Finite Group Databasebeta

← #13156513

Subgroups — #13156513

Order 1728.

734 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 106 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#11723524C211yesmaximal, characteristic
3#11722036C212yesmaximal, complemented
4C3×#63563C212yesmaximal, complemented
5C3×#63553C211yesmaximal, characteristic, complemented
6#10591402—31nomaximal
7#10590878—32nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C32×D12C2211yescharacteristic
10C2×#6296—21no—
11C2×#6291—21no—
12C2×#6291—21no—
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C32×C3⋊C8C2212yes—
16#88815—21no—
17#64149—31no—
18#63919—31no—
19#63918—32no—
20#63911—32no—
21#63911—61no—
22#63648—32no—
23C3×#832—32no—
24C3×#821—32no—
25C3×C8×Dic3—32no—
26#63564—32no—
27#63564—32no—
28#63564—62no—
29#63563D312yescomplemented
30#63553D311yescharacteristic, complemented
31C22×C3⋊(C3⋊D3)—21no—
32C22×C32×D3—21no—
33C2×C33×C4C2311yescharacteristic
34C4⋊(C3⋊(C3⋊D3))—22no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C32×D12—22no—
39#6296—42no—
40#6291—42no—
41#6291—42no—
42C32×C3⋊C8—22no—
43#4760—92no—
44#4694—91no—
45C2×C32×D4—31no—
46C2×#638—61no—
47C2×#638—61no—
48C2×C4⋊(C3⋊D3)—31no—
49C2×C4⋊(C3⋊D3)—32no—
50C2×C4⋊(C3⋊D3)—62no—
51C2×C4⋊(C3⋊D3)—61no—
52C2×C4⋊(C3⋊D3)—121no—
53C2×C3×D12—21no—
54C2×C3×D12C2×D312yes—
55C2×#627—62no—
56C2×#627—61no—
57C2×#626—62no—
58C2×#626—62no—
59C2×#626—62no—
60C2×#626—62no—
61C2×C32×C8—32no—
62C4⋊(C3⋊Dic3)C2×D312yes—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C3×C4⋊Dic3—32no—
65C3×C4⋊Dic3—32no—
66C3×C4⋊Dic3—62no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—22no—
70C2×C3×C3⋊C8C2×D312yes—
71C2×C3×C3⋊C8C2×D312yes—
72#3724—61no—
73Dic3⋊Dic3—62no—
74C22×C33C2×D411yescharacteristic
75C2×C3⋊(C3⋊D3)—41no—
76C2×C3⋊(C3⋊D3)—41no—
77C2×C3⋊(C3⋊D3)—41no—
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C2×C32×D3—41no—
81C33×C4#4112yes—
82C3×C3⋊Dic3—42no—
83C3×C3⋊Dic3D812yes—
84C3×C3⋊Dic3SD1612yes—
85#940—92no—
86#940—181no—
87#888—92no—
88#874—91no—
89#869—91no—
90C3×#102—92no—
91C3×C4×C8—91no—
92#832—32no—
93#821—32no—
94#821—92no—
95#821—92no—
96#821—182no—
97C8×Dic3—32no—
98#809—92no—
99C23×C32—61no—
100C22×C3⋊D3—61no—
101C22×C3⋊D3—62no—
102C22×C3⋊D3—62no—
103C22×C3⋊D3—122no—
104C22×C3⋊D3—122no—
105C22×C3×D3—22no—
106C22×C3×D3—22no—
107C32×D4—62no—
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4—41no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C32×C4C22×D312yes—
113#638—122no—
114#638—122no—
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3D1211yescharacteristic
117C2×C3⋊Dic3D1211yescharacteristic
118C4⋊(C3⋊D3)—62no—
119C4⋊(C3⋊D3)—64no—
120C4⋊(C3⋊D3)—62no—
121C4⋊(C3⋊D3)—62no—
122C4⋊(C3⋊D3)—122no—
