Finite Group Databasebeta

← #10592875⋊C3 #13143368

Subgroups — #10592875⋊C3 #13143368

Order 1728.

572 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 100 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×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64156⋊C3C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C211yesmaximal, characteristic, complemented
4#64149⋊C3C211yesmaximal, characteristic, complemented
5C2×#3753⋊C3C211yesmaximal, characteristic, complemented
6C2×#3745⋊C3C211yesmaximal, characteristic, complemented
7#63642⋊C3C211yesmaximal, characteristic, complemented
8#10592875C311yesmaximal, characteristic, complemented
9C3×#5176—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—22no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15SD16×H(3,𝔽3)—22no—
16C2×C8×H(3,𝔽3)C2211yescharacteristic
17#3755⋊C3—22no—
18#3753⋊C3—22no—
19#3752⋊C3—22no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3745⋊C3—22no—
23C2×C32×SD16—21no—
24C2×C32×SD16D311yescharacteristic, complemented
25C2×C32×SD16C2×C311yescharacteristic, complemented
26#64156C2×C311yescharacteristic, complemented
27C2×#3755C2×C311yescharacteristic, complemented
28#64149C2×C311yescharacteristic, complemented
29C2×#3753C2×C311yescharacteristic, complemented
30C2×#3745C2×C311yescharacteristic, complemented
31C3×#947—31no—
32C2×C3×#215—31no—
33C3×#940—31no—
34C2×C3×#213—31no—
35C2×C3×D24—31no—
36#63642C2×C311yescharacteristic, complemented
37C3×#815—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3—21no—
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)—21no—
44D4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47#638⋊C3—44no—
48C2×(C3⋊Dic3)⋊C3D412yes—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53C8×H(3,𝔽3)D412yescomplemented
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55C3×#496—91no—
56#5176—31no—
57#5176—91no—
58C2×C32×Q8—21no—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C32×Q8C22×C311yescharacteristic
61C2×C32×D4—21no—
62C2×C32×D4C2×D311yescharacteristic
63C2×C32×D4C22×C311yescharacteristic
64C2×#638—22no—
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
67C2×C3×#67—62no—
68C2×C3×D12—31no—
69C2×C3×D12—31no—
70C32×SD16—24no—
71C32×SD16—22no—
72C32×SD16—22no—
73C2×C32×C8—21no—
74C2×C32×C8C2×D311yescharacteristic
75C2×C32×C8C22×C311yescharacteristic
76#3755—22no—
77#3753—22no—
78#3752—22no—
79C4×C3⋊Dic3C22×C311yescharacteristic
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81#3745—22no—
82C3×#215—62no—
83C3×#213—62no—
84C3×(C2×D3)⋊C4—62no—
85C3×C4×Dic3—31no—
86C2×C3×C3⋊C8—31no—
87C3×D24—62no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—42no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—42no—
94C2×(C3⋊D3)⋊C3—42no—
95C4×H(3,𝔽3)—41no—
96C4×H(3,𝔽3)C2×D411yescharacteristic
97C4×H(3,𝔽3)C2×D411yescharacteristic
98(C3⋊Dic3)⋊C3—22no—
99C2×C3×SD16—31no—
100C2×C3×SD16—31no—
101C2×C3×SD16—61no—
102C2×C3×SD16—91no—
103C2×C3×SD16C3×D311yescharacteristic
104C2×C3×D8—91no—
105C2×C3×D8—91no—
106C3×#127—91no—
107C3×#124—91no—
108#947—31no—
109#947—91no—
110C2×#215—31no—
111C2×#215—91no—
112#940—31no—
113#940—91no—
114C2×#213—31no—
115C2×#213—91no—
116C2×D24—31no—
117C2×D24—91no—
118C3×#97—91no—
119#815—31no—
120#815—91no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—41no—
124C22×C3⋊D3—21no—
125C22×C3⋊D3—21no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C32×Q8—22no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—21no—
133C32×Q8—41no—
134C32×D4—22no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—21no—
138C32×D4—21no—
139C32×D4—41no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—41no—
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C23×C311yescharacteristic
146#638—44no—
147C2×C3⋊Dic3C3×D412yes—
148C4⋊(C3⋊D3)—21no—
149C4⋊(C3⋊D3)—21no—
150C4⋊(C3⋊D3)—21no—
151C4⋊(C3⋊D3)—21no—
152C3×#67—124no—
153C2×C3×Dic3—32no—
154C3×D12—61no—
155C3×D12—61no—
156C3×D12—61no—
157C3×D12—61no—
158C32×C8—22no—
159C32×C8#6712yescomplemented
160C32×C8C3×D412yescomplemented
161C3⋊(C3⋊C8)C3×D412yescomplemented
162C3×C3⋊C8—32no—
163#496—91noSylow
164C2×H(3,𝔽3)—81no—
