Finite Group Databasebeta

← #13161109

Subgroups — #13161109

Order 1728.

1220 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1220 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, complemented
2C2×#88861C211yesmaximal, characteristic, complemented
3C3×#64147C211yesmaximal, characteristic
4C2×C32×#214C211yesmaximal, characteristic, complemented
5#11722042C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592869—31nomaximal
9#10591517—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6293—21no—
16C2×Dic3×C3⋊D3—21no—
17C3⋊D3×Dic6—22no—
18C3⋊D3×Dic6—22no—
19#88861—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×#3757—21no—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C32×#214—22no—
24C2×C32×C3⋊C8C2211yescharacteristic
25C2×Q8×C3⋊D3—31no—
26C2×D3×Dic6—34no—
27C2×C32×SD16—31no—
28#64147D311yescharacteristic
29C2×#3754—31no—
30C2×#3754—34no—
31C2×#3754—64no—
32C3×#938—34no—
33C2×C3×#214D314yescomplemented
34#63662—31no—
35#63647—31no—
36#63643—31no—
37#63567—34no—
38#63565—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C33×D4—21no—
43C33×D4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46#6308—21no—
47#6308D412yescomplemented
48C3×#638—42no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C2×C32×Dic3—21no—
54C32×Dic6—21no—
55C32×Dic6D412yescomplemented
56#6293—24no—
57Dic3×C3⋊D3—24no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5170—94no—
61#5040—91no—
62C2×C32×Q8—31no—
63C2×C32×D4—24no—
64C2×C32×D4C2×D314yes—
65C2×C32×D4C2×D311yescharacteristic
66C2×#638—21no—
67Q8×C3⋊D3—62no—
68Q8×C3⋊D3—62no—
69C2×C4×C3⋊D3—61no—
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C2×#634—31no—
72C2×#634—34no—
73C2×#634—61no—
74C2×#634—64no—
75C2×C3×#67—64no—
76C2×C3×C4×D3—34no—
77C2×C3×Dic6C2×D314yes—
78C2×#627—64no—
79C2×D3×Dic3—64no—
80D3×Dic6—68no—
81D3×Dic6—68no—
82C32×SD16—62no—
83C2×C32×C8—31no—
84#3757—21no—
85#3754—62no—
86#3754—68no—
87#3754—616no—
88C4⋊(C3⋊Dic3)C2×D311yescharacteristic
89C2×C3⋊(C3⋊C8)—31no—
90C2×C3⋊(C3⋊C8)—34no—
91C2×C3⋊(C3⋊C8)—64no—
92C3×#217—64no—
93C3×#214—28no—
94C3×C4⋊Dic3—34no—
95C2×C3×C3⋊C8C2×D314yes—
96C22×C33—41no—
97C22×C33—41no—
98C22×C33—41no—
99C22×C33C2×D411yescharacteristic
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3SD1612yescomplemented
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3⋊(C3⋊Dic3)—22no—
106C3×C3⋊Dic3—21no—
107C3×C3⋊Dic3—41no—
108C32×Dic3—22no—
109C2×D3×Q8—94no—
110C22×Dic6—91no—
111C2×C3×SD16—34no—
112C3×#121—91no—
113#938—34no—
114C2×#214—94no—
115C2×#214—91no—
116C2×#214—184no—
117C2×#214C3⋊D311yescharacteristic, complemented
118#835—94no—
119#830—91no—
120(C2×D3)⋊C8—94no—
121#816—94no—
122#810—91no—
123#810—91no—
124C23×C32—21no—
125C23×C32—24no—
126C23×C32—44no—
127C22×C3⋊D3#6711yescharacteristic
128C22×C3×D3—34no—
129C32×Q8—32no—
130C32×Q8—61no—
131C32×D4—28no—
132C32×D4—24no—
133C32×D4—21no—
134C32×D4—21no—
135C32×D4—24no—
136C32×D4—44no—
137C2×C32×C4—24no—
138C2×C32×C4—61no—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C22×D311yescharacteristic
