Finite Group Databasebeta

← #13161213

Subgroups — #13161213

Order 1728.

1568 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1568 conjugacy classes of subgroups — too many to draw. Only its 168 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×#89080C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic, complemented
3#11723456C211yesmaximal, characteristic, complemented
4C2×#88795C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592886—31nomaximal
9#10591602—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11#89080—22no—
12#89080—22no—
13C2×C4⋊(C3⋊(C3⋊D3))—21no—
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)—21no—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×#216—22no—
21C2×C32×C3⋊C8C2211yescharacteristic
22#88807—21no—
23#88804—21no—
24#88795—22no—
25C2×#3832—31no—
26C2×#3832—34no—
27C2×#3832—64no—
28C2×C32×Q16—31no—
29#64155—31no—
30C2×#3755—31no—
31C2×C3×#216D314yescomplemented
32#63871—34no—
33C2×#3717—34no—
34#63662D311yescharacteristic, complemented
35#63648—31no—
36#63647—31no—
37C3×#835—34no—
38#63564—34no—
39#63557—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3⋊(C3⋊D3)D411yescharacteristic
42C33×Q8—21no—
43C33×Q8D412yes—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)D411yescharacteristic
47C4⋊(C3⋊(C3⋊D3))—24no—
48C4⋊(C3⋊(C3⋊D3))—21no—
49C4⋊(C3⋊(C3⋊D3))D412yes—
50C4×C3⋊(C3⋊D3)—24no—
51C4×C3⋊(C3⋊D3)—21no—
52C4×C3⋊(C3⋊D3)—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#5187—94no—
60#4808—91no—
61C2×C32×Q8—24no—
62C2×C32×Q8—31no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C2×D314yes—
65#3832—62no—
66#3832—68no—
67#3832—616no—
68#3832—616no—
69#3832—62no—
70#3832—68no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—34no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—64no—
75C2×C4⋊(C3⋊D3)—64no—
76C2×C4⋊(C3⋊D3)—68no—
77C2×C4×C3⋊D3—31no—
78C2×C4×C3⋊D3—34no—
79C2×C4×C3⋊D3—61no—
80C2×C4×C3⋊D3—64no—
81C2×C4×C3⋊D3—64no—
82C2×C4×C3⋊D3—68no—
83C2×C3×Dic6C2×D314yes—
84C32×Q16—62no—
85C2×C32×C8—31no—
86#3755—62no—
87#3752—61no—
88C4⋊(C3⋊Dic3)C2×D311yescharacteristic
89#3750—61no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C3×#216—28no—
92C3×C4⋊Dic3—34no—
93C2×C3×C3⋊C8—34no—
94C2×C3×C3⋊C8C2×D314yes—
95Dic3⋊Dic3—64no—
96#3722—64no—
97#3717—68no—
98C22×C33C2×D411yescharacteristic
99C2×C3⋊(C3⋊D3)—21no—
100C2×C3⋊(C3⋊D3)—22no—
101C2×C3⋊(C3⋊D3)—21no—
102C2×C3⋊(C3⋊D3)—21no—
103C2×C3⋊(C3⋊D3)—41no—
104C2×C3⋊(C3⋊D3)—41no—
105C33×C4—22no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3⋊(C3⋊Dic3)—21no—
109C3×C3⋊Dic3—41no—
110C32×Dic3—41no—
111C2×#239—94no—
112C2×#239—91no—
113C2×#239—184no—
114C2×C3×Q16—34no—
115#946—94no—
116C2×#216C3⋊D311yescharacteristic, complemented
117C2×#215—94no—
118C2×#204—91no—
119#897—91no—
120C3×#103—91no—
121#835—34no—
122#821—94no—
123(C2×D3)⋊C8—94no—
124(C2×D3)⋊C8—91no—
125#809—91no—
126C22×C3⋊D3—31no—
127C22×C3⋊D3—34no—
128C22×C3⋊D3—64no—
129C22×C3⋊D3—68no—
130C22×C3⋊D3—64no—
131C22×C3⋊D3—61no—
132C32×Q8—28no—
133C32×Q8—24no—
134C32×Q8—21no—
135C32×Q8—28no—
136C32×Q8—61no—
137C32×Q8—61no—
138C32×Q8D1212yes—
139C32×Q8#6718yes—
140C2×C32×C4—28no—
141C2×C32×C4—24no—
142C2×C32×C4—21no—