123C4⋊(C3⋊D3)—122no—
124C4⋊(C3⋊D3)—124no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—122no—
132C3×D12—22no—
133C3×D12—22no—
134C3×D12—42no—
135#627—124no—
136#627—122no—
137#626—124no—
138#626—124no—
139#626—124no—
140#626—124no—
141C32×C8—62no—
142C3×C3⋊C8—22no—
143C3×C3⋊C8—22no—
144C3×C3⋊C8—42no—
145#486—271noSylow
146C2×C33#12411yescharacteristic
147C2×C33C2×D811yescharacteristic
148C2×C33C2×SD1611yescharacteristic
149C3⋊(C3⋊D3)—81no—
150C32×D3—81no—
151C2×C3×D4—32no—
152C2×C3×D4—61no—
153C2×#67—182no—
154C2×#67—182no—
155C2×#67—182no—
156C2×#67—182no—
157C2×D12—92no—
158C2×D12—91no—
159C2×D12—182no—
160C2×D12—181no—
161C2×D12—181no—
162C2×D12—181no—
163C2×D12—361no—
164C2×D12D3211yescharacteristic
165C2×Dic6—182no—
166C2×Dic6—181no—
167C2×C3×C8—32no—
168C2×C3×C8—32no—
169C2×C3×C8—62no—
170C3×C4⋊C4—92no—
171C3×C42—91no—
172C4⋊Dic3—32no—
173C4⋊Dic3—32no—
174C4⋊Dic3—62no—
175C4⋊Dic3—181no—
176Dic3⋊C4—182no—
177C4×Dic3—32no—
178C4×Dic3—61no—
179C2×C3⋊C8D3212yes—
180C22×C32—22no—
181C22×C32—21no—
182C22×C32—41no—
183C22×C32—121no—
184C22×C32—121no—
185C22×C32—121no—
186C22×C32C2×D1211yescharacteristic
187C22×C32D3×D412yes—
188C2×C3⋊D3—124no—
189C2×C3⋊D3—122no—
190C2×C3⋊D3—124no—
191C2×C3⋊D3—121no—
192C2×C3⋊D3—122no—
193C2×C3⋊D3—121no—
194C2×C3⋊D3—121no—
195C2×C3⋊D3—122no—
196C2×C3⋊D3—122no—
197C2×C3⋊D3—122no—
198C2×C3⋊D3—122no—
199C2×C3⋊D3—242no—
200C2×C3⋊D3—242no—
201C2×C3⋊D3—242no—
202C2×C3⋊D3—242no—
203C2×C3×D3—42no—
204C2×C3×D3—42no—
205C2×C3×D3—42no—
206C2×C3×D3—42no—
207C2×C3×D3—42no—
208C2×C3×D3—42no—
209C32×C4—22no—
210C32×C4—22no—
211C32×C4—22no—
212C32×C4—42no—
213C32×C4#23512yes—
214C32×C4#23512yes—
215C32×C4#23712yes—
216C3⋊Dic3—42no—
217C3⋊Dic3#20412yes—
218C3⋊Dic3D2412yes—
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—271no—
228#102—92no—
229#102—272no—
230C4×C8—91no—
231C33#48611yescharacteristic, complemented, Sylow
232C23×C3—62no—
233C23×C3—62no—
234C22×D3—21no—
235C22×D3—182no—
236C22×D3—182no—
237C22×D3—181no—
238C22×D3—362no—
239C22×D3—362no—
240C3×D4—62no—
241C3×D4—62no—
242C3×D4—122no—
243C2×C3×C4—21no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4—91no—
247C2×C3×C4—91no—
248C2×C3×C4—182no—
249C2×C3×C4C2×D3212yes—
250C2×C3×C4C2×D3211yescharacteristic
251#67—364no—
252#67—364no—
253#67—364no—
254#67—364no—
255C2×Dic3—32no—
256C2×Dic3—32no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—62no—
260C2×Dic3—62no—
261C2×Dic3—122no—
262D12—22no—
263D12—184no—
264D12—182no—
265D12—182no—
266D12—182no—
267D12—362no—
268D12—362no—
269D12—364no—
270D12—362no—
271D12—362no—
272Dic6—364no—
273Dic6—362no—
274C3×C8—62no—
275C3×C8—62no—
276C3×C8—122no—
277C3⋊C8—22no—
278C2×C32—22no—
279C2×C32—22no—
280C2×C32—22no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32—241no—
