Finite Group Databasebeta

← #13161144

Subgroups — #13161144

Order 1728.

1524 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1524 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×Q8×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723413C211yesmaximal, characteristic
3C2×#88788C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722042C211yesmaximal, characteristic, complemented
7#11722041C211yesmaximal, characteristic, complemented
8#10592501—31nomaximal
9#10591562—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11Q8×C3⋊(C3⋊D3)—22no—
12Q8×C3⋊(C3⋊D3)C414yescomplemented
13C2×C4×C3⋊(C3⋊D3)—21no—
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×#6308—21no—
16C2×#6308C2211yescharacteristic
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#88788—22no—
24C2×Q8×C3⋊D3—31no—
25C2×Q8×C3⋊D3—34no—
26C2×Q8×C3⋊D3—64no—
27C2×#3743—31no—
28#64101—31no—
29#63849—34no—
30C2×#3712—34no—
31C32×#103—31no—
32#63662D311yescharacteristic, complemented
33#63643—31no—
34#63637—31no—
35C3×#835—34no—
36C3×#835D314yescomplemented
37#63567—34no—
38#63567—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C33×Q8—21no—
41C33×Q8C2×C412yes—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46C4×C3⋊(C3⋊D3)—24no—
47C4×C3⋊(C3⋊D3)C2×C412yes—
48C4×C3⋊(C3⋊D3)D412yes—
49#6308—21no—
50#6308—24no—
51#6308C2×C412yes—
52C2×C3×C3⋊Dic3—21no—
53C2×C32×Dic3—21no—
54C3×C3⋊(C3⋊C8)—21no—
55C32×C3⋊C8—21no—
56#4802—94no—
57#4802—91no—
58C2×C32×Q8—24no—
59C2×C32×Q8C2×D314yes—
60C2×C32×Q8C2×D311yescharacteristic
61Q8×C3⋊D3—34no—
62Q8×C3⋊D3—316no—
63Q8×C3⋊D3—62no—
64Q8×C3⋊D3—68no—
65Q8×C3⋊D3—616no—
66Q8×C3⋊D3—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×#634—31no—
74C2×#634—34no—
75C2×#634—61no—
76C2×#634—64no—
77C2×#634—64no—
78C2×#634—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
85#3743—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#3712—68no—
93C22×C33C2×D411yescharacteristic
94C2×C3⋊(C3⋊D3)—21no—
95C2×C3⋊(C3⋊D3)—21no—
96C2×C3⋊(C3⋊D3)C22⋊C412yes—
97C33×C4—22no—
98C33×C4C22×C411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C3⋊(C3⋊Dic3)—22no—
101C3⋊(C3⋊Dic3)C22⋊C412yes—
102C3×C3⋊Dic3—41no—
103C32×Dic3—41no—
104C2×D3×Q8—94no—
105C2×D3×Q8—91no—
106C2×D3×Q8—184no—
107C2×ΓL(1,𝔽25)—94no—
108C2×ΓL(1,𝔽25)—91no—
109#892—94no—
110#892—91no—
111C3×#103—34no—
112C3×#103—91no—
113#835—34no—
114#835C3⋊D311yescharacteristic, complemented
115#816—94no—
116#816—91no—
117#810—94no—
118#810—91no—
119C22×C3⋊D3—31no—
120C22×C3⋊D3—34no—
121C22×C3⋊D3—64no—
122C32×Q8—28no—
123C32×Q8—24no—
124C32×Q8—21no—
125C32×Q8—28no—
126C32×Q8C4×D318yes—
127C32×Q8C4×D312yes—
128C2×C32×C4—28no—
129C2×C32×C4—24no—
130C2×C32×C4—21no—
131C2×C32×C4—24no—
132C2×C32×C4—61no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—31no—
137C2×C3⋊Dic3—34no—