141#638—42no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3—61no—
144C2×C3⋊Dic3—64no—
145C2×C3⋊Dic3—124no—
146C2×C3⋊Dic3#6711yescharacteristic
147C4×C3⋊D3—21no—
148C4×C3⋊D3—21no—
149C4×C3⋊D3—64no—
150#634—32no—
151#634—38no—
152#634—68no—
153#634—64no—
154#634—61no—
155#634—64no—
156#634—124no—
157C3×#67—128no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3—34no—
160C2×C3×Dic3—64no—
161C3×C4×D3—64no—
162C3×C4×D3—64no—
163C3×Dic6—24no—
164C3×Dic6#6718yescomplemented
165#627—616no—
166D3×Dic3—616no—
167C32×C8—61no—
168C3⋊(C3⋊C8)—68no—
169C3⋊(C3⋊C8)—64no—
170C3⋊(C3⋊C8)—61no—
171C3×C3⋊C8—24no—
172#448—271noSylow
173C2×C33—81no—
174C2×C33#12011yescharacteristic
175C2×C33C2×SD1611yescharacteristic
176C2×C33#13711yescharacteristic
177C3×C3⋊D3—22no—
178C2×C3×Q8—34no—
179C2×C3×D4—24no—
180C2×C3×D4—181no—
181C2×C3×D4D3214yes—
182C2×C3×D4C2×C3⋊D311yescharacteristic
183C22×C3×C4—91no—
184C2×#67—64no—
185C22×Dic3—181no—
186D3×Q8—188no—
187D3×Q8—188no—
188C2×C4×D3—34no—
189C2×C4×D3—184no—
190C2×Dic6—94no—
191C2×Dic6—91no—
192C2×Dic6—184no—
193C2×Dic6—182no—
194C2×Dic6—181no—
195C2×Dic6—184no—
196C2×Dic6—182no—
197C2×Dic6C2×C3⋊D311yescharacteristic
198C3×SD16—68no—
199C2×C3×C8—34no—
200C3×C4⋊C4—91no—
201C3×C22⋊C4—181no—
202#217—64no—
203#214—22no—
204#214—188no—
205#214—182no—
206#214—1816no—
207C4⋊Dic3—34no—
208C2×C3⋊C8—94no—
209C2×C3⋊C8—91no—
210C2×C3⋊C8—184no—
211C2×C3⋊C8C2×C3⋊D311yescharacteristic
212C22×C32—24no—
213C22×C32—48no—
214C22×C32—41no—
215C22×C32—44no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—44no—
219C22×C32—44no—
220C22×C32—84no—
221C22×C32—84no—
222C22×C32D3×D414yes—
223C22×C32C2×#6711yescharacteristic
224C2×C3⋊D3—21no—
225C2×C3⋊D3—21no—
226C2×C3⋊D3#21412yescomplemented
227C2×C3×D3—38no—
228C2×C3×D3—64no—
229C2×C3×D3—64no—
230C32×C4—24no—
231C32×C4—24no—
232C32×C4—62no—
233C32×C4D3×D414yes—
234C32×C4D3×D411yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4C2×#6714yes—
237C3⋊Dic3—21no—
238C3⋊Dic3—41no—
239C3⋊Dic3—62no—
240C3⋊Dic3—68no—
241C3⋊Dic3—128no—
242C3×Dic3—28no—
243C3×Dic3—64no—
244C3×Dic3—124no—
245C22×Q8—271no—
246C2×SD16—31no—
247C2×SD16—271no—
248#121—91no—
249#103—271no—
250#103—271no—
251C22⋊C8—271no—
252C33#44811yescharacteristic, complemented, Sylow
253C23×C3—24no—
254C23×C3—21no—
255C23×C3—44no—
256C23×C3—91no—
257C22×D3—34no—
258C3×Q8—38no—
259C3×Q8—64no—
260C3×D4—28no—
261C3×D4—24no—
262C3×D4—24no—
263C3×D4—21no—
264C3×D4—21no—
265C3×D4—44no—
266C3×D4—362no—
267C2×C3×C4—24no—
268C2×C3×C4—64no—
269C2×C3×C4—91no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4C2×D3214yes—
274C2×C3×C4C22×C3⋊D311yescharacteristic
275#67—128no—
276C2×Dic3—21no—
277C2×Dic3—34no—
278C2×Dic3—64no—
279C2×Dic3—184no—
280C2×Dic3—184no—
281C2×Dic3—181no—
282C2×Dic3—364no—
283C4×D3—64no—
284C4×D3—64no—
285C4×D3—1816no—
286Dic6—21no—
287Dic6—98no—
288Dic6—92no—
289Dic6—188no—
290Dic6—184no—
291Dic6—1816no—
292Dic6—184no—
293Dic6—181no—
294Dic6—364no—
295Dic6#63812yescomplemented
296C3×C8—64no—
297C3⋊C8—21no—
298C3⋊C8—188no—
299C3⋊C8—184no—
