Finite Group Databasebeta

← #13161341

Subgroups — #13161341

Order 1728.

1208 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1208 conjugacy classes of subgroups — too many to draw. Only its 158 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×#89017C211yesmaximal, characteristic
2#11723987C211yesmaximal, characteristic, complemented
3C2×C3×#3755C211yesmaximal, characteristic, complemented
4C2×C32×#216C211yesmaximal, characteristic, complemented
5#11722038C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592886—31nomaximal
9#10591656—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6292—21no—
15C2×Dic3×C3⋊D3—21no—
16#89017—22no—
17#89017—22no—
18#88859—21no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20#88857—21no—
21C3×#3755—22no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×#216—22no—
24C2×C32×C3⋊C8C2211yescharacteristic
25C2×#3832—31no—
26C2×#3796—34no—
27C2×C32×Q16—31no—
28#64155—31no—
29#64155—34no—
30#64155—64no—
31C2×#3755D311yescharacteristic, complemented
32C2×C3×#216D314yescomplemented
33C2×C3×#215—34no—
34#63662—31no—
35#63648—31no—
36#63647—31no—
37#63566—34no—
38#63563—34no—
39#63557—34no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C22×C3×C3⋊D3—21no—
42C33×Q8—21no—
43C33×Q8—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—21no—
47C2×C3⋊(C3⋊Dic3)D411yescharacteristic
48C4×C3⋊(C3⋊D3)—21no—
49C4×C3⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)D412yes—
52C2×C32×Dic3—21no—
53C32×Dic6—21no—
54C32×Dic6D412yes—
55#6292—24no—
56Dic3×C3⋊D3—24no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#5187—94no—
60#5173—91no—
61C2×C32×Q8—24no—
62C2×C32×Q8—31no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D311yescharacteristic
65#3832—62no—
66#3832—62no—
67C2×C4⋊(C3⋊D3)—61no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4×C3⋊D3—31no—
70C2×C4×C3⋊D3—34no—
71C2×C4×C3⋊D3—61no—
72C2×C4×C3⋊D3—64no—
73C2×C3×D12—34no—
74C2×C3×Dic6C2×D314yes—
75C2×#626—64no—
76C2×D3×Dic3—64no—
77#3796—68no—
78#3796—68no—
79C32×Q16—62no—
80C2×C32×C8—31no—
81#3755—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×#215—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)—21no—
100C2×C3×C3⋊D3—22no—
101C2×C3×C3⋊D3—41no—
102C2×C3×C3⋊D3—41no—
103C33×C4—41no—
104C33×C4C2×D411yescharacteristic
105C33×C4C2×D411yescharacteristic
106C3⋊(C3⋊Dic3)—21no—
107C3⋊(C3⋊Dic3)—41no—
108C32×Dic3—22no—
109C2×#239—94no—
110C2×#237—91no—
111C2×C3×Q16—34no—
112C2×C3×SD16—91no—
113#946—94no—
114#946—91no—
115#946—184no—
116C2×#216C3⋊D311yescharacteristic, complemented
117C2×#215—34no—
118#835—94no—
119#832—91no—
120#821—94no—
121(C2×D3)⋊C8—94no—
122(C2×D3)⋊C8—91no—
123#816—91no—
124C22×C3⋊D3—21no—
125C22×C3⋊D3—31no—
126C22×C3⋊D3—34no—
127C22×C3⋊D3—64no—
128C22×C3×D3—64no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—24no—
132C32×Q8—24no—
133C32×Q8—32no—
134C32×Q8—44no—
135C32×Q8—44no—
136C32×Q8—61no—
137C2×C32×C4—28no—
138C2×C32×C4—24no—
139C2×C32×C4—21no—
140C2×C32×C4—24no—
141C2×C32×C4—61no—
142C2×C32×C4C22×D314yes—