138C2×C3⋊Dic3—64no—
139C2×C3⋊Dic3—68no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—61no—
142C4×C3⋊D3—32no—
143C4×C3⋊D3—32no—
144C4×C3⋊D3—38no—
145C4×C3⋊D3—38no—
146C4×C3⋊D3—68no—
147C4×C3⋊D3—68no—
148C4×C3⋊D3—64no—
149C4×C3⋊D3—616no—
150C4×C3⋊D3—632no—
151#634—32no—
152#634—38no—
153#634—68no—
154#634—64no—
155#634—616no—
156#634—632no—
157#634—68no—
158#634—64no—
159#634—61no—
160C2×C3×Dic3—24no—
161C2×C3×Dic3—64no—
162C32×C8—61no—
163C3⋊(C3⋊C8)—21no—
164C3×C3⋊C8—24no—
165C3×C3⋊C8—64no—
166#419—271noSylow
167C2×C33C2×C22⋊C411yescharacteristic
168C2×C33#13711yescharacteristic
169C2×C33#13711yescharacteristic
170C3⋊(C3⋊D3)—22no—
171C2×C3×Q8—24no—
172C2×C3×Q8D3214yes—
173C2×C3×Q8C2×C3⋊D311yescharacteristic
174D3×Q8—916no—
175D3×Q8—94no—
176D3×Q8—1816no—
177D3×Q8—188no—
178D3×Q8—182no—
179D3×Q8—1816no—
180C2×C4×D3—94no—
181C2×C4×D3—91no—
182C2×C4×D3—184no—
183C2×C4×D3—184no—
184C2×C4×D3—181no—
185C2×C4×D3—188no—
186C2×Dic6—94no—
187C2×Dic6—91no—
188C2×Dic6—184no—
189C2×Dic6—184no—
190C2×Dic6—181no—
191C2×Dic6—188no—
192C2×C3×C8—34no—
193C2×C3×C8—91no—
194C3×C4⋊C4—34no—
195C3×C4⋊C4—91no—
196(C2×D3)⋊C4—181no—
197(C2×D3)⋊C4—184no—
198C4⋊Dic3—34no—
199C4⋊Dic3C2×C3⋊D311yescharacteristic
200C4×Dic3—181no—
201C4×Dic3—184no—
202C2×C3⋊C8—34no—
203C2×C3⋊C8C2×C3⋊D311yescharacteristic
204ΓL(1,𝔽25)—182no—
205ΓL(1,𝔽25)—188no—
206C22×C32—24no—
207C22×C32D3×D414yes—
208C22×C32D3×D411yescharacteristic
209C2×C3⋊D3—32no—
210C2×C3⋊D3—38no—
211C2×C3⋊D3—68no—
212C2×C3⋊D3—68no—
213C2×C3⋊D3—68no—
214C2×C3⋊D3—64no—
215C2×C3⋊D3—61no—
216C2×C3⋊D3—64no—
217C2×C3⋊D3—61no—
218C32×C4—216no—
219C32×C4—28no—
220C32×C4—22no—
221C32×C4—24no—
222C32×C4—24no—
223C32×C4—121no—
224C32×C4C2×C4×D314yes—
225C32×C4C2×C4×D311yescharacteristic
226C32×C4D3×D414yes—
227C32×C4D3×D411yescharacteristic
228C3⋊Dic3—32no—
229C3⋊Dic3—38no—
230C3⋊Dic3—41no—
231C3⋊Dic3—68no—
232C3⋊Dic3—616no—
233C3⋊Dic3—68no—
234C3⋊Dic3—62no—
235C3×Dic3—44no—
236C3×Dic3—124no—
237C22×Q8—271no—
238C2×M16—271no—
239#117—271no—
240#103—31no—
241#103—91no—
242#103—271no—
243#103—271no—
244C33#41911yescharacteristic, complemented, Sylow
245C22×D3—94no—
246C22×D3—91no—
247C22×D3—184no—
248C3×Q8—28no—
249C3×Q8—24no—
250C3×Q8—21no—
251C3×Q8—28no—
252C3×Q8#62418yes—
253C3×Q8C4×C3⋊D312yes—
254C2×C3×C4—28no—
255C2×C3×C4—24no—
256C2×C3×C4—21no—
257C2×C3×C4—24no—
258C2×C3×C4—64no—
259C2×C3×C4—181no—
260C2×C3×C4C2×D3214yes—
261C2×C3×C4C22×C3⋊D311yescharacteristic
262C2×Dic3—21no—
263C2×Dic3—64no—
264C2×Dic3—94no—
265C2×Dic3—91no—
266C2×Dic3—184no—
267C2×Dic3—188no—
268C2×Dic3—184no—
269C2×Dic3—181no—
270C4×D3—98no—
271C4×D3—98no—
272C4×D3—92no—
273C4×D3—92no—
274C4×D3—188no—
275C4×D3—188no—
276C4×D3—184no—
277C4×D3—1816no—
278C4×D3—1832no—
279Dic6—98no—
280Dic6—92no—
281Dic6—188no—
282Dic6—184no—
283Dic6—1816no—
