Finite Group Databasebeta

← #13161343

Subgroups — #13161343

Order 1728.

1184 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C2×#89018C211yesmaximal, characteristic
2C2×#88862C211yesmaximal, characteristic, complemented
3C3×#64155C211yesmaximal, characteristic, complemented
4C2×C32×#216C211yesmaximal, characteristic, complemented
5#11722037C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592886—31nomaximal
9#10591657—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6292—21no—
15C2×#6289—21no—
16#89018—22no—
17#89018—22no—
18#88862—22no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×#3752—21no—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C3×#3750—21no—
23C32×#216—22no—
24C2×C32×C3⋊C8C2211yescharacteristic
25C2×#3832—31no—
26C2×#3799—34no—
27C2×C32×Q16—31no—
28#64155D311yescharacteristic, complemented
29C2×#3755—31no—
30C2×#3755—34no—
31C2×#3755—64no—
32C3×#946—34no—
33C2×C3×#216D314yescomplemented
34#63662—31no—
35#63648—31no—
36#63647—31no—
37#63565—34no—
38#63564—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C33×Q8—21no—
43C33×Q8—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4⋊(C3⋊(C3⋊D3))D412yes—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4×C3⋊D3—21no—
51C3×C4×C3⋊D3—21no—
52C2×C32×Dic3—21no—
53C32×Dic6—21no—
54C32×Dic6D412yes—
55#6292—24no—
56#6289—24no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5187—94no—
60#5047—91no—
61C2×C32×Q8—24no—
62C2×C32×Q8—31no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D311yescharacteristic
65#3832—62no—
66#3832—62no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—34no—
69C2×C4⋊(C3⋊D3)—61no—
70C2×C4⋊(C3⋊D3)—64no—
71C2×C4×C3⋊D3—61no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×C3×C4×D3—34no—
74C2×C3×Dic6C2×D314yes—
75C2×#626—64no—
76C2×#624—64no—
77#3799—68no—
78#3799—68no—
79C32×Q16—62no—
80C2×C32×C8—31no—
81#3755—62no—
82#3755—68no—
83#3755—616no—
84#3752—21no—
85C4⋊(C3⋊Dic3)C2×D311yescharacteristic
86#3750—21no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—34no—
89C2×C3⋊(C3⋊C8)—64no—
90C3×#216—28no—
91C3×(C2×D3)⋊C4—64no—
92C3×C4⋊Dic3—34no—
93C3×Dic3⋊C4—64no—
94C2×C3×C3⋊C8C2×D314yes—
95C22×C33C2×D411yescharacteristic
96C2×C3⋊(C3⋊D3)—22no—
97C2×C3⋊(C3⋊D3)—41no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C33×C4—41no—
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3×C3⋊Dic3—21no—
106C3×C3⋊Dic3—41no—
107C32×Dic3—22no—
108C2×#239—94no—
109C2×#235—91no—
110C2×C3×Q16—34no—
111C3×C22⋊Q8—91no—
112#946—34no—
113C2×#216C3⋊D311yescharacteristic, complemented
114C2×#215—94no—
115C2×#215—91no—
116C2×#215—184no—
117#835—94no—
118#830—91no—
119#821—94no—
120(C2×D3)⋊C8—94no—
121#810—91no—
122#809—91no—
123C22×C3⋊D3—61no—
124C22×C3⋊D3—64no—
125C22×C3⋊D3—124no—
126C22×C3⋊D3#6711yescharacteristic
127C22×C3×D3—34no—
128C32×Q8—28no—
129C32×Q8—24no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—24no—
133C32×Q8—32no—
134C32×Q8—44no—
135C32×Q8—61no—