143C2×C32×C4—24no—
144C2×C32×C4—61no—
145C2×C32×C4C22×D314yes—
146C2×C32×C4C22×D311yescharacteristic
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3—31no—
149C2×C3⋊Dic3—34no—
150C2×C3⋊Dic3—64no—
151C4⋊(C3⋊D3)—32no—
152C4⋊(C3⋊D3)—38no—
153C4⋊(C3⋊D3)—68no—
154C4⋊(C3⋊D3)—632no—
155C4⋊(C3⋊D3)—64no—
156C4⋊(C3⋊D3)—616no—
157C4⋊(C3⋊D3)—68no—
158C4⋊(C3⋊D3)—61no—
159C4⋊(C3⋊D3)—64no—
160C4×C3⋊D3—632no—
161C4×C3⋊D3—64no—
162C4×C3⋊D3—616no—
163C4×C3⋊D3—68no—
164C4×C3⋊D3—68no—
165C4×C3⋊D3—61no—
166C4×C3⋊D3—64no—
167C4×C3⋊D3—61no—
168C4×C3⋊D3—64no—
169C2×C3×Dic3—24no—
170C2×C3×Dic3—64no—
171C3×Dic6—24no—
172C3×Dic6—24no—
173C32×C8—61no—
174C3⋊(C3⋊C8)—21no—
175C3×C3⋊C8—24no—
176C3×C3⋊C8—64no—
177#452—271noSylow
178C2×C33#12011yescharacteristic
179C2×C33#13511yescharacteristic
180C2×C33#13711yescharacteristic
181C3⋊(C3⋊D3)—41no—
182C3⋊(C3⋊D3)—42no—
183C2×C3×Q8—24no—
184C2×C3×Q8—34no—
185C2×C3×Q8D3214yes—
186C2×C3×Q8C2×C3⋊D311yescharacteristic
187#239—188no—
188#239—182no—
189#239—1816no—
190#239—1816no—
191#239—188no—
192#239—182no—
193C2×D12—94no—
194C2×D12—91no—
195C2×D12—184no—
196C2×D12—184no—
197C2×D12—181no—
198C2×D12—188no—
199C2×C4×D3—94no—
200C2×C4×D3—91no—
201C2×C4×D3—184no—
202C2×C4×D3—184no—
203C2×C4×D3—181no—
204C2×C4×D3—188no—
205C2×Dic6C2×C3⋊D311yescharacteristic
206C3×Q16—68no—
207C2×C3×C8—34no—
208C2×C3×C8—91no—
209C3×C4⋊C4—91no—
210#216—22no—
211#215—188no—
212(C2×D3)⋊C4—181no—
213(C2×D3)⋊C4—184no—
214C4⋊Dic3—34no—
215C4⋊Dic3—181no—
216Dic3⋊C4—184no—
217C2×C3⋊C8—34no—
218C2×C3⋊C8C2×C3⋊D311yescharacteristic
219#204—182no—
220C22×C32—24no—
221C22×C32D3×D414yes—
222C22×C32D3×D411yescharacteristic
223C2×C3⋊D3—616no—
224C2×C3⋊D3—68no—
225C2×C3⋊D3—62no—
226C2×C3⋊D3—61no—
227C2×C3⋊D3—61no—
228C2×C3⋊D3—61no—
229C2×C3⋊D3—68no—
230C2×C3⋊D3—64no—
231C2×C3⋊D3—64no—
232C2×C3⋊D3—64no—
233C2×C3⋊D3—124no—
234C2×C3⋊D3—124no—
235C2×C3⋊D3—128no—
236C2×C3⋊D3—128no—
237C2×C3⋊D3—121no—
238C2×C3⋊D3—124no—
239C2×C3⋊D3—121no—
240C2×C3⋊D3—124no—
241C32×C4—216no—
242C32×C4—22no—
243C32×C4—28no—
244C32×C4—24no—
245C32×C4—24no—
246C32×C4—121no—
247C32×C4C2×D1211yescharacteristic
248C32×C4D3×D414yes—
249C32×C4D3×D411yescharacteristic
250C32×C4C2×#6714yes—
251C3⋊Dic3—41no—
252C3⋊Dic3—68no—
253C3⋊Dic3—61no—
254C3⋊Dic3—64no—
255C3×Dic3—44no—
256C3×Dic3—124no—
257C2×#41—271no—
258C2×Q16—31no—
259C2×SD16—271no—
260C22⋊Q8—271no—
261#103—91no—
262#102—271no—
263C22⋊C8—271no—
264C33#45211yescharacteristic, complemented, Sylow
265C22×D3—94no—
266C22×D3—91no—
267C22×D3—184no—
268C22×D3—188no—
269C22×D3—184no—
270C22×D3—181no—
271C3×Q8—28no—
272C3×Q8—24no—
273C3×Q8—21no—
274C3×Q8—28no—
275C3×Q8—64no—
276C3×Q8—64no—
277C3×Q8#62618yes—
278C3×Q8#63812yes—
279C2×C3×C4—28no—
280C2×C3×C4—24no—
281C2×C3×C4—21no—
282C2×C3×C4—24no—
283C2×C3×C4—64no—
284C2×C3×C4—181no—
285C2×C3×C4C2×D3214yes—
286C2×C3×C4C22×C3⋊D311yescharacteristic
287C2×Dic3—21no—
288C2×Dic3—64no—
289C2×Dic3—94no—
290C2×Dic3—91no—
291C2×Dic3—184no—
292D12—98no—
293D12—92no—
294D12—188no—
295D12—1832no—
296D12—184no—
297D12—1816no—
298D12—188no—
299D12—184no—
300D12—181no—
301C4×D3—1832no—
302C4×D3—184no—
303C4×D3—1816no—
304C4×D3—188no—
305C4×D3—188no—
