Finite Group Databasebeta

← #13161339

Subgroups — #13161339

Order 1728.

1184 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 160 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×#89019C211yesmaximal, characteristic
2#11723987C211yesmaximal, characteristic
3C2×C3×#3756C211yesmaximal, characteristic, complemented
4C2×C32×#216C211yesmaximal, characteristic, complemented
5#11722040C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic
8#10592885—31nomaximal
9#10591654—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6293—21no—
15C2×#6289—21no—
16#89019—22no—
17#89019—22no—
18#88859—21no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20#88857—21no—
21C3×#3756—22no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×#216—22no—
24C2×C32×C3⋊C8C2211yescharacteristic
25C2×Q8×C3⋊D3—31no—
26C2×#3797—34no—
27C2×C32×Q16—31no—
28#64155—31no—
29#64155—34no—
30#64155—64no—
31C2×#3756D311yescharacteristic, complemented
32C2×C3×#216—34no—
33C2×C3×#216D314yescomplemented
34#63662—31no—
35#63647—31no—
36#63643—31no—
37#63566—34no—
38#63566—34no—
39#63557—34no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C33×Q8—21no—
42C33×Q8—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46C2×C3⋊(C3⋊Dic3)D411yescharacteristic
47C4×C3⋊(C3⋊D3)—21no—
48C4×C3⋊(C3⋊D3)—21no—
49C2×C3×C3⋊Dic3—21no—
50C3×#634—21no—
51C3×#634D412yes—
52C2×C32×Dic3—21no—
53C32×Dic6—21no—
54C32×Dic6D412yes—
55#6293—24no—
56#6289—24no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#5186—94no—
60#5186—91no—
61C2×C32×Q8—24no—
62C2×C32×Q8—31no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D311yescharacteristic
65Q8×C3⋊D3—62no—
66Q8×C3⋊D3—62no—
67C2×C4×C3⋊D3—31no—
68C2×C4×C3⋊D3—34no—
69C2×C4×C3⋊D3—61no—
70C2×C4×C3⋊D3—64no—
71C2×#634—61no—
72C2×#634C2×D311yescharacteristic
73C2×C3×Dic6—34no—
74C2×C3×Dic6C2×D314yes—
75C2×#627—64no—
76C2×#624—64no—
77#3797—68no—
78#3797—68no—
79C32×Q16—62no—
80C2×C32×C8—31no—
81#3756—22no—
82#3752—61no—
83#3752—64no—
84#3752—68no—
85C4⋊(C3⋊Dic3)—31no—
86C4⋊(C3⋊Dic3)—34no—
87C4⋊(C3⋊Dic3)—64no—
88#3750—61no—
89#3750—64no—
90#3750—68no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#216—28no—
93C3×#216—68no—
94C2×C3×C3⋊C8—34no—
95C2×C3×C3⋊C8C2×D314yes—
96C22×C33C2×D411yescharacteristic
97C2×C3⋊(C3⋊D3)—21no—
98C2×C3⋊(C3⋊D3)—21no—
99C2×C3⋊(C3⋊D3)Q1612yescomplemented
100C33×C4—41no—
101C33×C4C2×D411yescharacteristic
102C33×C4C2×D411yescharacteristic
103C3⋊(C3⋊Dic3)—21no—
104C3⋊(C3⋊Dic3)—41no—
105C3×C3⋊Dic3—22no—
106C32×Dic3—22no—
107C2×D3×Q8—94no—
108C2×D3×Q8—91no—
109C2×C3×Q16—34no—
110C2×C3×Q16—91no—
111#946—94no—
112#946—91no—
113#946—184no—
114C2×#216—34no—
115C2×#216C3⋊D311yescharacteristic, complemented
116#835—94no—
117#835—91no—
118(C2×D3)⋊C8—94no—
119(C2×D3)⋊C8—91no—
120#816—94no—
121#816—91no—
122C22×C3⋊D3—31no—
123C22×C3⋊D3—34no—
