Finite Group Databasebeta

← #13160068

Subgroups — #13160068

Order 1728.

1148 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 178 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×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
3#11723404C211yesmaximal, characteristic, complemented
4#11722191C211yesmaximal, characteristic, complemented
5C32×#816C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8#10592503—31nomaximal
9#10590690—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6292—21no—
14C2×#6289—21no—
15#89018—22no—
16#89018C414yescomplemented
17C2×C33×C8C2211yescharacteristic
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C3×C8×C3⋊D3—22no—
21C32×C4⋊Dic3C2211yescharacteristic
22#88803—21no—
23Dic3×C3⋊Dic3—21no—
24C2×#3832—31no—
25C2×#3799—34no—
26C2×C8×C3⋊D3D311yescharacteristic, complemented
27#64101—31no—
28C2×C3×D3×C8—34no—
29#63841—34no—
30C32×#103—31no—
31#63662—31no—
32#63648—31no—
33#63648—34no—
34#63648—64no—
35#63636—31no—
36C3×#816D314yescomplemented
37#63566—34no—
38#63564—34no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C3×C3⋊D3D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))C2×C412yes—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C3×C4×C3⋊D3C2×C412yes—
47C3×C4×C3⋊D3D412yes—
48C2×C32×Dic3—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6C2×C412yes—
52#6292—24no—
53#6289—24no—
54C33×C8—21no—
55C3×C3⋊(C3⋊C8)—21no—
56#5113—91no—
57#4804—94no—
58C2×C32×Q8—31no—
59#3832—34no—
60#3832—62no—
61C2×C4⋊(C3⋊D3)—31no—
62C2×C4⋊(C3⋊D3)—34no—
63C2×C4⋊(C3⋊D3)—61no—
64C2×C4⋊(C3⋊D3)—64no—
65C2×C4×C3⋊D3—61no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×C3×C4×D3—34no—
68C2×C3×Dic6C2×D314yes—
69C2×#626—64no—
70C2×#624—64no—
71#3799—316no—
72#3799—68no—
73C2×C32×C8—24no—
74C2×C32×C8C2×D314yes—
75C2×C32×C8C2×D311yescharacteristic
76C32×C4⋊C4—31no—
77#3752—61no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—34no—
80C4⋊(C3⋊Dic3)—64no—
81C4×C3⋊Dic3—61no—
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
83C8×C3⋊D3—22no—
84C3×C4⋊Dic3C2×D314yes—
85C2×C3×C3⋊C8—34no—
86C3×D3×C8—68no—
87(C2×D3)⋊Dic3—64no—
88Dic32—64no—
89C22×C33C2×D411yescharacteristic
90C2×C3⋊(C3⋊D3)—22no—
91C2×C3⋊(C3⋊D3)—41no—
92C2×C3⋊(C3⋊D3)—41no—
93C2×C3×C3⋊D3—21no—
94C2×C3×C3⋊D3—21no—
95C2×C3×C3⋊D3C22⋊C412yes—
96C33×C4C22×C411yescharacteristic
97C33×C4C2×D411yescharacteristic
98C3⋊(C3⋊Dic3)—41no—
99C3×C3⋊Dic3C22⋊C412yes—
100C32×Dic3—22no—
101C32×Dic3—41no—
102C2×#239—94no—
103C2×#235—91no—
104C22×C3×C8—91no—
105#926—91no—
106C2×D3×C8—34no—
107#892—94no—
108C3×#103—34no—
109#835—94no—
110#821—94no—
111#821—91no—
112#821—91no—
113#821—184no—
114#816—91no—
115#816C3⋊D311yescharacteristic, complemented
116#809—94no—
117C22×C3⋊D3—61no—
118C22×C3⋊D3—64no—
119C22×C3⋊D3—124no—
120C22×C3⋊D3D1211yescharacteristic
121C22×C3×D3—34no—
122C32×Q8—32no—
123C32×Q8—61no—
124C2×C32×C4—24no—
125C2×C32×C4—61no—