143C2×C32×C4C22×D311yescharacteristic
144C2×C3⋊Dic3—31no—
145C2×C3⋊Dic3—34no—
146C2×C3⋊Dic3—64no—
147C2×C3⋊Dic3—68no—
148C2×C3⋊Dic3—64no—
149C2×C3⋊Dic3—61no—
150C4⋊(C3⋊D3)—21no—
151C4⋊(C3⋊D3)—64no—
152C4⋊(C3⋊D3)#6712yes—
153C4×C3⋊D3—64no—
154C4×C3⋊D3—61no—
155C4×C3⋊D3—61no—
156C4×C3⋊D3—64no—
157C4×C3⋊D3—64no—
158C4×C3⋊D3—124no—
159C4×C3⋊D3—124no—
160C2×C3×Dic3—24no—
161C3×D12—38no—
162C3×D12—64no—
163C3×Dic6—24no—
164C3×Dic6#6718yes—
165#626—616no—
166D3×Dic3—616no—
167C32×C8—61no—
168C3⋊(C3⋊C8)—21no—
169C3×C3⋊C8—24no—
170C3×C3⋊C8—64no—
171#452—271noSylow
172C2×C33#12011yescharacteristic
173C2×C33#13511yescharacteristic
174C2×C33#13711yescharacteristic
175C3⋊(C3⋊D3)—41no—
176C3×C3⋊D3—42no—
177C2×C3×Q8—24no—
178C2×C3×Q8—34no—
179C2×C3×Q8D3214yes—
180C2×C3×Q8C2×C3⋊D311yescharacteristic
181C2×C3×D4—91no—
182C2×#67—181no—
183C22×Dic3—181no—
184#239—188no—
185#239—188no—
186#237—182no—
187#237—182no—
188C2×D12—34no—
189C2×D12—184no—
190C2×C4×D3—94no—
191C2×C4×D3—91no—
192C2×C4×D3—184no—
193C2×C4×D3—184no—
194C2×Dic6C2×C3⋊D311yescharacteristic
195C3×Q16—68no—
196C3×SD16—182no—
197C2×C3×C8—34no—
198C2×C3×C8—91no—
199#216—22no—
200#215—68no—
201(C2×D3)⋊C4—184no—
202(C2×D3)⋊C4—181no—
203(C2×D3)⋊C4—188no—
204C4⋊Dic3—94no—
205C4⋊Dic3—91no—
206C4⋊Dic3—184no—
207Dic3⋊C4—184no—
208Dic3⋊C4—181no—
209Dic3⋊C4—188no—
210C2×C3⋊C8—34no—
211C2×C3⋊C8C2×C3⋊D311yescharacteristic
212C22×C32—24no—
213C22×C32D3×D414yes—
214C22×C32D3×D411yescharacteristic
215C2×C3⋊D3—22no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—68no—
219C2×C3⋊D3—61no—
220C2×C3⋊D3—61no—
221C2×C3⋊D3—64no—
222C2×C3⋊D3—61no—
223C2×C3⋊D3—64no—
224C2×C3⋊D3—64no—
225C2×C3⋊D3—124no—
226C2×C3⋊D3—124no—
227C2×C3×D3—68no—
228C2×C3×D3—124no—
229C2×C3×D3—124no—
230C32×C4—24no—
231C32×C4—24no—
232C32×C4—48no—
233C32×C4—44no—
234C32×C4—41no—
235C32×C4—62no—
236C32×C4D3×D414yes—
237C32×C4D3×D411yescharacteristic
238C32×C4C2×#6714yes—
239C32×C4C2×#6711yescharacteristic
240C3⋊Dic3—61no—
241C3⋊Dic3—64no—
242C3⋊Dic3—124no—
243C3⋊Dic3—128no—
244C3⋊Dic3—121no—
245C3⋊Dic3—124no—
246C3×Dic3—28no—
247C2×#41—271no—
248C2×Q16—31no—
249C2×SD16—91no—
250C22⋊Q8—271no—
251#103—271no—
252#102—271no—
253C22⋊C8—271no—
254C33#45211yescharacteristic, complemented, Sylow
255C23×C3—181no—
256C22×D3—64no—
257C22×D3—94no—
258C22×D3—91no—
259C22×D3—184no—
260C3×Q8—24no—
261C3×Q8—24no—
262C3×Q8—21no—
263C3×Q8—21no—
264C3×Q8—38no—
265C3×Q8—44no—
266C3×Q8—44no—
267C3×Q8—64no—
268C3×D4—92no—
269C3×D4—181no—
270C2×C3×C4—28no—
271C2×C3×C4—24no—
272C2×C3×C4—21no—
273C2×C3×C4—24no—
274C2×C3×C4—64no—
275C2×C3×C4C2×D3214yes—
276C2×C3×C4C22×C3⋊D311yescharacteristic
277#67—184no—
278C2×Dic3—21no—
279C2×Dic3—94no—
280C2×Dic3—91no—
281C2×Dic3—184no—
282C2×Dic3—184no—
283C2×Dic3—188no—
284C2×Dic3—184no—
285C2×Dic3—181no—
286D12—38no—
287D12—64no—
288D12—1816no—
289C4×D3—1816no—
290C4×D3—184no—
291C4×D3—184no—
292C4×D3—181no—
293C4×D3—181no—
294C4×D3—364no—
295C4×D3—364no—
296Dic6—21no—
297Dic6#63812yes—
298C3×C8—64no—
299C3×C8—181no—
300C3⋊C8—21no—