306C4×D3—184no—
307C4×D3—184no—
308C4×D3—181no—
309C4×D3—181no—
310Dic6—21no—
311Dic6—21no—
312C3×C8—64no—
313C3×C8—181no—
314C3⋊C8—21no—
315C3⋊C8—64no—
316C2×C32—24no—
317C2×C32—24no—
318C2×C32—24no—
319C2×C32#88211yescharacteristic
320C2×C32#91611yescharacteristic
321C2×C32#91814yes—
322C2×C32#91811yescharacteristic
323C2×C32#91914yes—
324C2×C32#93714yes—
325C3⋊D3—128no—
326C3⋊D3—1216no—
327C3⋊D3—124no—
328C3⋊D3—128no—
329C3⋊D3—121no—
330C3⋊D3—122no—
331#41—542no—
332#41—542no—
333C2×Q8—31no—
334C2×Q8D3×C3⋊D311yescharacteristic
335C2×D4—271no—
336C2×D4—541no—
337C22×C4—271no—
338C22×C4—541no—
339Q16—62no—
340SD16—542no—
341C2×C8—31no—
342C2×C8—91no—
343C4⋊C4—91no—
344C4⋊C4—541no—
345C22⋊C4—541no—
346C22×C3—24no—
347C22×C3#381114yes—
348C22×C3D4×C3⋊D311yescharacteristic
349C2×D3—1816no—
350C2×D3—188no—
351C2×D3—181no—
352C2×D3—181no—
353C2×D3—182no—
354C2×D3—181no—
355C2×D3—184no—
356C2×D3—184no—
357C2×D3—188no—
358C2×D3—184no—
359C2×D3—364no—
360C2×D3—364no—
361C2×D3—368no—
362C2×D3—368no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—364no—
366C2×D3—364no—
367C3×C4—216no—
368C3×C4—22no—
369C3×C4—28no—
370C3×C4—24no—
371C3×C4—24no—
372C3×C4—124no—
373C3×C4—361no—
374C3×C4C2×#62614yes—
375C3×C4#381114yes—
376C3×C4D4×C3⋊D311yescharacteristic
377C3×C4C2×#63811yescharacteristic
378Dic3—41no—
379Dic3—124no—
380Dic3—188no—
381Dic3—181no—
382Dic3—184no—
383C32—24no—
384C32#480811yescharacteristic, complemented
385C32#518714yescomplemented
386C23—271no—
387C23—541no—
388Q8—21no—
389Q8—61no—
390Q8—61no—
391Q8#629112yes—
392D4—272no—
393D4—544no—
394D4—541no—
395C2×C4—21no—
396C2×C4—61no—
397C2×C4—181no—
398C2×C4—271no—
399C2×C4—544no—
400C2×C4—541no—
401C2×C4—541no—
402C2×C4C2×D3×C3⋊D311yescharacteristic
403C8—61no—
404C8—181no—
405C2×C3—24no—
406C2×C3—24no—
407C2×C3—24no—
408C2×C3#6394514yes—
409C2×C3#6395014yes—
410C2×C3#6398114yes—
411C2×C3#6412711yescharacteristic
412C2×C3#6412811yescharacteristic
413C2×C3#6414611yescharacteristic
414D3—368no—
415D3—3616no—
416D3—364no—
417D3—368no—
418D3—361no—
419D3—362no—
420C22—541no—
421C22—541no—
422C22—542no—
423C22—541no—
424C22—1081no—
425C22—1081no—
426C22#8903911yescharacteristic
427C4—22no—
428C4—121no—
429C4—361no—
430C4—541no—
431C4C2×#629111yescharacteristic
432C4#8903911yescharacteristic
433C3—24no—
434C3#1059160214yescomplemented
435C3#1059288611yescharacteristic, complemented
436C2—1081no—
437C2—1082no—
438C2#1172357811yescharacteristic
439C2#1172358911yescharacteristic
440C2#1172363811yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#89080C2
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. C2×C32×#216C2semidirect
3. #11723456C2semidirect
4. C2×#88795C2semidirect
5. C3×#63662C2semidirect
6. #11722036C2semidirect
7. #11722024C2semidirect
31. C2×C3×#216D3semidirect
34. #63662D3semidirect
116. C2×#216C3⋊D3semidirect
264. C33#452semidirect
384. C32#4808semidirect
385. C32#5187semidirect
434. C3#10591602semidirect
435. C3#10592886semidirect

Maximal subgroups

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

subgroupindex
C2×#890802
C2×C32×#2162
#117234562
C2×#887952
C3×#636622
#117220362
#117220242
#105928863
#105916023