126C2×C32×C4—61no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—61no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—124no—
132C2×C3⋊Dic3D1211yescharacteristic
133C4⋊(C3⋊D3)—32no—
134C4⋊(C3⋊D3)—38no—
135C4⋊(C3⋊D3)—68no—
136C4⋊(C3⋊D3)—64no—
137C4⋊(C3⋊D3)—61no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—124no—
140C4×C3⋊D3—64no—
141C4×C3⋊D3C4×D312yes—
142C4×C3⋊D3#6712yes—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—34no—
146C3×C4×D3—38no—
147C3×C4×D3—38no—
148C3×Dic6—24no—
149C3×Dic6C4×D318yes—
150#626—616no—
151#624—616no—
152C32×C8—21no—
153C32×C8—24no—
154C32×C8—44no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—64no—
157#416—271noSylow
158C2×C33C2×C22⋊C411yescharacteristic
159C2×C33#13511yescharacteristic
160C2×C33#13511yescharacteristic
161C3⋊(C3⋊D3)—42no—
162C3×C3⋊D3—22no—
163C2×C3×Q8—34no—
164C22×C3×C4—91no—
165C2×#67—181no—
166#239—916no—
167#239—188no—
168#235—94no—
169#235—182no—
170C2×D12—94no—
171C2×D12—91no—
172C2×D12—184no—
173C2×D12—184no—
174C2×C4×D3—34no—
175C2×C4×D3—181no—
176C2×C4×D3—184no—
177C2×Dic6C2×C3⋊D311yescharacteristic
178C2×C3×C8—24no—
179C2×C3×C8—91no—
180C2×C3×C8—182no—
181C2×C3×C8D3214yes—
182C2×C3×C8C2×C3⋊D311yescharacteristic
183C3×C4⋊C4—34no—
184#217—181no—
185(C2×D3)⋊C4—184no—
186C4⋊Dic3—94no—
187C4⋊Dic3—91no—
188C4⋊Dic3—184no—
189C4⋊Dic3C2×C3⋊D311yescharacteristic
190C4×Dic3—181no—
191C4×Dic3—184no—
192C2×C3⋊C8—34no—
193D3×C8—68no—
194C22×C32—24no—
195C22×C32C2×D1211yescharacteristic
196C22×C32D3×D414yes—
197C2×C3⋊D3—21no—
198C2×C3⋊D3—21no—
199C2×C3⋊D3—62no—
200C2×C3⋊D3—68no—
201C2×C3⋊D3—128no—
202C2×C3⋊D3—121no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—124no—
205C2×C3⋊D3—124no—
206C2×C3⋊D3—244no—
207C2×C3⋊D3—244no—
208C2×C3⋊D3(C2×D3)⋊C412yes—
209C2×C3×D3—38no—
210C2×C3×D3—64no—
211C2×C3×D3—64no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—62no—
215C32×C4—121no—
216C32×C4C2×C4×D314yes—
217C32×C4C2×C4×D311yescharacteristic
218C32×C4D3×D414yes—
219C32×C4C2×#6711yescharacteristic
220C3⋊Dic3—121no—
221C3⋊Dic3—124no—
222C3⋊Dic3—244no—
223C3⋊Dic3(C2×D3)⋊C412yes—
224C3×Dic3—28no—
225C3×Dic3—38no—
226C3×Dic3—44no—
227C2×#41—271no—
228C22×C8—91no—
229#117—271no—
230#103—31no—
231#103—271no—
232#102—271no—
233#102—271no—
234C33#41611yescharacteristic, complemented, Sylow
235C23×C3—91no—
236C22×D3—34no—
237C22×D3—184no—
238C22×D3—181no—
239C22×D3—364no—
240C3×Q8—38no—
241C3×Q8—64no—
242C2×C3×C4—24no—
243C2×C3×C4—64no—
244C2×C3×C4—64no—
245C2×C3×C4—91no—
246C2×C3×C4—92no—
247C2×C3×C4—92no—
248C2×C3×C4C2×D3214yes—
249C2×C3×C4C22×C3⋊D311yescharacteristic
250#67—184no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—34no—
254C2×Dic3—184no—
255C2×Dic3—181no—
256C2×Dic3—364no—
257D12—98no—
258D12—92no—
259D12—188no—
260D12—1816no—
261D12—184no—
262D12—181no—
263D12—364no—
264C4×D3—38no—
265C4×D3—38no—
266C4×D3—184no—
267C4×D3—1816no—
268Dic6—21no—
269Dic6C4×C3⋊D312yes—
270C3×C8—24no—
271C3×C8—21no—
272C3×C8—44no—
273C3×C8—181no—