301C3⋊C8—64no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32—24no—
305C2×C32#91511yescharacteristic
306C2×C32#91611yescharacteristic
307C2×C32#91814yes—
308C2×C32#91914yes—
309C2×C32#93714yes—
310C2×C32#93711yescharacteristic
311C3⋊D3—42no—
312C3⋊D3—128no—
313C3⋊D3—121no—
314C3⋊D3—124no—
315C3×D3—128no—
316#41—542no—
317#41—542no—
318C2×Q8—31no—
319C2×Q8D3×C3⋊D311yescharacteristic
320C2×D4—91no—
321C2×D4—541no—
322C22×C4—271no—
323C22×C4—541no—
324Q16—62no—
325SD16—182no—
326C2×C8—31no—
327C2×C8—91no—
328C4⋊C4—271no—
329C4⋊C4—541no—
330C22⋊C4—541no—
331C22×C3—24no—
332C22×C3—182no—
333C22×C3—361no—
334C22×C3—361no—
335C22×C3#381114yes—
336C22×C3D4×C3⋊D311yescharacteristic
337C2×D3—68no—
338C2×D3—124no—
339C2×D3—124no—
340C2×D3—188no—
341C2×D3—181no—
342C2×D3—181no—
343C2×D3—181no—
344C2×D3—184no—
345C2×D3—184no—
346C2×D3—184no—
347C2×D3—364no—
348C2×D3—364no—
349C3×C4—24no—
350C3×C4—24no—
351C3×C4—48no—
352C3×C4—41no—
353C3×C4—44no—
354C3×C4—68no—
355C3×C4D3×#6714yes—
356C3×C4D3×#6714yes—
357C3×C4D4×C3⋊D311yescharacteristic
358C3×C4C2×#63811yescharacteristic
359Dic3—22no—
360Dic3—181no—
361Dic3—184no—
362Dic3—364no—
363Dic3—368no—
364Dic3—361no—
365Dic3—364no—
366C32—24no—
367C32#517311yescharacteristic, complemented
368C32#518714yescomplemented
369C23—181no—
370C23—271no—
371Q8—21no—
372Q8—21no—
373Q8—32no—
374Q8—61no—
375D4—92no—
376D4—181no—
377D4—544no—
378C2×C4—21no—
379C2×C4—61no—
380C2×C4—271no—
381C2×C4—544no—
382C2×C4—541no—
383C2×C4—541no—
384C2×C4—541no—
385C2×C4C2×D3×C3⋊D311yescharacteristic
386C8—61no—
387C8—181no—
388C2×C3—24no—
389C2×C3—24no—
390C2×C3—24no—
391C2×C3—362no—
392C2×C3#6395014yes—
393C2×C3#6395114yes—
394C2×C3#6398314yes—
395C2×C3#6412711yescharacteristic
396C2×C3#6412811yescharacteristic
397C2×C3#6414611yescharacteristic
398D3—128no—
399D3—368no—
400D3—361no—
401D3—364no—
402C22—182no—
403C22—361no—
404C22—361no—
405C22—541no—
406C22—541no—
407C22—541no—
408C22#8903911yescharacteristic
409C4—41no—
410C4—62no—
411C4—541no—
412C4—1081no—
413C4D3×#63811yescharacteristic
414C4C3⋊D3×#6711yescharacteristic
415C3—24no—
416C3#1059165614yescomplemented
417C3#1059288611yescharacteristic, complemented
418C2—362no—
419C2—1081no—
420C2#1172358911yescharacteristic
421C2#1172359011yescharacteristic
422C2#1172364311yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#89017C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#452
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. #11723987C2semidirect
3. C2×C3×#3755C2semidirect
4. C2×C32×#216C2semidirect
5. #11722038C2semidirect
6. #11722034C2semidirect
7. #11722024C2semidirect
31. C2×#3755D3semidirect
32. C2×C3×#216D3semidirect
116. C2×#216C3⋊D3semidirect
254. C33#452semidirect
367. C32#5173semidirect
368. C32#5187semidirect
416. C3#10591656semidirect
417. C3#10592886semidirect

Maximal subgroups

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

subgroupindex
C2×#890172
#117239872
C2×C3×#37552
C2×C32×#2162
#117220382
#117220342
#117220242
#105928863
#105916563