Finite Group Databasebeta

← #13161110

Subgroups — #13161110

Order 1728.

1220 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 170 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×D3×#634C211yesmaximal, characteristic, complemented
2C2×#88861C211yesmaximal, characteristic, complemented
3C2×C3×#3754C211yesmaximal, characteristic, complemented
4C32×#938C211yesmaximal, characteristic
5#11722041C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592739—31nomaximal
9#10591517—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×#67—21no—
14C2×C32×C4×D3C2211yescharacteristic
15C2×#6293—21no—
16C2×D3×C3⋊Dic3—21no—
17D3×#634—22no—
18D3×#634—22no—
19#88861—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×#3754—22no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×#217—21no—
24C32×C4⋊Dic3C2211yescharacteristic
25C22×#634—31no—
26C2×D3×Dic6—34no—
27C32×#121—31no—
28C2×#3754—31no—
29C2×#3754—34no—
30C2×#3754—64no—
31C2×#3754D311yescharacteristic, complemented
32C3×#938D314yes—
33C2×C3×#214—34no—
34#63657—31no—
35#63637—31no—
36#63637—31no—
37#63567—34no—
38#63565—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C32×D3D411yescharacteristic
42C33×D4—21no—
43C33×D4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46#6308—21no—
47#6308D412yescomplemented
48C2×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634D412yescomplemented
51C32×#67—42no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C32×C4×D3—21no—
55C32×C4×D3—21no—
56#6293—24no—
57D3×C3⋊Dic3—24no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5170—91no—
61#5040—94no—
62C2×C32×D4—24no—
63C2×C32×D4—61no—
64C2×C32×D4C2×D314yes—
65C2×C32×D4C2×D311yescharacteristic
66C22×C32×C4—31no—
67C22×C3⋊Dic3—61no—
68C2×#634—31no—
69C2×#634—34no—
70C2×#634—62no—
71C2×#634—61no—
72C2×#634—62no—
73C2×#634—64no—
74C2×#634C2×D311yescharacteristic
75C2×C3×#67—24no—
76C2×C3×C4×D3C2×D314yes—
77C2×C3×Dic6—34no—
78C2×#627—64no—
79C2×D3×Dic3—64no—
80D3×Dic6—68no—
81D3×Dic6—68no—
82C32×C4⋊C4—31no—
83C32×C22⋊C4—61no—
84#3754—22no—
85#3754—62no—
86#3754—68no—
87#3754—616no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—34no—
90C2×C3⋊(C3⋊C8)—64no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#217—24no—
93C3×#214—68no—
94C3×C4⋊Dic3C2×D314yes—
95C2×C3×C3⋊C8—34no—
96C22×C33—41no—
97C22×C33—41no—
98C22×C33—41no—
99C22×C33C2×D411yescharacteristic
100C2×C32×D3—21no—
101C2×C32×D3—21no—
102C2×C32×D3SD1612yescomplemented
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3⋊(C3⋊Dic3)—22no—
106C3×C3⋊Dic3—22no—
107C32×Dic3—21no—
108C32×Dic3—41no—
109C2×D3×Q8—91no—
110C22×Dic6—94no—
111C2×C3×SD16—91no—
112C3×#121—34no—
113#938C3⋊D311yescharacteristic
114C2×#214—34no—
115C2×#214—94no—
116C2×#214—91no—
117C2×#214—184no—
118#835—91no—
119#830—94no—
120(C2×D3)⋊C8—91no—
121#816—91no—
122#810—94no—
123#810—94no—
124C23×C32—21no—
125C23×C32—24no—
126C23×C32—31no—
127C23×C32—44no—
128C22×C3×D3#6714yes—
129C32×D4—28no—
130C32×D4—24no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—24no—
134C32×D4—44no—
135C32×D4—122no—
136C2×C32×C4—24no—
137C2×C32×C4—31no—
