Finite Group Databasebeta

← #13161337

Subgroups — #13161337

Order 1728.

1128 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 162 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×C3⋊D3×Dic6C211yesmaximal, characteristic
2C2×#88863C211yesmaximal, characteristic, complemented
3C3×#64155C211yesmaximal, characteristic
4C2×C32×#216C211yesmaximal, characteristic, complemented
5#11722042C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic
8#10592885—31nomaximal
9#10591653—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6293—21no—
15C2×Dic3×C3⋊D3—21no—
16C3⋊D3×Dic6—22no—
17C3⋊D3×Dic6—22no—
18#88863—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×Q8×C3⋊D3—31no—
26C2×D3×Dic6—34no—
27C2×C32×Q16—31no—
28#64155D311yescharacteristic
29C2×#3756—31no—
30C2×#3756—34no—
31C2×#3756—64no—
32C3×#946—34no—
33C2×C3×#216D314yescomplemented
34#63662—31no—
35#63647—31no—
36#63643—31no—
37#63567—34no—
38#63565—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3×C3⋊D3D411yescharacteristic
41C33×Q8—21no—
42C33×Q8—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46#6308—21no—
47#6308D412yes—
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#6293—24no—
56Dic3×C3⋊D3—24no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5186—94no—
60#5050—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—61no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×#634—31no—
70C2×#634—34no—
71C2×#634—61no—
72C2×#634—64no—
73C2×C3×C4×D3—34no—
74C2×C3×Dic6C2×D314yes—
75C2×#627—64no—
76C2×D3×Dic3—64no—
77D3×Dic6—68no—
78D3×Dic6—68no—
79C32×Q16—62no—
80C2×C32×C8—31no—
81#3756—62no—
82#3756—68no—
83#3756—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—21no—
97C2×C3×C3⋊D3—21no—
98C2×C3×C3⋊D3Q1612yescomplemented
99C33×C4—41no—
100C33×C4C2×D411yescharacteristic
101C33×C4C2×D411yescharacteristic
102C3⋊(C3⋊Dic3)—22no—
103C3×C3⋊Dic3—21no—
104C3×C3⋊Dic3—41no—
105C32×Dic3—22no—
106C2×D3×Q8—94no—
107C22×Dic6—91no—
108C2×C3×Q16—34no—
109C3×C22⋊Q8—91no—
110#946—34no—
111C2×#216—94no—
112C2×#216—91no—
113C2×#216—184no—
114C2×#216C3⋊D311yescharacteristic, complemented
115#835—94no—
116#830—91no—
117(C2×D3)⋊C8—94no—
118#816—94no—
119#810—91no—
120#810—91no—
121C22×C3⋊D3#6711yescharacteristic
122C22×C3×D3—34no—
123C32×Q8—28no—
124C32×Q8—24no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—24no—
128C32×Q8—32no—
129C32×Q8—44no—
130C32×Q8—61no—
131C2×C32×C4—21no—
132C2×C32×C4—24no—
133C2×C32×C4—24no—
134C2×C32×C4—44no—
135C2×C32×C4—61no—
136C2×C32×C4C22×D314yes—
137C2×C32×C4C22×D311yescharacteristic
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3—61no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—124no—
142C2×C3⋊Dic3#6711yescharacteristic
143C4×C3⋊D3—21no—
144C4×C3⋊D3—21no—
145C4×C3⋊D3—64no—
146#634—32no—
147#634—38no—
148#634—68no—
149#634—64no—
150#634—61no—
151#634—64no—
152#634—124no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—34no—
155C2×C3×Dic3—64no—
156C3×C4×D3—64no—
157C3×C4×D3—64no—
158C3×Dic6—24no—
159C3×Dic6#6718yes—
160#627—616no—
161D3×Dic3—616no—
162C32×C8—61no—
163C3⋊(C3⋊C8)—68no—
164C3⋊(C3⋊C8)—64no—
165C3⋊(C3⋊C8)—61no—
166C3×C3⋊C8—24no—
167#451—271noSylow
168C2×C33#12011yescharacteristic
169C2×C33C2×Q1611yescharacteristic
170C2×C33#13711yescharacteristic
171C3×C3⋊D3—22no—
172C2×C3×Q8—24no—
173C2×C3×Q8—34no—
174C2×C3×Q8D3214yes—
175C2×C3×Q8C2×C3⋊D311yescharacteristic
176C22×C3×C4—91no—
177C22×Dic3—181no—
178D3×Q8—188no—
179D3×Q8—188no—
180C2×C4×D3—34no—
181C2×C4×D3—184no—
182C2×Dic6—94no—
183C2×Dic6—91no—
184C2×Dic6—184no—
185C2×Dic6—182no—
186C2×Dic6—181no—
187C2×Dic6—184no—
188C2×Dic6—182no—
189C2×Dic6C2×C3⋊D311yescharacteristic
190C3×Q16—68no—
191C2×C3×C8—34no—
192C3×C4⋊C4—91no—
193C3×C4⋊C4—181no—
194C3×C22⋊C4—181no—
195#216—22no—
196#216—188no—
197#216—182no—
198#216—1816no—
199(C2×D3)⋊C4—64no—
200C4⋊Dic3—34no—
201Dic3⋊C4—64no—
202C2×C3⋊C8—94no—
203C2×C3⋊C8—91no—
204C2×C3⋊C8—184no—
205C2×C3⋊C8C2×C3⋊D311yescharacteristic
206C22×C32—24no—
207C22×C32D3×D414yes—
208C22×C32C2×#6711yescharacteristic
