Finite Group Databasebeta

← #13161336

Subgroups — #13161336

Order 1728.

1424 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1424 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×#89026C211yesmaximal, characteristic
2#11723987C211yesmaximal, characteristic, complemented
3C2×C3×#3755C211yesmaximal, characteristic, complemented
4C2×C32×#215C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592868—31nomaximal
9#10591652—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C33×Q8C2211yescharacteristic, complemented
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6290—21no—
16#89026—22no—
17#89026—22no—
18#88859—21no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
20#88857—21no—
21C3×#3755—22no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
23C32×#215—22no—
24C2×C32×C3⋊C8C2211yescharacteristic, complemented
25C2×D4×C3⋊D3—31no—
26C2×#3802—34no—
27C2×C32×SD16—31no—
28#64155—31no—
29#64155—34no—
30#64155—64no—
31C2×#3755D311yescharacteristic, complemented
32C2×C3×#215—34no—
33C2×C3×#215D314yescomplemented
34#63659—31no—
35#63648—31no—
36#63647—31no—
37#63563—34no—
38#63563—34no—
39#63557—34no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C22×C3×C3⋊D3—21no—
42C22×C32×D3—21no—
43C2×D3×C3⋊D3—24no—
44C2×D3×C3⋊D3—42no—
45C2×D3×C3⋊D3—42no—
46C33×Q8—21no—
47C33×Q8—21no—
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)—21no—
51C2×C3⋊(C3⋊Dic3)D411yescharacteristic
52C4×C3⋊(C3⋊D3)—21no—
53C4×C3⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)D412yes—
56C32×D12—21no—
57C32×D12D412yes—
58#6290—24no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61#5169—94no—
62#5169—91no—
63C23×C3⋊D3—61no—
64C22×D32—64no—
65C2×C32×Q8—24no—
66C2×C32×Q8C2×D314yescomplemented
67C2×C32×Q8C2×D311yescharacteristic, complemented
68C2×C32×D4—31no—
69C2×#638—61no—
70D4×C3⋊D3—62no—
71D4×C3⋊D3—62no—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
73C2×C4×C3⋊D3—31no—
74C2×C4×C3⋊D3—34no—
75C2×C4×C3⋊D3—64no—
76C2×C3×D12—34no—
77C2×C3×D12C2×D314yes—
78C2×#625—64no—
79#3802—68no—
80#3802—68no—
81C32×SD16—62no—
82C2×C32×C8—31no—
83#3755—22no—
84#3752—61no—
85#3752—64no—
86#3752—68no—
87C4⋊(C3⋊Dic3)—31no—
88C4⋊(C3⋊Dic3)—34no—
89C4⋊(C3⋊Dic3)—64no—
90#3750—61no—
91#3750—64no—
92#3750—68no—
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
94C3×#215—28no—
95C3×#215—68no—
96C2×C3×C3⋊C8—34no—
97C2×C3×C3⋊C8C2×D314yescomplemented
98C22×C33C2×D411yescharacteristic
99C2×C3⋊(C3⋊D3)—21no—
100C2×C3⋊(C3⋊D3)—21no—
101C2×C3⋊(C3⋊D3)SD1612yescomplemented
102C2×C3×C3⋊D3—22no—
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3—41no—
