Finite Group Databasebeta

← #13161142

Subgroups — #13161142

Order 1728.

1604 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1604 conjugacy classes of subgroups — too many to draw. Only its 180 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×#89080C211yesmaximal, characteristic
2#11723413C211yesmaximal, characteristic, complemented
3C2×#88785C211yesmaximal, characteristic, complemented
4C3×#63662C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8#10592503—31nomaximal
9#10591560—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11#89080—22no—
12#89080C414yescomplemented
13C2×C4⋊(C3⋊(C3⋊D3))—21no—
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)—21no—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88804—21no—
22Dic3×C3⋊Dic3—21no—
23#88785—22no—
24C2×#3832—31no—
25C2×#3832—34no—
26C2×#3832—64no—
27C2×C8×C3⋊D3—31no—
28#64101—31no—
29#63849—34no—
30C2×#3710—34no—
31C32×#103—31no—
32#63662D311yescharacteristic, complemented
33#63648—31no—
34#63636—31no—
35C3×#835—34no—
36C3×#835D314yescomplemented
37#63564—34no—
38#63564—34no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C33×Q8—21no—
42C33×Q8C2×C412yes—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46C4⋊(C3⋊(C3⋊D3))—24no—
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))C2×C412yes—
49C4×C3⋊(C3⋊D3)—24no—
50C4×C3⋊(C3⋊D3)C2×C412yes—
51C4×C3⋊(C3⋊D3)D412yes—
52C2×C3×C3⋊Dic3—21no—
53C2×C32×Dic3—21no—
54C3×C3⋊(C3⋊C8)—21no—
55C32×C3⋊C8—21no—
56#4804—94no—
57#4804—91no—
58C2×C32×Q8—24no—
59C2×C32×Q8C2×D314yes—
60C2×C32×Q8C2×D311yescharacteristic
61#3832—34no—
62#3832—316no—
63#3832—62no—
64#3832—68no—
65#3832—616no—
66#3832—616no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—34no—
69C2×C4⋊(C3⋊D3)—61no—
70C2×C4⋊(C3⋊D3)—64no—
71C2×C4⋊(C3⋊D3)—64no—
72C2×C4⋊(C3⋊D3)—68no—
73C2×C4×C3⋊D3—31no—
74C2×C4×C3⋊D3—34no—
75C2×C4×C3⋊D3—61no—
76C2×C4×C3⋊D3—64no—
77C2×C4×C3⋊D3—64no—
78C2×C4×C3⋊D3—68no—
79C2×C32×C8—31no—
80C32×C4⋊C4—31no—
81#3752—61no—
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic
83C4×C3⋊Dic3—61no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C8×C3⋊D3—62no—
86C3×C4⋊Dic3—34no—
87C3×C4⋊Dic3C2×D314yes—
88C2×C3×C3⋊C8—34no—
89C2×C3×C3⋊C8C2×D314yes—
90#3722—64no—
91Dic32—64no—
92#3710—68no—
93C22×C33C2×D411yescharacteristic
94C2×C3⋊(C3⋊D3)—21no—
95C2×C3⋊(C3⋊D3)—21no—
96C2×C3⋊(C3⋊D3)—22no—
97C2×C3⋊(C3⋊D3)—41no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C3⋊(C3⋊D3)C22⋊C412yes—
100C33×C4—22no—
101C33×C4C22×C411yescharacteristic
102C33×C4C2×D411yescharacteristic
103C3⋊(C3⋊Dic3)C22⋊C412yes—
104C3×C3⋊Dic3—41no—
105C32×Dic3—41no—
106C2×#239—94no—
107C2×#239—91no—
108C2×#239—184no—
109C2×D3×C8—94no—
110C2×D3×C8—91no—
111#892—94no—
112#892—91no—
113C3×#103—34no—
114C3×#103—91no—
115#835—34no—
116#835C3⋊D311yescharacteristic, complemented
117#821—94no—
118#821—91no—
119#809—94no—
120#809—91no—
121C22×C3⋊D3—31no—
122C22×C3⋊D3—34no—
123C22×C3⋊D3—64no—
124C22×C3⋊D3—68no—
125C22×C3⋊D3—64no—
126C22×C3⋊D3—61no—
127C32×Q8—28no—
128C32×Q8—24no—
129C32×Q8—21no—
130C32×Q8—28no—
131C32×Q8C4×D318yes—
132C32×Q8C4×D312yes—
133C2×C32×C4—28no—
134C2×C32×C4—24no—
135C2×C32×C4—21no—
