharacteristic
206C2×C3⋊D3—22no—
207C2×C3⋊D3—42no—
208C2×C3×D3—22no—
209C2×C3×D3—22no—
210C2×C3×D3—42no—
211C2×C3×D3—64no—
212C2×C3×D3—122no—
213C2×C3×D3—124no—
214C2×C3×D3—124no—
215C2×C3×D3—244no—
216D32—82no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—22no—
220C32×C4—42no—
221C32×C4—62no—
222C32×C4—121no—
223C32×C4D3×D412yes—
224C32×C4C2×#6712yes—
225C32×C4C2×#6712yes—
226C3⋊Dic3—121no—
227C3⋊Dic3—122no—
228C3⋊Dic3—122no—
229C3⋊Dic3—124no—
230C3⋊Dic3—124no—
231C3×Dic3—22no—
232C3×Dic3—22no—
233C3×Dic3—42no—
234C3×Dic3—42no—
235C3×Dic3—42no—
236H(5,𝔽2)—91no—
237#137—92no—
238#124—271no—
239#104—272no—
240#100—271no—
241C33#45511yescharacteristic, complemented, Sylow
242C23×C3—92no—
243C23×C3—182no—
244C22×D3—34no—
245C22×D3—62no—
246C22×D3—64no—
247C3×Q8—32no—
248C3×Q8—32no—
249C3×Q8—62no—
250C3×Q8—62no—
251C3×Q8—62no—
252C3×Q8—122no—
253C3×D4—32no—
254C3×D4—32no—
255C3×D4—62no—
256C3×D4—62no—
257C3×D4—62no—
258C3×D4—122no—
259C3×D4—184no—
260C3×D4—182no—
261C2×C3×C4—22no—
262C2×C3×C4—21no—
263C2×C3×C4—41no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—62no—
267C2×C3×C4—62no—
268C2×C3×C4—122no—
269C2×C3×C4—182no—
270C2×C3×C4C2×D3212yes—
271C2×C3×C4C2×D3211yescharacteristic
272#67—22no—
273#67—64no—
274C2×Dic3—21no—
275C2×Dic3—182no—
276C2×Dic3—182no—
277C2×Dic3—181no—
278C2×Dic3—362no—
279C2×Dic3—362no—
280D12—64no—
281D12—62no—
282D12—62no—
283D12—122no—
284D12#62512yes—
285C4×D3—22no—
286C4×D3—64no—
287Dic6—22no—
288Dic6#62512yes—
289C3×C8—182no—
290C3⋊C8—62no—
291C3⋊C8—62no—
292C3⋊C8—122no—
293C2×C32—21no—
294C2×C32—22no—
295C2×C32—22no—
296C2×C32—22no—
297C2×C32—21no—
298C2×C32—41no—
299C2×C32—42no—
300C2×C32—42no—
301C2×C32—122no—
302C2×C32#93712yes—
303C2×C32#95311yescharacteristic
304C3⋊D3—42no—
305C3×D3—42no—
306C3×D3—42no—
307C3×D3—82no—
308C3×D3—124no—
309C3×D3—242no—
310#41—32no—
311#41—182no—
312C2×Q8—31no—
313C2×D4—91no—
314C2×D4—182no—
315C2×D4—182no—
316Q16—182no—
317SD16—182no—
318M16—92no—
319C22⋊C4—542no—
320C42—271no—
321C22×C3—21no—
322C22×C3—22no—
323C22×C3—41no—
324C22×C3—62no—
325C22×C3—62no—
326C22×C3—122no—
327C22×C3—182no—
328C22×C3—362no—
329C22×C3—722no—
330C22×C3D3×#6712yes—
331C22×C3#381111yescharacteristic
332C2×D3—22no—
333C2×D3—64no—
334C2×D3—122no—
335C2×D3—124no—
336C2×D3—124no—
337C2×D3—244no—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—42no—
342C3×C4—62no—
343C3×C4—62no—
344C3×C4—122no—
345C3×C4—122no—
346C3×C4—122no—
347C3×C4C2×#62512yes—
348C3×C4D3×#6712yes—
349C3×C4D3×#6712yes—
350Dic3—22no—
351Dic3—41no—
352Dic3—362no—
353Dic3—361no—
354Dic3—362no—
355Dic3—364no—
356Dic3—364no—
357C32—21no—
358C32—22no—
359C32—41no—
360C32#520212yescomplemented
361C32#524211yescharacteristic, complemented
362C23—92no—
363C23—182no—
364Q8—32no—
365Q8—62no—
366D4—32no—
367D4—62no—
368D4—184no—
369D4—182no—
370C2×C4—62no—
371C2×C4—61no—
372C2×C4—182no—
373C2×C4—541no—
374C2×C4D3311yescharacteristic
375C8—182no—
376C2×C3—22no—
377C2×C3—21no—
378C2×C3—22no—
379C2×C3—21no—
380C2×C3—22no—
381C2×C3—42no—
382C2×C3—42no—
383C2×C3—41no—
384C2×C3—122no—
385C2×C3—122no—
386C2×C3—242no—
387C2×C3—362no—
388C2×C3#6397612yes—
389C2×C3#6398311yescharacteristic
390D3—42no—
391D3—124no—
392D3—242no—
393C22—62no—
394C22—182no—
395C22—362no—
396C22—722no—
397C22#8895211yescharacteristic
398C4—62no—
399C4—121no—
400C4—1081no—
401C4D3×#62512yes—
402C3—21no—
403C3—22no—
404C3—41no—
405C3#1059165811yescharacteristic, complemented
406C3#1059166412yescomplemented
407C2—21no—
408C2—122no—
409C2—362no—
410C2#1172294611yescharacteristic
subgroupG/H
centerC2#11722946
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC2×C33#120
prime radicalC3×#64308C2
hypocenterC33#455
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723436C2semidirect
3. C3×#63827C2semidirect
4. #11722073C2semidirect
5. #11722043C2semidirect
10. C2×C32×Dic6C22semidirect
14. C4×C3⋊(C3⋊Dic3)C22semidirect
15. C3×#3748C22semidirect
26. #63827D3semidirect
63. C2×C3×Dic6C2×D3semidirect
73. #3748C2×D3semidirect
181. C2×Dic6D32semidirect
241. C33#455semidirect
360. C32#5202semidirect
361. C32#5242semidirect
405. C3#10591658semidirect
406. C3#10591664semidirect

Maximal subgroups

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

subgroupindex
C3×#643082
#117234362
C3×#638272
#117220732
#117220432
#105916643
#105916583