105C2×C32×D3—22no—
106C2×C32×D3—41no—
107C2×C32×D3—41no—
108D3×C3⋊D3—48no—
109C33×C4—41no—
110C33×C4C2×D411yescharacteristic
111C33×C4C2×D411yescharacteristic
112C3⋊(C3⋊Dic3)—21no—
113C3⋊(C3⋊Dic3)—41no—
114C2×D3×D4—94no—
115C2×D3×D4—91no—
116C2×C3×SD16—34no—
117C2×C3×SD16—91no—
118#946—94no—
119#946—91no—
120#946—184no—
121C2×#215—34no—
122C2×#215C3⋊D311yescharacteristic, complemented
123#832—94no—
124#832—91no—
125#821—94no—
126#821—91no—
127(C2×D3)⋊C8—94no—
128(C2×D3)⋊C8—91no—
129C23×C32—61no—
130C22×C3⋊D3—21no—
131C22×C3⋊D3—31no—
132C22×C3⋊D3—34no—
133C22×C3⋊D3—64no—
134C22×C3⋊D3—64no—
135C22×C3⋊D3—122no—
136C22×C3⋊D3—122no—
137C22×C3×D3—24no—
138C22×C3×D3—64no—
139C2×D32—616no—
140C2×D32—128no—
141C2×D32—128no—
142C32×Q8—21no—
143C32×Q8—21no—
144C32×Q8—24no—
145C32×Q8—24no—
146C32×Q8—44no—
147C32×Q8—44no—
148C32×D4—32no—
149C32×D4—61no—
150C2×C32×C4—28no—
151C2×C32×C4—24no—
152C2×C32×C4—21no—
153C2×C32×C4—24no—
154C2×C32×C4C22×D314yes—
155C2×C32×C4C22×D311yescharacteristic
156#638—64no—
157C2×C3⋊Dic3—31no—
158C2×C3⋊Dic3—34no—
159C2×C3⋊Dic3—64no—
160C2×C3⋊Dic3—68no—
161C2×C3⋊Dic3—64no—
162C2×C3⋊Dic3—61no—
163C4⋊(C3⋊D3)—21no—
164C4⋊(C3⋊D3)#6712yes—
165C4×C3⋊D3—61no—
166C4×C3⋊D3—61no—
167C4×C3⋊D3—64no—
168C4×C3⋊D3—64no—
169C4×C3⋊D3—124no—
170C4×C3⋊D3—124no—
171C3×D12—24no—
172C3×D12—38no—
173C3×D12—64no—
174C3×D12#6718yes—
175#625—616no—
176C32×C8—61no—
177C3⋊(C3⋊C8)—21no—
178C3×C3⋊C8—24no—
179C3×C3⋊C8—64no—
180#450—271noSylow
181C2×C33#12011yescharacteristic
182C2×C33C2×SD1611yescharacteristic
183C2×C33AGL(1,ℤ/8)11yescharacteristic
184C3⋊(C3⋊D3)—22no—
185C3×C3⋊D3—42no—
186C32×D3—42no—
187C23×D3—181no—
188C23×D3—184no—
189C2×C3×Q8—24no—
190C2×C3×Q8D3214yescomplemented
191C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
192C2×C3×D4—34no—
193C2×C3×D4—91no—
194C2×#67—181no—
195C2×#67—184no—
196D3×D4—182no—
197D3×D4—188no—
198D3×D4—188no—
199D3×D4—182no—
200C2×D12—34no—
201C2×D12C2×C3⋊D311yescharacteristic
202C2×C4×D3—94no—
203C2×C4×D3—91no—
204C2×C4×D3—184no—
205C3×SD16—68no—
206C3×SD16—182no—
207C2×C3×C8—34no—
208C2×C3×C8—91no—
209#215—22no—
210#215—68no—
211(C2×D3)⋊C4—184no—
212(C2×D3)⋊C4—181no—
213(C2×D3)⋊C4—188no—
214C4⋊Dic3—94no—
215C4⋊Dic3—91no—
216C4⋊Dic3—184no—
217Dic3⋊C4—184no—
218Dic3⋊C4—181no—
219Dic3⋊C4—188no—
220C2×C3⋊C8—34no—
221C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
222C22×C32—24no—
223C22×C32—62no—
224C22×C32—121no—
225C22×C32—121no—
226C22×C32D3×D414yes—
227C22×C32D3×D411yescharacteristic
228C2×C3⋊D3—22no—
229C2×C3⋊D3—32no—
230C2×C3⋊D3—38no—
231C2×C3⋊D3—41no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3—68no—