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3#21612yescomplemented
212C2×C3×D3—38no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—48no—
218C32×C4—44no—
219C32×C4—41no—
220C32×C4—62no—
221C32×C4D3×D414yes—
222C32×C4D3×D411yescharacteristic
223C32×C4D3×D411yescharacteristic
224C32×C4C2×#6714yes—
225C3⋊Dic3—21no—
226C3⋊Dic3—41no—
227C3⋊Dic3—62no—
228C3⋊Dic3—68no—
229C3⋊Dic3—128no—
230C3×Dic3—28no—
231C3×Dic3—64no—
232C3×Dic3—124no—
233C22×Q8—271no—
234C2×Q16—31no—
235C2×Q16—271no—
236C22⋊Q8—91no—
237#103—271no—
238#103—271no—
239C22⋊C8—271no—
240C33#45111yescharacteristic, complemented, Sylow
241C23×C3—91no—
242C22×D3—34no—
243C3×Q8—28no—
244C3×Q8—24no—
245C3×Q8—24no—
246C3×Q8—21no—
247C3×Q8—21no—
248C3×Q8—38no—
249C3×Q8—44no—
250C3×Q8—64no—
251C2×C3×C4—24no—
252C2×C3×C4—21no—
253C2×C3×C4—24no—
254C2×C3×C4—44no—
255C2×C3×C4—64no—
256C2×C3×C4—91no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4C2×D3214yes—
261C2×C3×C4C22×C3⋊D311yescharacteristic
262C2×Dic3—21no—
263C2×Dic3—34no—
264C2×Dic3—64no—
265C2×Dic3—184no—
266C2×Dic3—184no—
267C2×Dic3—181no—
268C2×Dic3—364no—
269C4×D3—64no—
270C4×D3—64no—
271C4×D3—1816no—
272Dic6—21no—
273Dic6—98no—
274Dic6—92no—
275Dic6—188no—
276Dic6—184no—
277Dic6—1816no—
278Dic6—184no—
279Dic6—181no—
280Dic6—364no—
281Dic6#63812yes—
282C3×C8—64no—
283C3⋊C8—21no—
284C3⋊C8—188no—
285C3⋊C8—184no—
286C3⋊C8—181no—
287C2×C32—24no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32D3×Q1614yes—
291C2×C32#91814yes—
292C2×C32#93714yes—
293C2×C32#93711yescharacteristic
294C2×C32C2×#21611yescharacteristic
295C2×C32#95111yescharacteristic
296C3⋊D3—22no—
297C3×D3—68no—
298C2×Q8—31no—
299C2×Q8—271no—
300C2×Q8—541no—
301C2×Q8—542no—
302C2×Q8—542no—
303C2×Q8D3×C3⋊D311yescharacteristic
304C22×C4—91no—
305C22×C4—541no—
306Q16—62no—
307Q16—542no—
308C2×C8—31no—
309C2×C8—271no—
310C4⋊C4—91no—
311C4⋊C4—181no—
312C22⋊C4—181no—
313C22×C3—24no—
314C22×C3—92no—
315C22×C3—181no—
316C22×C3—181no—
317C22×C3D3×#6714yes—
318C22×C3D4×C3⋊D311yescharacteristic
319C2×D3—38no—
320C2×D3—64no—
321C2×D3—64no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—48no—
325C3×C4—41no—
326C3×C4—44no—
327C3×C4—68no—
328C3×C4—181no—
329C3×C4—361no—
330C3×C4D3×#6714yes—
331C3×C4#381114yes—
332C3×C4D4×C3⋊D311yescharacteristic
333C3×C4C2×#63811yescharacteristic
334Dic3—22no—
335Dic3—64no—
336Dic3—124no—
337Dic3—182no—
338Dic3—188no—
339Dic3—368no—
340C32—24no—
341C32#505011yescharacteristic, complemented
342C32#518614yescomplemented
343C23—91no—
344Q8—21no—
345Q8—21no—
346Q8—32no—
347Q8—61no—
348Q8—272no—
349Q8—544no—
350Q8—541no—
351C2×C4—21no—
352C2×C4—61no—
353C2×C4—91no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4—544no—
358C2×C4—541no—
359C2×C4C2×D3×C3⋊D311yescharacteristic
360C8—61no—
361C8—541no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—182no—
366C2×C3D3×#21614yes—
367C2×C3#6394314yes—
368C2×C3#6398314yes—
369C2×C3Q16×C3⋊D311yescharacteristic
370C2×C3#6412711yescharacteristic
371C2×C3#6414611yescharacteristic
372D3—68no—
373C22—92no—
374C22—181no—
375C22—181no—
376C22C3⋊D3×#6711yescharacteristic
377C4—41no—
378C4—62no—
379C4—181no—
380C4—361no—
381C4—542no—
382C4D3×#63811yescharacteristic
383C4#8903911yescharacteristic
384C3—24no—
385C3#1059165314yescomplemented
386C3#1059288511yescharacteristic, complemented
387C2—182no—
388C2C3⋊D3×#21611yescharacteristic
389C2#1172357611yescharacteristic
390C2#1172364311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
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
2. C2×#88863C2semidirect
4. C2×C32×#216C2semidirect
5. #11722042C2semidirect
6. #11722037C2semidirect
33. C2×C3×#216D3semidirect
98. C2×C3×C3⋊D3Q16semidirect
114. C2×#216C3⋊D3semidirect
211. C2×C3⋊D3#216semidirect
240. C33#451semidirect
341. C32#5050semidirect
342. C32#5186semidirect
385. C3#10591653semidirect
386. C3#10592885semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C2×#888632
C3×#641552
C2×C32×#2162
#117220422
#117220372
#117220232
#105928853
#105916533