136C2×C32×C4—24no—
137C2×C32×C4—61no—
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D311yescharacteristic
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—31no—
142C2×C3⋊Dic3—34no—
143C2×C3⋊Dic3—64no—
144C4⋊(C3⋊D3)—32no—
145C4⋊(C3⋊D3)—38no—
146C4⋊(C3⋊D3)—68no—
147C4⋊(C3⋊D3)—64no—
148C4⋊(C3⋊D3)—616no—
149C4⋊(C3⋊D3)—632no—
150C4⋊(C3⋊D3)—68no—
151C4⋊(C3⋊D3)—64no—
152C4⋊(C3⋊D3)—61no—
153C4×C3⋊D3—32no—
154C4×C3⋊D3—32no—
155C4×C3⋊D3—38no—
156C4×C3⋊D3—38no—
157C4×C3⋊D3—68no—
158C4×C3⋊D3—68no—
159C4×C3⋊D3—64no—
160C4×C3⋊D3—616no—
161C4×C3⋊D3—632no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—64no—
164C32×C8—61no—
165C3⋊(C3⋊C8)—21no—
166C3×C3⋊C8—24no—
167C3×C3⋊C8—64no—
168#416—271noSylow
169C2×C33C2×C22⋊C411yescharacteristic
170C2×C33#13511yescharacteristic
171C2×C33#13511yescharacteristic
172C3⋊(C3⋊D3)—22no—
173C3⋊(C3⋊D3)—42no—
174C2×C3×Q8—24no—
175C2×C3×Q8D3214yes—
176C2×C3×Q8C2×C3⋊D311yescharacteristic
177#239—916no—
178#239—94no—
179#239—1816no—
180#239—188no—
181#239—182no—
182#239—1816no—
183C2×D12—94no—
184C2×D12—91no—
185C2×D12—184no—
186C2×D12—184no—
187C2×D12—181no—
188C2×D12—188no—
189C2×C4×D3—94no—
190C2×C4×D3—91no—
191C2×C4×D3—184no—
192C2×C4×D3—184no—
193C2×C4×D3—181no—
194C2×C4×D3—188no—
195C2×C3×C8—34no—
196C2×C3×C8—91no—
197C3×C4⋊C4—34no—
198C3×C4⋊C4—91no—
199(C2×D3)⋊C4—181no—
200(C2×D3)⋊C4—184no—
201C4⋊Dic3—34no—
202C4⋊Dic3C2×C3⋊D311yescharacteristic
203C4×Dic3—181no—
204C4×Dic3—184no—
205C2×C3⋊C8—34no—
206C2×C3⋊C8C2×C3⋊D311yescharacteristic
207D3×C8—182no—
208D3×C8—188no—
209C22×C32—24no—
210C22×C32D3×D414yes—
211C22×C32D3×D411yescharacteristic
212C2×C3⋊D3—32no—
213C2×C3⋊D3—38no—
214C2×C3⋊D3—68no—
215C2×C3⋊D3—61no—
216C2×C3⋊D3—61no—
217C2×C3⋊D3—616no—
218C2×C3⋊D3—68no—
219C2×C3⋊D3—62no—
220C2×C3⋊D3—64no—
221C2×C3⋊D3—64no—
222C2×C3⋊D3—124no—
223C2×C3⋊D3—124no—
224C2×C3⋊D3—128no—
225C2×C3⋊D3—128no—
226C2×C3⋊D3—124no—
227C2×C3⋊D3—121no—
228C2×C3⋊D3—124no—
229C2×C3⋊D3—121no—
230C32×C4—216no—
231C32×C4—28no—
232C32×C4—22no—
233C32×C4—24no—
234C32×C4—24no—
235C32×C4—121no—
236C32×C4C2×C4×D314yes—
237C32×C4C2×C4×D311yescharacteristic
238C32×C4D3×D414yes—
239C32×C4D3×D411yescharacteristic
240C3⋊Dic3—32no—
241C3⋊Dic3—38no—
242C3⋊Dic3—41no—
243C3⋊Dic3—68no—
244C3×Dic3—44no—
245C3×Dic3—124no—
246C2×#41—271no—
247C22×C8—271no—
248#117—271no—
249#103—31no—
250#103—91no—
251#102—271no—
252#102—271no—
253C33#41611yescharacteristic, complemented, Sylow
254C22×D3—94no—
255C22×D3—91no—
256C22×D3—184no—
257C22×D3—188no—
258C22×D3—184no—
259C22×D3—181no—
260C3×Q8—28no—
261C3×Q8—24no—
262C3×Q8—21no—
263C3×Q8—28no—
264C3×Q8#62418yes—
265C3×Q8C4×C3⋊D312yes—
266C2×C3×C4—28no—
267C2×C3×C4—24no—
268C2×C3×C4—21no—
269C2×C3×C4—24no—
270C2×C3×C4—64no—
271C2×C3×C4—181no—
272C2×C3×C4C2×D3214yes—
273C2×C3×C4C22×C3⋊D311yescharacteristic
274C2×Dic3—21no—
275C2×Dic3—64no—
276C2×Dic3—94no—
277C2×Dic3—91no—
278C2×Dic3—184no—
279D12—98no—
280D12—92no—
281D12—188no—
282D12—184no—
283D12—1816no—
284D12—1832no—
285D12—188no—
286D12—184no—
287D12—181no—
288C4×D3—98no—
289C4×D3—98no—
290C4×D3—92no—
291C4×D3—92no—
292C4×D3—188no—
293C4×D3—188no—
294C4×D3—184no—
295C4×D3—1816no—