300C3⋊C8—181no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32—88no—
305C2×C32—84no—
306C2×C32—81no—
307C2×C32D3×SD1614yes—
308C2×C32#91514yes—
309C2×C32C2×#21411yescharacteristic
310C2×C32#93714yes—
311C2×C32#93711yescharacteristic
312C2×C32#95111yescharacteristic
313C3⋊D3—22no—
314C3×D3—68no—
315C2×Q8—31no—
316C2×Q8—271no—
317C2×Q8—541no—
318C2×Q8—542no—
319C2×Q8—542no—
320C2×D4—181no—
321C2×D4D3×C3⋊D311yescharacteristic
322C22×C4—91no—
323C22×C4—541no—
324SD16—62no—
325SD16—542no—
326C2×C8—31no—
327C2×C8—271no—
328C4⋊C4—91no—
329C22⋊C4—181no—
330C22×C3—24no—
331C22×C3—48no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—44no—
336C22×C3—44no—
337C22×C3—44no—
338C22×C3—84no—
339C22×C3—84no—
340C22×C3—92no—
341C22×C3—181no—
342C22×C3—181no—
343C22×C3D3×#6714yes—
344C22×C3D4×C3⋊D311yescharacteristic
345C2×D3—38no—
346C2×D3—64no—
347C2×D3—64no—
348C3×C4—24no—
349C3×C4—24no—
350C3×C4—68no—
351C3×C4—181no—
352C3×C4—361no—
353C3×C4D3×#6714yes—
354C3×C4#381114yes—
355C3×C4D4×C3⋊D311yescharacteristic
356C3×C4C2×#63811yescharacteristic
357Dic3—22no—
358Dic3—64no—
359Dic3—124no—
360Dic3—182no—
361Dic3—188no—
362Dic3—368no—
363C32—24no—
364C32#504011yescharacteristic, complemented
365C32#517014yescomplemented
366C23—21no—
367C23—91no—
368Q8—32no—
369Q8—61no—
370Q8—272no—
371Q8—544no—
372Q8—541no—
373D4—21no—
374D4—21no—
375D4—362no—
376C2×C4—61no—
377C2×C4—91no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—544no—
382C2×C4—541no—
383C2×C4C2×D3×C3⋊D311yescharacteristic
384C8—61no—
385C8—541no—
386C2×C3—24no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—88no—
390C2×C3—84no—
391C2×C3—81no—
392C2×C3—182no—
393C2×C3D3×#21414yes—
394C2×C3#6392914yes—
395C2×C3#6398314yes—
396C2×C3SD16×C3⋊D311yescharacteristic
397C2×C3#6412411yescharacteristic
398C2×C3#6414611yescharacteristic
399D3—68no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—92no—
404C22—181no—
405C22—181no—
406C22C3⋊D3×#6711yescharacteristic
407C4—62no—
408C4—181no—
409C4—361no—
410C4—542no—
411C4D3×#63811yescharacteristic
412C4#8903911yescharacteristic
413C3—24no—
414C3#1059151714yescomplemented
415C3#1059286911yescharacteristic, complemented
416C2—81no—
417C2—182no—
418C2C3⋊D3×#21411yescharacteristic
419C2#1172355211yescharacteristic
420C2#1172364311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64147C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#448
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×Dic6C2semidirect
2. C2×#88861C2semidirect
4. C2×C32×#214C2semidirect
5. #11722042C2semidirect
6. #11722037C2semidirect
7. #11722023C2semidirect
33. C2×C3×#214D3semidirect
47. #6308D4semidirect
55. C32×Dic6D4semidirect
102. C2×C3×C3⋊D3SD16semidirect
117. C2×#214C3⋊D3semidirect
164. C3×Dic6#67semidirect
226. C2×C3⋊D3#214semidirect
252. C33#448semidirect
295. Dic6#638semidirect
364. C32#5040semidirect
365. C32#5170semidirect
414. C3#10591517semidirect
415. C3#10592869semidirect

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×#888612
C3×#641472
C2×C32×#2142
#117220422
#117220372
#117220232
#105928693
#105915173