124C22×C3⋊D3—64no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—24no—
128C32×Q8—24no—
129C32×Q8—32no—
130C32×Q8—44no—
131C32×Q8—44no—
132C32×Q8—61no—
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—
144C2×C3⋊Dic3—68no—
145C2×C3⋊Dic3—64no—
146C2×C3⋊Dic3—61no—
147C4×C3⋊D3—64no—
148C4×C3⋊D3—61no—
149C4×C3⋊D3—61no—
150C4×C3⋊D3—64no—
151C4×C3⋊D3—64no—
152C4×C3⋊D3—124no—
153C4×C3⋊D3—124no—
154#634—21no—
155#634—64no—
156#634#6712yes—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—64no—
159C3×Dic6—24no—
160C3×Dic6—38no—
161C3×Dic6—64no—
162C3×Dic6#6718yes—
163#627—616no—
164#624—616no—
165C32×C8—61no—
166C3⋊(C3⋊C8)—21no—
167C3×C3⋊C8—24no—
168C3×C3⋊C8—64no—
169#451—271noSylow
170C2×C33#12011yescharacteristic
171C2×C33C2×Q1611yescharacteristic
172C2×C33#13711yescharacteristic
173C3⋊(C3⋊D3)—22no—
174C2×C3×Q8—24no—
175C2×C3×Q8—34no—
176C2×C3×Q8—91no—
177C2×C3×Q8D3214yes—
178C2×C3×Q8C2×C3⋊D311yescharacteristic
179D3×Q8—182no—
180D3×Q8—188no—
181D3×Q8—188no—
182D3×Q8—182no—
183C2×C4×D3—94no—
184C2×C4×D3—91no—
185C2×C4×D3—184no—
186C2×C4×D3—181no—
187C2×C4×D3—184no—
188C2×Dic6—34no—
189C2×Dic6—181no—
190C2×Dic6—184no—
191C2×Dic6C2×C3⋊D311yescharacteristic
192C3×Q16—68no—
193C3×Q16—182no—
194C2×C3×C8—34no—
195C2×C3×C8—91no—
196#216—22no—
197#216—68no—
198(C2×D3)⋊C4—184no—
199(C2×D3)⋊C4—181no—
200(C2×D3)⋊C4—188no—
201C4⋊Dic3—94no—
202C4⋊Dic3—91no—
203C4⋊Dic3—184no—
204Dic3⋊C4—184no—
205Dic3⋊C4—181no—
206Dic3⋊C4—188no—
207C2×C3⋊C8—34no—
208C2×C3⋊C8C2×C3⋊D311yescharacteristic
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—64no—
218C2×C3⋊D3—64no—
219C2×C3⋊D3—124no—
220C2×C3⋊D3—124no—
221C32×C4—24no—
222C32×C4—24no—
223C32×C4—48no—
224C32×C4—44no—
225C32×C4—41no—
226C32×C4—62no—
227C32×C4D3×D414yes—
228C32×C4D3×D411yescharacteristic
229C32×C4C2×#6714yes—
230C32×C4C2×#6711yescharacteristic
231C3⋊Dic3—22no—
232C3⋊Dic3—61no—
233C3⋊Dic3—64no—
234C3⋊Dic3—124no—
235C3⋊Dic3—128no—
236C3⋊Dic3—124no—
237C3⋊Dic3—121no—
238C3×Dic3—28no—
239C3×Dic3—68no—
240C22×Q8—271no—
241C2×Q16—31no—
242C2×Q16—91no—
243C22⋊Q8—271no—
244#103—271no—
245#103—271no—
246C22⋊C8—271no—
247C33#45111yescharacteristic, complemented, Sylow
248C22×D3—94no—
249C22×D3—91no—
250C22×D3—184no—
251C3×Q8—24no—
252C3×Q8—24no—
253C3×Q8—21no—
254C3×Q8—21no—
255C3×Q8—38no—
256C3×Q8—44no—
257C3×Q8—44no—
258C3×Q8—64no—
259C3×Q8—92no—
260C3×Q8—181no—
261C2×C3×C4—28no—
262C2×C3×C4—24no—
263C2×C3×C4—21no—
264C2×C3×C4—24no—
265C2×C3×C4—64no—
266C2×C3×C4—181no—
267C2×C3×C4C2×D3214yes—
268C2×C3×C4C22×C3⋊D311yescharacteristic
269C2×Dic3—21no—
270C2×Dic3—64no—
271C2×Dic3—94no—
272C2×Dic3—91no—
273C2×Dic3—184no—
274C2×Dic3—188no—
275C2×Dic3—184no—
276C2×Dic3—181no—
277C4×D3—184no—
278C4×D3—1816no—
279C4×D3—184no—
280C4×D3—184no—
281C4×D3—181no—
282C4×D3—181no—
283C4×D3—364no—
284C4×D3—364no—
285Dic6—21no—
286Dic6—38no—
287Dic6—64no—
288Dic6—184no—