296C4×D3—1832no—
297C3×C8—64no—
298C3×C8—181no—
299C3⋊C8—21no—
300C3⋊C8—64no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32D3×C22⋊C414yes—
305C2×C32D3×C22⋊C411yescharacteristic
306C2×C32#91614yes—
307C2×C32#91611yescharacteristic
308C2×C32#91914yes—
309C2×C32#91911yescharacteristic
310C3⋊D3—62no—
311C3⋊D3—68no—
312C3⋊D3—128no—
313C3⋊D3—1216no—
314C3⋊D3—128no—
315C3⋊D3—122no—
316#41—274no—
317#41—542no—
318C2×Q8D3×C3⋊D311yescharacteristic
319C2×D4—271no—
320C2×D4—541no—
321C22×C4—271no—
322C22×C4—541no—
323C2×C8—31no—
324C2×C8—91no—
325C2×C8—542no—
326C4⋊C4—31no—
327C4⋊C4—91no—
328C22⋊C4—541no—
329C42—541no—
330C22×C3—24no—
331C22×C3#381114yes—
332C22×C3D4×C3⋊D311yescharacteristic
333C2×D3—92no—
334C2×D3—98no—
335C2×D3—188no—
336C2×D3—181no—
337C2×D3—181no—
338C2×D3—184no—
339C2×D3—184no—
340C2×D3—1816no—
341C2×D3—182no—
342C2×D3—188no—
343C2×D3—364no—
344C2×D3—364no—
345C2×D3—368no—
346C2×D3—368no—
347C2×D3—361no—
348C2×D3—361no—
349C2×D3—364no—
350C2×D3—364no—
351C3×C4—216no—
352C3×C4—22no—
353C3×C4—28no—
354C3×C4—24no—
355C3×C4—24no—
356C3×C4—124no—
357C3×C4—361no—
358C3×C4C2×#62414yes—
359C3×C4#381114yes—
360C3×C4C2×C4×C3⋊D311yescharacteristic
361C3×C4D4×C3⋊D311yescharacteristic
362Dic3—41no—
363Dic3—92no—
364Dic3—98no—
365Dic3—124no—
366Dic3—188no—
367C32—24no—
368C32#480414yescomplemented
369C32#480411yescharacteristic, complemented
370C23—271no—
371C23—541no—
372Q8—21no—
373Q8#628912yes—
374D4—272no—
375D4—544no—
376D4—541no—
377C2×C4—21no—
378C2×C4—61no—
379C2×C4—181no—
380C2×C4—271no—
381C2×C4—272no—
382C2×C4—272no—
383C2×C4—544no—
384C2×C4C2×D3×C3⋊D311yescharacteristic
385C8—61no—
386C8—181no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—24no—
390C2×C3#6395014yes—
391C2×C3#6395014yes—
392C2×C3#6397414yes—
393C2×C3C22⋊C4×C3⋊D311yescharacteristic
394C2×C3#6412511yescharacteristic
395C2×C3#6412811yescharacteristic
396D3—188no—
397D3—182no—
398D3—368no—
399D3—3616no—
400D3—368no—
401D3—362no—
402C22—272no—
403C22—541no—
404C22—541no—
405C22—542no—
406C22—1081no—
407C22—1081no—
408C22#8903911yescharacteristic
409C4—22no—
410C4—121no—
411C4—272no—
412C4—361no—
413C4C2×#628911yescharacteristic
414C4#8903911yescharacteristic
415C3—24no—
416C3#1059156014yescomplemented
417C3#1059250311yescharacteristic, complemented
418C2—542no—
419C2—1082no—
420C2#1172358711yescharacteristic
421C2#1172358911yescharacteristic
422C2#1172362711yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#89080C2
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#416
LCSG ▷ 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. #11723413C2semidirect
3. C2×#88785C2semidirect
4. C3×#63662C2semidirect
5. C32×#835C2semidirect
6. #11722036C2semidirect
7. #11722035C2semidirect
12. #89080C4semidirect
32. #63662D3semidirect
36. C3×#835D3semidirect
116. #835C3⋊D3semidirect
253. C33#416semidirect
368. C32#4804semidirect
369. C32#4804semidirect
416. C3#10591560semidirect
417. C3#10592503semidirect

Maximal subgroups

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

subgroupindex
C2×#890802
#117234132
C2×#887852
C3×#636622
C32×#8352
#117220362
#117220352
#105925033
#105915603