Finite Group Databasebeta

← #13159422

Subgroups — #13159422

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#11723938C211yesmaximal, characteristic, complemented
3C2×#88796C211yesmaximal, characteristic, complemented
4C2×#88794C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592475—31nomaximal
9#10590597—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×#6308C2211yescharacteristic, complemented
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6290—21no—
16#89026—22no—
17#89026—22no—
18C33×C4⋊C4C2211yescharacteristic, complemented
19#88859—21no—
20C4⋊(C3⋊(C3⋊Dic3))—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
22C2×C32×C3⋊C8C2211yescharacteristic, complemented
23#88796—22no—
24#88794—22no—
25C2×D4×C3⋊D3—31no—
26C2×#3802—34no—
27C2×#3754—31no—
28C2×#3744—31no—
29#64106—31no—
30#64106—34no—
31#64106—64no—
32C2×#3716—34no—
33C2×#3716—34no—
34C32×#102—31no—
35#63647—31no—
36#63636D311yescharacteristic, complemented
37C3×#809—34no—
38C3×#809D314yescomplemented
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—
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C4×C3⋊(C3⋊D3)—21no—
51C4×C3⋊(C3⋊D3)—21no—
52#6308—21no—
53#6308—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#4776—94no—
62#4776—91no—
63C23×C3⋊D3—61no—
64C22×D32—64no—
65C2×C32×D4—31no—
66C2×#638—61no—
67D4×C3⋊D3—62no—
68D4×C3⋊D3—62no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×C4×C3⋊D3—31no—
71C2×C4×C3⋊D3—34no—
72C2×C4×C3⋊D3—64no—
73C2×#634—31no—
74C2×#634—34no—
75C2×#634—64no—
76C2×C3×D12—34no—
77C2×C3×D12C2×D314yes—
78C2×#625—64no—
79#3802—68no—
80#3802—68no—
81C2×C32×C8—31no—
82C32×C4⋊C4—24no—
83C32×C4⋊C4C2×D314yescomplemented
84C32×C4⋊C4C2×D311yescharacteristic, complemented
85#3754—62no—
86#3752—61no—
87#3752—64no—
88#3752—68no—
89C4⋊(C3⋊Dic3)—61no—
90C4⋊(C3⋊Dic3)—64no—
91C4⋊(C3⋊Dic3)—68no—
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
93#3744—62no—
94C2×C3×C3⋊C8—34no—
95C2×C3×C3⋊C8C2×D314yescomplemented
96#3716—68no—
97#3716—68no—
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×#214—94no—
117C2×#214—91no—
118C2×#204—94no—
119C2×#204—91no—
120#897—94no—
121#897—91no—
122#897—184no—
123C3×#102—34no—
124C3×#102—91no—
125(C2×D3)⋊C8—94no—
126(C2×D3)⋊C8—91no—
127#809—34no—
128#809C3⋊D311yescharacteristic, complemented
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×D4—32no—
143C32×D4—61no—
144C2×C32×C4—28no—
145C2×C32×C4—24no—
146C2×C32×C4—21no—
147C2×C32×C4—24no—
148C2×C32×C4C22×D314yes—
149C2×C32×C4C22×D311yescharacteristic
150#638—64no—
151C2×C3⋊Dic3—31no—
152C2×C3⋊Dic3—34no—
153C2×C3⋊Dic3—64no—
154C2×C3⋊Dic3—68no—
155C2×C3⋊Dic3—64no—
156C2×C3⋊Dic3—61no—
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)D1212yes—
159C4×C3⋊D3—61no—
160C4×C3⋊D3—61no—
161C4×C3⋊D3—64no—
162C4×C3⋊D3—64no—
163C4×C3⋊D3—124no—
164C4×C3⋊D3—124no—
165#634—61no—
166#634—61no—
167#634—64no—
168#634—64no—
169#634—124no—
170#634—124no—
171C3×D12—24no—
172C3×D12—38no—
173C3×D12—64no—
174C3×D12D1218yes—
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×D4—34no—
190C2×C3×D4—91no—
191C2×#67—181no—
192C2×#67—184no—
193D3×D4—182no—
194D3×D4—188no—
195D3×D4—188no—
196D3×D4—182no—
197C2×D12—34no—
198C2×D12C2×C3⋊D311yescharacteristic
199C2×C4×D3—94no—
200C2×C4×D3—91no—
201C2×C4×D3—184no—
202C2×Dic6—94no—
203C2×Dic6—91no—
204C2×Dic6—184no—
205C2×C3×C8—34no—
206C2×C3×C8—91no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4D3214yescomplemented
209C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
210#214—182no—
211#214—188no—
212(C2×D3)⋊C4—184no—
213(C2×D3)⋊C4—181no—
214(C2×D3)⋊C4—188no—
215C4⋊Dic3—184no—
216C4⋊Dic3—181no—
217C4⋊Dic3—188no—
218C2×C3⋊C8—34no—
219C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