284Dic6—1832no—
285Dic6—188no—
286Dic6—184no—
287Dic6—181no—
288C3×C8—64no—
289C3×C8—181no—
290C3⋊C8—21no—
291C3⋊C8—64no—
292C2×C32—24no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32D3×C22⋊C414yes—
296C2×C32D3×C22⋊C411yescharacteristic
297C2×C32#91514yes—
298C2×C32#91511yescharacteristic
299C2×C32#91814yes—
300C2×C32#91811yescharacteristic
301C3⋊D3—616no—
302C3⋊D3—68no—
303C3⋊D3—62no—
304C2×Q8—274no—
305C2×Q8—271no—
306C2×Q8—541no—
307C2×Q8—542no—
308C2×Q8D3×C3⋊D311yescharacteristic
309C22×C4—271no—
310C22×C4—541no—
311M16—542no—
312C2×C8—31no—
313C2×C8—91no—
314C4⋊C4—31no—
315C4⋊C4—91no—
316C22⋊C4—541no—
317C42—541no—
318C22×C3—24no—
319C22×C3#381114yes—
320C22×C3D4×C3⋊D311yescharacteristic
321C2×D3—92no—
322C2×D3—98no—
323C2×D3—188no—
324C2×D3—188no—
325C2×D3—188no—
326C2×D3—181no—
327C2×D3—181no—
328C2×D3—184no—
329C2×D3—184no—
330C3×C4—216no—
331C3×C4—22no—
332C3×C4—28no—
333C3×C4—24no—
334C3×C4—24no—
335C3×C4—124no—
336C3×C4—361no—
337C3×C4C2×#62414yes—
338C3×C4#381114yes—
339C3×C4C2×C4×C3⋊D311yescharacteristic
340C3×C4D4×C3⋊D311yescharacteristic
341Dic3—41no—
342Dic3—92no—
343Dic3—98no—
344Dic3—124no—
345Dic3—188no—
346Dic3—1816no—
347Dic3—182no—
348Dic3—188no—
349C32—24no—
350C32#480214yescomplemented
351C32#480211yescharacteristic, complemented
352C23—271no—
353Q8—21no—
354Q8—272no—
355Q8—544no—
356Q8—541no—
357Q8#628912yes—
358C2×C4—21no—
359C2×C4—61no—
360C2×C4—181no—
361C2×C4—271no—
362C2×C4—272no—
363C2×C4—272no—
364C2×C4—544no—
365C2×C4—541no—
366C2×C4C2×D3×C3⋊D311yescharacteristic
367C8—61no—
368C8—181no—
369C2×C3—24no—
370C2×C3—24no—
371C2×C3—24no—
372C2×C3#6395114yes—
373C2×C3#6395114yes—
374C2×C3#6397414yes—
375C2×C3C22⋊C4×C3⋊D311yescharacteristic
376C2×C3#6412411yescharacteristic
377C2×C3#6412711yescharacteristic
378D3—1816no—
379D3—188no—
380D3—182no—
381C22—272no—
382C22—541no—
383C22—541no—
384C22#8903911yescharacteristic
385C4—22no—
386C4—121no—
387C4—272no—
388C4—361no—
389C4—542no—
390C4C2×#628911yescharacteristic
391C4#8903911yescharacteristic
392C3—24no—
393C3#1059156214yescomplemented
394C3#1059250111yescharacteristic, complemented
395C2—542no—
396C2#1172358811yescharacteristic
397C2#1172359011yescharacteristic
398C2#1172362711yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#419
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
4. C3×#63662C2semidirect
5. C32×#835C2semidirect
6. #11722042C2semidirect
7. #11722041C2semidirect
12. Q8×C3⋊(C3⋊D3)C4semidirect
32. #63662D3semidirect
36. C3×#835D3semidirect
114. #835C3⋊D3semidirect
244. C33#419semidirect
350. C32#4802semidirect
351. C32#4802semidirect
393. C3#10591562semidirect
394. C3#10592501semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊(C3⋊D3)2
#117234132
C2×#887882
C3×#636622
C32×#8352
#117220422
#117220412
#105925013
#105915623