234C2×C3⋊D3—61no—
235C2×C3⋊D3—61no—
236C2×C3⋊D3—64no—
237C2×C3⋊D3—64no—
238C2×C3⋊D3—124no—
239C2×C3⋊D3—124no—
240C2×C3⋊D3—128no—
241C2×C3×D3—28no—
242C2×C3×D3—44no—
243C2×C3×D3—44no—
244C2×C3×D3—68no—
245C2×C3×D3—124no—
246C2×C3×D3—124no—
247D32—1232no—
248C32×C4—24no—
249C32×C4—24no—
250C32×C4—48no—
251C32×C4—44no—
252C32×C4—41no—
253C32×C4D3×D414yes—
254C32×C4D3×D411yescharacteristic
255C32×C4C2×#6714yes—
256C32×C4C2×#6711yescharacteristic
257C3⋊Dic3—61no—
258C3⋊Dic3—64no—
259C3⋊Dic3—124no—
260C3⋊Dic3—128no—
261C3⋊Dic3—124no—
262C3⋊Dic3—121no—
263C22×D4—271no—
264C2×SD16—31no—
265C2×SD16—91no—
266C22⋊Q8—271no—
267#102—271no—
268#102—271no—
269C22⋊C8—271no—
270C33#45011yescharacteristic, complemented, Sylow
271C23×C3—64no—
272C23×C3—181no—
273C22×D3—21no—
274C22×D3—64no—
275C22×D3—94no—
276C22×D3—91no—
277C22×D3—184no—
278C22×D3—184no—
279C22×D3—1816no—
280C22×D3—362no—
281C22×D3—362no—
282C22×D3—368no—
283C22×D3—368no—
284C3×Q8—24no—
285C3×Q8—24no—
286C3×Q8—21no—
287C3×Q8—21no—
288C3×Q8—44no—
289C3×Q8—44no—
290C3×D4—38no—
291C3×D4—64no—
292C3×D4—92no—
293C3×D4—181no—
294C2×C3×C4—28no—
295C2×C3×C4—24no—
296C2×C3×C4—21no—
297C2×C3×C4—24no—
298C2×C3×C4C2×D3214yes—
299C2×C3×C4C22×C3⋊D311yescharacteristic
300#67—184no—
301#67—1816no—
302C2×Dic3—94no—
303C2×Dic3—91no—
304C2×Dic3—184no—
305C2×Dic3—188no—
306C2×Dic3—184no—
307C2×Dic3—181no—
308D12—21no—
309D12—38no—
310D12—64no—
311D12#63812yes—
312C4×D3—184no—
313C4×D3—184no—
314C4×D3—181no—
315C4×D3—181no—
316C4×D3—364no—
317C4×D3—364no—
318C3×C8—64no—
319C3×C8—181no—
320C3⋊C8—21no—
321C3⋊C8—64no—
322C2×C32—24no—
323C2×C32—24no—
324C2×C32—24no—
325C2×C32—122no—
326C2×C32D3×SD1614yes—
327C2×C32D3×SD1611yescharacteristic
328C2×C32#91414yes—
329C2×C32#91411yescharacteristic
330C2×C32#93714yes—
331C2×C32#93711yescharacteristic
332C3⋊D3—42no—
333C3⋊D3—62no—
334C3⋊D3—68no—
335C3⋊D3—128no—
336C3×D3—48no—
337C3×D3—128no—
338C24—541no—
339C2×Q8D3×C3⋊D311yescharacteristic, complemented
340C2×D4—31no—
341C2×D4—91no—
342C2×D4—541no—
343C2×D4—542no—
344C2×D4—542no—
345C22×C4—271no—
346SD16—62no—
347SD16—182no—
348C2×C8—31no—
349C2×C8—91no—
350C4⋊C4—271no—
351C4⋊C4—541no—
352C22⋊C4—541no—
353C22×C3—24no—
354C22×C3—68no—
355C22×C3—124no—
356C22×C3—124no—
357C22×C3—182no—
358C22×C3—361no—
359C22×C3—361no—
360C22×C3#381114yes—
361C22×C3D4×C3⋊D311yescharacteristic
362C2×D3—22no—
363C2×D3—41no—
364C2×D3—41no—
365C2×D3—68no—
366C2×D3—92no—
367C2×D3—98no—
368C2×D3—124no—
369C2×D3—124no—
370C2×D3—188no—
371C2×D3—181no—
372C2×D3—181no—
373C2×D3—184no—
374C2×D3—184no—