220#204—188no—
221#204—182no—
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×C4C2×D1214yes—
254C32×C4C2×D1211yescharacteristic
255C32×C4D3×D414yes—
256C32×C4D3×D411yescharacteristic
257C3⋊Dic3—61no—
258C3⋊Dic3—64no—
259C3⋊Dic3—124no—
260C3⋊Dic3—128no—
261C3⋊Dic3—124no—
262C3⋊Dic3—121no—
263C22×D4—271no—
264C2×SD16—271no—
265C2×SD16—271no—
266C22⋊Q8—271no—
267#102—31no—
268#102—91no—
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×D4—38no—
285C3×D4—64no—
286C3×D4—92no—
287C3×D4—181no—
288C2×C3×C4—28no—
289C2×C3×C4—24no—
290C2×C3×C4—21no—
291C2×C3×C4—24no—
292C2×C3×C4C2×D3214yes—
293C2×C3×C4C22×C3⋊D311yescharacteristic
294#67—184no—
295#67—1816no—
296C2×Dic3—94no—
297C2×Dic3—91no—
298C2×Dic3—184no—
299C2×Dic3—188no—
300C2×Dic3—184no—
301C2×Dic3—181no—
302D12—21no—
303D12—38no—
304D12—64no—
305D12C4⋊(C3⋊D3)12yes—
306C4×D3—184no—
307C4×D3—184no—
308C4×D3—181no—
309C4×D3—181no—
310C4×D3—364no—
311C4×D3—364no—
312Dic6—184no—
313Dic6—184no—
314Dic6—181no—
315Dic6—181no—
316Dic6—364no—
317Dic6—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×C32#88214yes—
327C2×C32#88211yescharacteristic
328C2×C32#91114yes—
329C2×C32#91111yescharacteristic
330C2×C32D3×SD1614yes—
331C2×C32D3×SD1611yescharacteristic
332C3⋊D3—42no—
333C3⋊D3—62no—
334C3⋊D3—68no—
335C3⋊D3—128no—
336C3×D3—48no—
337C3×D3—128no—
338C24—541no—
339C2×Q8—271no—
340C2×D4—31no—
341C2×D4—91no—
342C2×D4—541no—
343C2×D4—542no—
344C2×D4—542no—
345C22×C4—271no—
346SD16—542no—
347SD16—542no—
348C2×C8—31no—
349C2×C8—91no—
350C4⋊C4—541no—
351C4⋊C4D3×C3⋊D311yescharacteristic, complemented
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×D1214yes—
385C3×C4D3×D1214yes—
386C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
387C3×C4D4×C3⋊D311yescharacteristic
388Dic3—181no—
389Dic3—184no—
390Dic3—364no—
391Dic3—368no—
392Dic3—361no—
393Dic3—364no—
394C32—24no—
395C32#477614yescomplemented
396C32#477611yescharacteristic, complemented
397C23—61no—
398C23—181no—
399C23—271no—
400C23—544no—
401C23—1082no—
402C23—1082no—
403Q8—541no—
404Q8—541no—
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#6392314yes—
424C2×C3#6392714yes—
425C2×C3#6394014yes—
426C2×C3#6409111yescharacteristic
427C2×C3#6412011yescharacteristic
428C2×C3SD16×C3⋊D311yescharacteristic
429D3—42no—
430D3—128no—
431D3—188no—
432D3—182no—
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×C4⋊(C3⋊D3)11yescharacteristic
449C4C3⋊D3×D1211yescharacteristic
450C3—24no—
451C3#1059059714yescomplemented
452C3#1059247511yescharacteristic, complemented
453C2—122no—
454C2—362no—
455C2—542no—
456C2#1172354211yescharacteristic
457C2#1172354811yescharacteristic
458C2#1172357111yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#89026C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#450
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11723938C2semidirect
3. C2×#88796C2semidirect
4. C2×#88794C2semidirect
5. C3×#63636C2semidirect
6. C32×#809C2semidirect
7. #11722024C2semidirect
12. C2×#6308C22semidirect
18. C33×C4⋊C4C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. C2×C32×C3⋊C8C22semidirect
36. #63636D3semidirect
38. C3×#809D3semidirect
83. C32×C4⋊C4C2×D3semidirect
84. C32×C4⋊C4C2×D3semidirect
92. C2×C3⋊(C3⋊C8)C2×D3semidirect
95. C2×C3×C3⋊C8C2×D3semidirect
101. C2×C3⋊(C3⋊D3)SD16semidirect
128. #809C3⋊D3semidirect
208. C3×C4⋊C4D32semidirect
209. C3×C4⋊C4C2×C3⋊D3semidirect
219. C2×C3⋊C8C2×C3⋊D3semidirect
270. C33#450semidirect
351. C4⋊C4D3×C3⋊D3semidirect
395. C32#4776semidirect
396. C32#4776semidirect
451. C3#10590597semidirect
452. C3#10592475semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
#117239382
C2×#887962
C2×#887942
C3×#636362
C32×#8092
#117220242
#105924753
#105905973