138C2×C32×C4—61no—
139C2×C32×C4—61no—
140C2×C32×C4—61no—
141C2×C32×C4C22×D314yes—
142C2×C32×C4C22×D311yescharacteristic
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—64no—
145C2×C3⋊Dic3—61no—
146C2×C3⋊Dic3—64no—
147C2×C3⋊Dic3—124no—
148#634—21no—
149#634—32no—
150#634—38no—
151#634—68no—
152#634—64no—
153#634—61no—
154#634—64no—
155#634—124no—
156#634#6712yescomplemented
157C3×#67—48no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3—64no—
160C2×C3×Dic3#6714yes—
161C3×C4×D3—24no—
162C3×C4×D3—24no—
163C3×Dic6—38no—
164C3×Dic6—64no—
165#627—616no—
166D3×Dic3—616no—
167C3⋊(C3⋊C8)—21no—
168C3⋊(C3⋊C8)—68no—
169C3⋊(C3⋊C8)—64no—
170C3⋊(C3⋊C8)—61no—
171C3×C3⋊C8—64no—
172#448—271noSylow
173C2×C33—81no—
174C2×C33#12011yescharacteristic
175C2×C33C2×SD1611yescharacteristic
176C2×C33#13711yescharacteristic
177C32×D3—22no—
178C2×C3×Q8—91no—
179C2×C3×D4—24no—
180C2×C3×D4—64no—
181C2×C3×D4D3214yes—
182C2×C3×D4C2×C3⋊D311yescharacteristic
183C22×C3×C4—34no—
184C2×#67—21no—
185C22×Dic3—184no—
186D3×Q8—182no—
187D3×Q8—182no—
188C2×C4×D3—181no—
189C2×C4×D3C2×C3⋊D311yescharacteristic
190C2×Dic6—34no—
191C2×Dic6—94no—
192C2×Dic6—91no—
193C2×Dic6—184no—
194C2×Dic6—188no—
195C2×Dic6—188no—
196C2×Dic6—184no—
197C2×Dic6—181no—
198C3×SD16—182no—
199C2×C3×C8—91no—
200C3×C4⋊C4—34no—
201C3×C22⋊C4—64no—
202#217—21no—
203#214—68no—
204#214—188no—
205#214—182no—
206#214—1816no—
207C4⋊Dic3C2×C3⋊D311yescharacteristic
208C2×C3⋊C8—34no—
209C2×C3⋊C8—94no—
210C2×C3⋊C8—91no—
211C2×C3⋊C8—184no—
212C22×C32—24no—
213C22×C32—32no—
214C22×C32—48no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—44no—
218C22×C32—41no—
219C22×C32—44no—
220C22×C32—44no—
221C22×C32—61no—
222C22×C32—61no—
223C22×C32—84no—
224C22×C32—84no—
225C22×C32D3×D411yescharacteristic
226C22×C32C2×#6714yes—
227C2×C3×D3—24no—
228C2×C3×D3—24no—
229C2×C3×D3#21418yescomplemented
230C32×C4—24no—
231C32×C4—24no—
232C32×C4—61no—
233C32×C4—121no—
234C32×C4D3×D414yes—
235C32×C4D3×D414yes—
236C32×C4D3×D411yescharacteristic
237C32×C4C2×#6711yescharacteristic
238C3⋊Dic3—22no—
239C3⋊Dic3—62no—
240C3⋊Dic3—68no—
241C3⋊Dic3—128no—
242C3×Dic3—24no—
243C3×Dic3—44no—
244C3×Dic3—68no—
245C22×Q8—271no—
246C2×SD16—91no—
247C2×SD16—271no—
248#121—31no—
249#103—271no—
250#103—271no—
251C22⋊C8—271no—
252C33#44811yescharacteristic, complemented, Sylow
253C23×C3—24no—
254C23×C3—21no—
255C23×C3—34no—
256C23×C3—44no—
257C22×D3#63811yescharacteristic
258C3×Q8—92no—
259C3×Q8—181no—
260C3×D4—28no—
261C3×D4—24no—
262C3×D4—24no—
263C3×D4—21no—
264C3×D4—21no—
265C3×D4—44no—
266C3×D4—128no—
267C2×C3×C4—24no—
268C2×C3×C4—34no—
269C2×C3×C4—64no—
270C2×C3×C4—64no—
271C2×C3×C4—64no—
272C2×C3×C4—181no—
273C2×C3×C4C2×D3214yes—
274C2×C3×C4C22×C3⋊D311yescharacteristic
275#67—42no—
276C2×Dic3—21no—
277C2×Dic3—64no—
278C2×Dic3—1816no—
279C2×Dic3—184no—
280C2×Dic3—181no—
281C2×Dic3—364no—
282C2×Dic3#63811yescharacteristic
283C4×D3—21no—
284C4×D3—21no—
285C4×D3—184no—
286Dic6—38no—
287Dic6—64no—
288Dic6—98no—
289Dic6—92no—
290Dic6—188no—
291Dic6—184no—
292Dic6—1816no—
293Dic6—184no—
294Dic6—181no—
295Dic6—364no—
296C3×C8—181no—