288C2×C32#87511yescharacteristic
289C2×C32#88512yes—
290C2×C32C2×#20411yescharacteristic
291C2×C32C2×D2411yescharacteristic
292C2×C32D3×D812yes—
293C2×C32D3×SD1612yes—
294C3⋊D3—242no—
295C3⋊D3—244no—
296C3⋊D3—244no—
297C3⋊D3—241no—
298C3⋊D3—242no—
299C3×D3—82no—
300C3×D3—82no—
301C2×Q8—541no—
302C2×D4—31no—
303C2×D4—271no—
304C2×D4—541no—
305C2×D4—541no—
306C2×C8—32no—
307C4⋊C4—92no—
308C4⋊C4—541no—
309C42—91no—
310C22×C3—21no—
311C22×C3—22no—
312C22×C3—41no—
313C22×C3—122no—
314C22×C3—122no—
315C22×C3—122no—
316C22×C3—122no—
317C22×C3—122no—
318C22×C3—122no—
319C22×C3D3×D1212yes—
320C22×C3#381111yescharacteristic
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—364no—
325C2×D3—362no—
326C2×D3—362no—
327C2×D3—362no—
328C2×D3—364no—
329C2×D3—362no—
330C2×D3—362no—
331C2×D3—362no—
332C2×D3—361no—
333C2×D3—361no—
334C2×D3—361no—
335C2×D3—722no—
336C2×D3—722no—
337C2×D3—722no—
338C2×D3—722no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—42no—
343C3×C4—92no—
344C3×C4—92no—
345C3×C4—362no—
346C3×C4#379812yes—
347C3×C4#380412yes—
348C3×C4#380412yes—
349Dic3—34no—
350Dic3—34no—
351Dic3—62no—
352Dic3—62no—
353Dic3—122no—
354Dic3—122no—
355Dic3—242no—
356C32—22no—
357C32—21no—
358C32—41no—
359C32#469411yescharacteristic, complemented
360C32#476012yescomplemented
361C23—61no—
362C23—541no—
363Q8—1082no—
364D4—62no—
365D4—542no—
366D4—1082no—
367D4—1082no—
368C2×C4—91no—
369C2×C4—91no—
370C2×C4—182no—
371C2×C4D3311yescharacteristic
372C8—62no—
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—242no—
383C2×C3—242no—
384C2×C3D3×#20412yes—
385C2×C3D3×D2412yes—
386C2×C3#6385212yes—
387C2×C3#6392611yescharacteristic
388C2×C3#6393011yescharacteristic
389C2×C3#6395311yescharacteristic
390D3—81no—
391D3—722no—
392D3—724no—
393D3—724no—
394D3—722no—
395D3—721no—
396C22—121no—
397C22—121no—
398C22—121no—
399C22—1081no—
400C22—1081no—
401C22—1081no—
402C22#8895511yescharacteristic
403C4—92no—
404C4—92no—
405C4—362no—
406C4#8895912yes—
407C3—21no—
408C3—22no—
409C3—41no—
410C3#1059087812yescomplemented
411C3#1059140211yescharacteristic, complemented
412C2—241no—
413C2—2161no—
414C2#1172283311yescharacteristic
415C2#1172284211yescharacteristic
416C2#1172300811yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723524C2
hypocenterC33#486
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
3. #11722036C2semidirect
4. C3×#63563C2semidirect
5. C3×#63553C2semidirect
29. #63563D3semidirect
30. #63553D3semidirect
231. C33#486semidirect
359. C32#4694semidirect
360. C32#4760semidirect
410. C3#10590878semidirect
411. C3#10591402semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
#117235242
#117220362
C3×#635632
C3×#635532
#105914023
#105908783