289Dic6—1816no—
290Dic6#63812yes—
291C3×C8—64no—
292C3×C8—181no—
293C3⋊C8—21no—
294C3⋊C8—64no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—24no—
298C2×C32D3×Q1614yes—
299C2×C32D3×Q1611yescharacteristic
300C2×C32#91814yes—
301C2×C32#91811yescharacteristic
302C2×C32#93714yes—
303C2×C32#93711yescharacteristic
304C3⋊D3—62no—
305C3⋊D3—68no—
306C3⋊D3—128no—
307C2×Q8—31no—
308C2×Q8—91no—
309C2×Q8—541no—
310C2×Q8—542no—
311C2×Q8—542no—
312C2×Q8D3×C3⋊D311yescharacteristic
313C22×C4—271no—
314C22×C4—541no—
315Q16—62no—
316Q16—182no—
317C2×C8—31no—
318C2×C8—91no—
319C4⋊C4—271no—
320C4⋊C4—541no—
321C22⋊C4—541no—
322C22×C3—24no—
323C22×C3#381114yes—
324C22×C3D4×C3⋊D311yescharacteristic
325C2×D3—92no—
326C2×D3—98no—
327C2×D3—188no—
328C2×D3—181no—
329C2×D3—181no—
330C2×D3—184no—
331C2×D3—184no—
332C2×D3—364no—
333C2×D3—364no—
334C3×C4—24no—
335C3×C4—24no—
336C3×C4—48no—
337C3×C4—41no—
338C3×C4—44no—
339C3×C4—68no—
340C3×C4—182no—
341C3×C4D3×#6714yes—
342C3×C4D3×#6714yes—
343C3×C4D4×C3⋊D311yescharacteristic
344C3×C4C2×#63811yescharacteristic
345Dic3—22no—
346Dic3—68no—
347Dic3—181no—
348Dic3—184no—
349Dic3—364no—
350Dic3—368no—
351Dic3—361no—
352Dic3—364no—
353C32—24no—
354C32#518614yescomplemented
355C32#518611yescharacteristic, complemented
356C23—271no—
357Q8—21no—
358Q8—21no—
359Q8—32no—
360Q8—61no—
361Q8—92no—
362Q8—181no—
363Q8—544no—
364C2×C4—21no—
365C2×C4—61no—
366C2×C4—181no—
367C2×C4—271no—
368C2×C4—544no—
369C2×C4—541no—
370C2×C4—541no—
371C2×C4—541no—
372C2×C4C2×D3×C3⋊D311yescharacteristic
373C8—61no—
374C8—181no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3#6394414yes—
379C2×C3#6394514yes—
380C2×C3#6398314yes—
381C2×C3Q16×C3⋊D311yescharacteristic
382C2×C3#6412711yescharacteristic
383C2×C3#6414611yescharacteristic
384D3—182no—
385D3—188no—
386D3—368no—
387C22—272no—
388C22—541no—
389C22—541no—
390C22#8903911yescharacteristic
391C4—41no—
392C4—62no—
393C4—182no—
394C4—541no—
395C4—1081no—
396C4D3×#63811yescharacteristic
397C4C3⋊D3×#6711yescharacteristic
398C3—24no—
399C3#1059165414yescomplemented
400C3#1059288511yescharacteristic, complemented
401C2—542no—
402C2#1172357711yescharacteristic
403C2#1172357811yescharacteristic
404C2#1172364311yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#451
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
3. C2×C3×#3756C2semidirect
4. C2×C32×#216C2semidirect
5. #11722040C2semidirect
6. #11722038C2semidirect
31. C2×#3756D3semidirect
33. C2×C3×#216D3semidirect
99. C2×C3⋊(C3⋊D3)Q16semidirect
115. C2×#216C3⋊D3semidirect
247. C33#451semidirect
354. C32#5186semidirect
355. C32#5186semidirect
399. C3#10591654semidirect
400. C3#10592885semidirect

Maximal subgroups

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

subgroupindex
C2×#890192
#117239872
C2×C3×#37562
C2×C32×#2162
#117220402
#117220382
#117220242
#105928853
#105916543