274C3⋊C8—64no—
275C2×C32—24no—
276C2×C32—24no—
277C2×C32—24no—
278C2×C32D3×C22⋊C414yes—
279C2×C32#90411yescharacteristic
280C2×C32#90411yescharacteristic
281C2×C32#91614yes—
282C2×C32#91914yes—
283C2×C32C2×(C2×D3)⋊C411yescharacteristic
284C3⋊D3—22no—
285C3⋊D3—122no—
286C3⋊D3—128no—
287C3⋊D3—248no—
288C3×D3—68no—
289#41—274no—
290#41—542no—
291C2×Q8—31no—
292C2×D4—271no—
293C2×D4—541no—
294C22×C4—91no—
295C22×C4—541no—
296C2×C8—91no—
297C2×C8—182no—
298C2×C8D3×C3⋊D311yescharacteristic
299C4⋊C4—31no—
300C4⋊C4—271no—
301C22⋊C4—541no—
302C42—541no—
303C22×C3—24no—
304C22×C3—92no—
305C22×C3—181no—
306C22×C3—181no—
307C22×C3D3×D1214yes—
308C22×C3D4×C3⋊D311yescharacteristic
309C2×D3—38no—
310C2×D3—64no—
311C2×D3—64no—
312C2×D3—182no—
313C2×D3—188no—
314C2×D3—368no—
315C2×D3—364no—
316C2×D3—364no—
317C2×D3—361no—
318C2×D3—361no—
319C2×D3—724no—
320C2×D3—724no—
321C3×C4—24no—
322C3×C4—24no—
323C3×C4—68no—
324C3×C4—92no—
325C3×C4—124no—
326C3×C4C4×D3214yes—
327C3×C4D3×#6714yes—
328C3×C4C2×C4×C3⋊D311yescharacteristic
329C3×C4D4×C3⋊D311yescharacteristic
330Dic3—22no—
331Dic3—38no—
332Dic3—41no—
333Dic3—364no—
334Dic3—361no—
335Dic3—724no—
336C32—24no—
337C32#480414yescomplemented
338C32#511311yescharacteristic, complemented
339C23—91no—
340C23—541no—
341Q8—32no—
342Q8—61no—
343D4—272no—
344D4—544no—
345D4—541no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—91no—
349C2×C4—92no—
350C2×C4—92no—
351C2×C4—544no—
352C2×C4—541no—
353C2×C4C2×D3×C3⋊D311yescharacteristic
354C8—21no—
355C8—181no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—182no—
360C2×C3#6380614yes—
361C2×C3#6380814yes—
362C2×C3D3×(C2×D3)⋊C414yes—
363C2×C3C22⋊C4×C3⋊D311yescharacteristic
364C2×C3#6412511yescharacteristic
365C2×C3#6412811yescharacteristic
366D3—68no—
367D3—368no—
368D3—362no—
369D3—728no—
370C22—92no—
371C22—181no—
372C22—181no—
373C22—542no—
374C22—1081no—
375C22—1081no—
376C22C3⋊D3×D1211yescharacteristic
377C4—62no—
378C4—92no—
379C4—121no—
380C4—1081no—
381C4C4×D3×C3⋊D311yescharacteristic
382C4C3⋊D3×#6711yescharacteristic
383C3—24no—
384C3#1059069014yescomplemented
385C3#1059250311yescharacteristic, complemented
386C2—182no—
387C2—1082no—
388C2#1172333811yescharacteristic
389C2#1172334111yescharacteristic
390C2C3⋊D3×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89018C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#416
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C8×C3⋊D3C2semidirect
3. #11723404C2semidirect
4. #11722191C2semidirect
5. C32×#816C2semidirect
6. #11722038C2semidirect
7. #11722035C2semidirect
16. #89018C4semidirect
26. C2×C8×C3⋊D3D3semidirect
36. C3×#816D3semidirect
115. #816C3⋊D3semidirect
234. C33#416semidirect
337. C32#4804semidirect
338. C32#5113semidirect
384. C3#10590690semidirect
385. C3#10592503semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
C2×C3×C8×C3⋊D32
#117234042
#117221912
C32×#8162
#117220382
#117220352
#105925033
#105906903