375C2×D3—364no—
376C2×D3—364no—
377C2×D3—368no—
378C2×D3—3632no—
379C3×C4—24no—
380C3×C4—24no—
381C3×C4—48no—
382C3×C4—41no—
383C3×C4—44no—
384C3×C4D3×#6714yes—
385C3×C4D3×#6714yes—
386C3×C4D4×C3⋊D311yescharacteristic
387C3×C4C2×#63811yescharacteristic
388Dic3—181no—
389Dic3—184no—
390Dic3—364no—
391Dic3—368no—
392Dic3—361no—
393Dic3—364no—
394C32—24no—
395C32#516914yescomplemented
396C32#516911yescharacteristic, complemented
397C23—61no—
398C23—181no—
399C23—271no—
400C23—544no—
401C23—1082no—
402C23—1082no—
403Q8—21no—
404Q8—21no—
405D4—32no—
406D4—61no—
407D4—92no—
408D4—181no—
409D4—544no—
410C2×C4—21no—
411C2×C4—271no—
412C2×C4—541no—
413C2×C4—541no—
414C2×C4—541no—
415C2×C4C2×D3×C3⋊D311yescharacteristic
416C8—61no—
417C8—181no—
418C2×C3—24no—
419C2×C3—24no—
420C2×C3—24no—
421C2×C3—128no—
422C2×C3—362no—
423C2×C3#6394014yes—
424C2×C3#6394114yes—
425C2×C3#6398314yes—
426C2×C3SD16×C3⋊D311yescharacteristic
427C2×C3#6412311yescharacteristic
428C2×C3#6414611yescharacteristic
429D3—42no—
430D3—128no—
431D3—182no—
432D3—188no—
433D3—368no—
434C22—62no—
435C22—121no—
436C22—121no—
437C22—182no—
438C22—272no—
439C22—361no—
440C22—361no—
441C22—541no—
442C22—541no—
443C22—1088no—
444C22#8903911yescharacteristic
445C4—41no—
446C4—541no—
447C4—1081no—
448C4D3×#63811yescharacteristic
449C4C3⋊D3×#6711yescharacteristic
450C3—24no—
451C3#1059165214yescomplemented
452C3#1059286811yescharacteristic, complemented
453C2—122no—
454C2—362no—
455C2—542no—
456C2#1172357111yescharacteristic
457C2#1172357211yescharacteristic
458C2#1172364311yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#89026C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#450
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×#215C2semidirect
5. #11722034C2semidirect
6. #11722032C2semidirect
7. #11722024C2semidirect
11. C2×C33×Q8C22semidirect
19. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
22. C2×C3×C3⋊(C3⋊C8)C22semidirect
24. C2×C32×C3⋊C8C22semidirect
31. C2×#3755D3semidirect
33. C2×C3×#215D3semidirect
66. C2×C32×Q8C2×D3semidirect
67. C2×C32×Q8C2×D3semidirect
93. C2×C3⋊(C3⋊C8)C2×D3semidirect
97. C2×C3×C3⋊C8C2×D3semidirect
101. C2×C3⋊(C3⋊D3)SD16semidirect
122. C2×#215C3⋊D3semidirect
190. C2×C3×Q8D32semidirect
191. C2×C3×Q8C2×C3⋊D3semidirect
221. C2×C3⋊C8C2×C3⋊D3semidirect
270. C33#450semidirect
339. C2×Q8D3×C3⋊D3semidirect
395. C32#5169semidirect
396. C32#5169semidirect
451. C3#10591652semidirect
452. C3#10592868semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
#117239872
C2×C3×#37552
C2×C32×#2152
#117220342
#117220322
#117220242
#105928683
#105916523