136C2×C32×C4—21no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—44no—
140C2×C32×C4—61no—
141C2×C32×C4C22×D314yes—
142C2×C32×C4C22×D311yescharacteristic
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3#6711yescharacteristic
145C4⋊(C3⋊D3)—32no—
146C4⋊(C3⋊D3)—38no—
147C4⋊(C3⋊D3)—68no—
148C4⋊(C3⋊D3)—64no—
149C4⋊(C3⋊D3)—61no—
150C4⋊(C3⋊D3)—64no—
151C4⋊(C3⋊D3)—124no—
152C4×C3⋊D3—21no—
153C4×C3⋊D3—21no—
154C4×C3⋊D3—64no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—64no—
158C3×C4×D3—64no—
159C3×C4×D3—64no—
160C3×Dic6—24no—
161C3×Dic6#6718yes—
162#626—616no—
163#624—616no—
164C32×C8—61no—
165C3⋊(C3⋊C8)—68no—
166C3⋊(C3⋊C8)—64no—
167C3⋊(C3⋊C8)—61no—
168C3×C3⋊C8—24no—
169#452—271noSylow
170C2×C33#12011yescharacteristic
171C2×C33#13511yescharacteristic
172C2×C33#13711yescharacteristic
173C3⋊(C3⋊D3)—42no—
174C3×C3⋊D3—41no—
175C2×C3×Q8—24no—
176C2×C3×Q8—34no—
177C2×C3×Q8D3214yes—
178C2×C3×Q8C2×C3⋊D311yescharacteristic
179C22×C3×C4—91no—
180C2×#67—181no—
181#239—188no—
182#239—188no—
183#235—182no—
184#235—182no—
185C2×D12—94no—
186C2×D12—91no—
187C2×D12—184no—
188C2×D12—184no—
189C2×C4×D3—34no—
190C2×C4×D3—181no—
191C2×C4×D3—184no—
192C2×Dic6C2×C3⋊D311yescharacteristic
193C3×Q16—68no—
194C2×C3×C8—34no—
195C3×C4⋊C4—91no—
196C3×C4⋊C4—181no—
197C3×C22⋊C4—181no—
198#216—22no—
199#215—188no—
200#215—182no—
201#215—1816no—
202(C2×D3)⋊C4—64no—
203C4⋊Dic3—34no—
204Dic3⋊C4—64no—
205C2×C3⋊C8—94no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—184no—
208C2×C3⋊C8C2×C3⋊D311yescharacteristic
209C22×C32—24no—
210C22×C32D3×D414yes—
211C22×C32C2×#6711yescharacteristic
212C2×C3⋊D3—21no—
213C2×C3⋊D3—21no—
214C2×C3⋊D3—21no—
215C2×C3⋊D3—62no—
216C2×C3⋊D3—68no—
217C2×C3⋊D3—128no—
218C2×C3⋊D3—121no—
219C2×C3⋊D3—121no—
220C2×C3⋊D3—124no—
221C2×C3⋊D3—124no—
222C2×C3⋊D3—244no—
223C2×C3⋊D3—244no—
224C2×C3×D3—64no—
225C2×C3×D3—64no—
226C2×C3×D3—64no—
227C32×C4—24no—
228C32×C4—24no—
229C32×C4—48no—
230C32×C4—44no—
231C32×C4—41no—
232C32×C4—62no—
233C32×C4D3×D414yes—
234C32×C4D3×D411yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4C2×#6714yes—
237C3⋊Dic3—21no—
238C3⋊Dic3—41no—
239C3×Dic3—28no—
240C3×Dic3—64no—
241C3×Dic3—124no—
242C2×#41—271no—
243C2×Q16—31no—
244C2×SD16—271no—
245C22⋊Q8—91no—
246#103—271no—
247#102—271no—
248C22⋊C8—271no—
249C33#45211yescharacteristic, complemented, Sylow
250C23×C3—91no—
251C22×D3—34no—
252C22×D3—184no—
253C22×D3—181no—
254C22×D3—364no—
255C3×Q8—28no—
256C3×Q8—24no—
257C3×Q8—24no—
258C3×Q8—21no—
259C3×Q8—21no—
260C3×Q8—38no—
261C3×Q8—44no—
262C3×Q8—64no—
263C2×C3×C4—24no—
264C2×C3×C4—21no—
265C2×C3×C4—24no—
266C2×C3×C4—44no—
267C2×C3×C4—64no—
268C2×C3×C4—91no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4C2×D3214yes—
273C2×C3×C4C22×C3⋊D311yescharacteristic
274#67—184no—
275C2×Dic3—21no—
276C2×Dic3—34no—
277C2×Dic3—64no—
278D12—98no—
279D12—92no—
280D12—188no—
281D12—1816no—
282D12—184no—
283D12—181no—
284D12—364no—
285C4×D3—64no—
286C4×D3—64no—
287C4×D3—184no—
288C4×D3—1816no—