297C3⋊C8—64no—
298C3⋊C8—188no—
299C3⋊C8—184no—
300C3⋊C8—181no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32—62no—
305C2×C32—88no—
306C2×C32—81no—
307C2×C32—84no—
308C2×C32D3×SD1611yescharacteristic
309C2×C32#91511yescharacteristic
310C2×C32C2×#21414yes—
311C2×C32#93714yes—
312C2×C32#93711yescharacteristic
313C2×C32#95114yes—
314C3×D3—28no—
315C2×Q8—91no—
316C2×Q8—271no—
317C2×Q8—541no—
318C2×Q8—542no—
319C2×Q8—542no—
320C2×D4—61no—
321C2×D4D3×C3⋊D311yescharacteristic
322C22×C4—31no—
323C22×C4—541no—
324SD16—182no—
325SD16—542no—
326C2×C8—91no—
327C2×C8—271no—
328C4⋊C4—31no—
329C22⋊C4—61no—
330C22×C3—24no—
331C22×C3—38no—
332C22×C3—48no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—44no—
337C22×C3—44no—
338C22×C3—44no—
339C22×C3—64no—
340C22×C3—64no—
341C22×C3—84no—
342C22×C3—84no—
343C22×C3D3×#6714yes—
344C22×C3C2×#63811yescharacteristic
345C2×D3—21no—
346C2×D3—21no—
347C2×D3#375412yescomplemented
348C3×C4—24no—
349C3×C4—24no—
350C3×C4—64no—
351C3×C4—124no—
352C3×C4—182no—
353C3×C4D3×#6714yes—
354C3×C4#381114yes—
355C3×C4D4×C3⋊D311yescharacteristic
356C3×C4D4×C3⋊D311yescharacteristic
357Dic3—21no—
358Dic3—41no—
359Dic3—68no—
360Dic3—182no—
361Dic3—188no—
362Dic3—368no—
363C32—24no—
364C32#504014yescomplemented
365C32#517011yescharacteristic, complemented
366C23—21no—
367C23—31no—
368Q8—92no—
369Q8—181no—
370Q8—272no—
371Q8—544no—
372Q8—541no—
373D4—21no—
374D4—21no—
375D4—122no—
376C2×C4—31no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—181no—
381C2×C4—544no—
382C2×C4—541no—
383C2×C4C2×D3×C3⋊D311yescharacteristic
384C8—181no—
385C8—541no—
386C2×C3—24no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—68no—
390C2×C3—88no—
391C2×C3—84no—
392C2×C3—81no—
393C2×C3D3×#21414yes—
394C2×C3#6392914yes—
395C2×C3#6398314yes—
396C2×C3C2×#375411yescharacteristic
397C2×C3#6414611yescharacteristic
398C2×C3#6416011yescharacteristic
399D3—22no—
400C22—32no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—61no—
405C22—61no—
406C22D3×#63811yescharacteristic
407C4—61no—
408C4—121no—
409C4—182no—
410C4—542no—
411C4C3⋊D3×#6711yescharacteristic
412C4#8903911yescharacteristic
413C3—24no—
414C3#1059151714yescomplemented
415C3#1059273911yescharacteristic, complemented
416C2—62no—
417C2—81no—
418C2D3×#375411yescharacteristic
419C2#1172355011yescharacteristic
420C2#1172364311yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC32×#938C2
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×D3×#634C2semidirect
2. C2×#88861C2semidirect
3. C2×C3×#3754C2semidirect
5. #11722041C2semidirect
6. #11722039C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
31. C2×#3754D3semidirect
47. #6308D4semidirect
50. C3×#634D4semidirect
102. C2×C32×D3SD16semidirect
156. #634#67semidirect
229. C2×C3×D3#214semidirect
252. C33#448semidirect
347. C2×D3#3754semidirect
364. C32#5040semidirect
365. C32#5170semidirect
414. C3#10591517semidirect
415. C3#10592739semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6342
C2×#888612
C2×C3×#37542
C32×#9382
#117220412
#117220392
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927393
#105915173