289Dic6—21no—
290Dic6#63812yes—
291C3×C8—64no—
292C3⋊C8—21no—
293C3⋊C8—188no—
294C3⋊C8—184no—
295C3⋊C8—181no—
296C2×C32—24no—
297C2×C32—24no—
298C2×C32—24no—
299C2×C32#91814yes—
300C2×C32#91914yes—
301C2×C32#93714yes—
302C2×C32#93711yescharacteristic
303C2×C32#94211yescharacteristic
304C2×C32#95011yescharacteristic
305C3⋊D3—41no—
306C3⋊D3—122no—
307C3⋊D3—128no—
308C3⋊D3—248no—
309C3×D3—124no—
310#41—542no—
311#41—542no—
312C2×Q8—31no—
313C2×Q8D3×C3⋊D311yescharacteristic
314C2×D4—271no—
315C2×D4—541no—
316C22×C4—91no—
317C22×C4—541no—
318Q16—62no—
319SD16—542no—
320C2×C8—31no—
321C2×C8—271no—
322C4⋊C4—91no—
323C4⋊C4—181no—
324C22⋊C4—181no—
325C22×C3—24no—
326C22×C3—181no—
327C22×C3—181no—
328C22×C3—181no—
329C22×C3D3×#6714yes—
330C22×C3D4×C3⋊D311yescharacteristic
331C2×D3—64no—
332C2×D3—64no—
333C2×D3—64no—
334C2×D3—182no—
335C2×D3—188no—
336C2×D3—368no—
337C2×D3—361no—
338C2×D3—361no—
339C2×D3—364no—
340C2×D3—364no—
341C2×D3—724no—
342C2×D3—724no—
343C3×C4—24no—
344C3×C4—24no—
345C3×C4—48no—
346C3×C4—41no—
347C3×C4—44no—
348C3×C4—68no—
349C3×C4—181no—
350C3×C4—361no—
351C3×C4D3×#6714yes—
352C3×C4#381114yes—
353C3×C4D4×C3⋊D311yescharacteristic
354C3×C4C2×#63811yescharacteristic
355Dic3—22no—
356Dic3—64no—
357Dic3—124no—
358C32—24no—
359C32#504711yescharacteristic, complemented
360C32#518714yescomplemented
361C23—91no—
362C23—541no—
363Q8—21no—
364Q8—21no—
365Q8—32no—
366Q8—61no—
367D4—272no—
368D4—544no—
369D4—541no—
370C2×C4—21no—
371C2×C4—61no—
372C2×C4—91no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—181no—
376C2×C4—544no—
377C2×C4C2×D3×C3⋊D311yescharacteristic
378C8—61no—
379C8—541no—
380C2×C3—24no—
381C2×C3—24no—
382C2×C3—24no—
383C2×C3—361no—
384C2×C3#6394814yes—
385C2×C3#6394914yes—
386C2×C3#6398314yes—
387C2×C3#6412711yescharacteristic
388C2×C3#6412811yescharacteristic
389C2×C3#6414611yescharacteristic
390D3—124no—
391D3—362no—
392D3—368no—
393D3—728no—
394C22—181no—
395C22—181no—
396C22—181no—
397C22—542no—
398C22—1081no—
399C22—1081no—
400C22C3⋊D3×#6711yescharacteristic
401C4—41no—
402C4—62no—
403C4—181no—
404C4—361no—
405C4D3×#63811yescharacteristic
406C4#8903911yescharacteristic
407C3—24no—
408C3#1059165714yescomplemented
409C3#1059288611yescharacteristic, complemented
410C2—361no—
411C2—1082no—
412C2#1172358511yescharacteristic
413C2#1172358611yescharacteristic
414C2#1172364311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#89018C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#88862C2semidirect
3. C3×#64155C2semidirect
4. C2×C32×#216C2semidirect
5. #11722037C2semidirect
6. #11722036C2semidirect
7. #11722023C2semidirect
28. #64155D3semidirect
33. C2×C3×#216D3semidirect
113. C2×#216C3⋊D3semidirect
249. C33#452semidirect
359. C32#5047semidirect
360. C32#5187semidirect
408. C3#10591657semidirect
409. C3#10592886semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
C2×#888622
C3×#641552
C2×C32×#2162
#117220372
#117220362
#117220232
#105928863
#105916573