165C2×H(3,𝔽3)#12711yescharacteristic
166C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
167(C3⋊D3)⋊C3—81no—
168(C3⋊D3)⋊C3—81no—
169C2×C3×Q8—31no—
170C2×C3×Q8—31no—
171C2×C3×Q8—61no—
172C2×C3×Q8C2×C3×D311yescharacteristic
173C2×C3×D4—31no—
174C2×C3×D4—31no—
175C2×C3×D4—61no—
176C2×C3×D4—91no—
177C2×C3×D4—91no—
178C2×C3×D4—182no—
179C2×C3×D4C2×C3×D311yescharacteristic
180C2×#67—62no—
181C2×#67—182no—
182C2×D12—31no—
183C2×D12—31no—
184C2×D12—91no—
185C2×D12—91no—
186C3×SD16—22no—
187C3×SD16—62no—
188C3×SD16—62no—
189C3×SD16—64no—
190C3×SD16—182no—
191C3×D8—182no—
192C3×D8—182no—
193C2×C3×C8—31no—
194C2×C3×C8—31no—
195C2×C3×C8—61no—
196C2×C3×C8—91no—
197C2×C3×C8C2×C3×D311yescharacteristic
198C3×C22⋊C4—182no—
199C3×C42—91no—
200#215—62no—
201#215—182no—
202#213—62no—
203#213—182no—
204(C2×D3)⋊C4—62no—
205(C2×D3)⋊C4—182no—
206C4×Dic3—31no—
207C4×Dic3—91no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—91no—
210D24—62no—
211D24—182no—
212C22×C32—21no—
213C22×C32—42no—
214C22×C32—42no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—42no—
218C22×C32—82no—
219C22×C32D3×D411yescharacteristic
220C22×C32C2×C3×D411yescharacteristic
221C2×C3⋊D3—41no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3—42no—
224C2×C3⋊D3—42no—
225C2×C3×D3—122no—
226C2×C3×D3—122no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—42no—
232C32×C4—41no—
233C32×C4—41no—
234C32×C4D3×D411yescharacteristic
235C32×C4C2×#6711yescharacteristic
236C32×C4C2×C3×D411yescharacteristic
237C32×C4C2×C3×D411yescharacteristic
238C3⋊Dic3—22no—
239C3×Dic3—62no—
240C2×SD16—91no—
241C2×SD16(C3⋊D3)⋊C311yescharacteristic, complemented
242C2×D8—91no—
243C2×D8—91no—
244#127—91no—
245#124—91no—
246#97—91no—
247H(3,𝔽3)#49611yescharacteristic, complemented, Sylow
248C23×C3—21no—
249C23×C3—61no—
250C23×C3—61no—
251C23×C3—121no—
252C23×C3—181no—
253C23×C3—181no—
254C22×D3—61no—
255C22×D3—61no—
256C22×D3—181no—
257C22×D3—181no—
258C3×Q8—21no—
259C3×Q8—21no—
260C3×Q8—61no—
261C3×Q8—61no—
262C3×Q8—61no—
263C3×Q8—61no—
264C3×Q8—62no—
265C3×Q8—121no—
266C3×D4—21no—
267C3×D4—21no—
268C3×D4—61no—
269C3×D4—61no—
270C3×D4—61no—
271C3×D4—61no—
272C3×D4—62no—
273C3×D4—121no—
274C3×D4—181no—
275C3×D4—181no—
276C3×D4—181no—
277C3×D4—181no—
278C3×D4—364no—
279C2×C3×C4—21no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—92no—
286C2×C3×C4—121no—
287C2×C3×C4C22×C3×D311yescharacteristic
288#67—124no—
289#67—364no—
290C2×Dic3—32no—
291C2×Dic3—92no—
292D12—61no—
293D12—61no—
294D12—61no—
295D12—61no—
296D12—181no—
297D12—181no—
298D12—181no—
299D12—181no—
300C3×C8—32no—
301C3×C8—32no—
302C3×C8—62no—
303C3×C8—92no—
304C3×C8C3×#6712yes—
305C3⋊C8—32no—
306C3⋊C8—92no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32—82no—
310C2×C32—81no—
311C2×C32—81no—
312C2×C32#91412yes—
313C2×C32#94011yescharacteristic
314C2×C32C3×#12711yescharacteristic
315C2×C32C3×AGL(1,ℤ/8)12yes—
316C3⋊D3—81no—
317C3⋊D3—81no—
318C3×D3—241no—
319C3×D3—241no—
320C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
321C2×D4—91no—
322C2×D4—91no—
323C2×D4—182no—
324C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
325SD16—22no—
326SD16—182no—
327D8—182no—
328D8—182no—
329C2×C8—91no—
330C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
331C22⋊C4—182no—
332C42—91no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—42no—
336C22×C3—41no—
337C22×C3—61no—
338C22×C3—122no—
339C22×C3—121no—
340C22×C3—121no—
341C22×C3—122no—
342C22×C3—122no—
343C22×C3—242no—
344C22×C3—361no—
345C22×C3—361no—
346C22×C3—362no—
347C22×C3—362no—
348C22×C3C3×D3×D411yescharacteristic
349C2×D3—122no—
350C2×D3—122no—
351C2×D3—121no—
352C2×D3—121no—
353C2×D3—361no—
354C2×D3—361no—
355C2×D3—362no—
356C2×D3—362no—
357C3×C4—31no—
358C3×C4—31no—
359C3×C4—31no—
360C3×C4—31no—
361C3×C4—41no—
362C3×C4—61no—
363C3×C4—61no—
364C3×C4—121no—
365C3×C4—121no—
366C3×C4—122no—
367C3×C4—182no—
368C3×C4C3×D3×D411yescharacteristic
369C3×C4C2×C3×#6711yescharacteristic
370Dic3—62no—
371Dic3—182no—
372C32—21no—
373C32#517611yescharacteristic, complemented
374C32C3×#49611yescharacteristic, complemented
375C23—21no—
376C23—181no—
377C23—181no—
378Q8—21no—
379Q8—21no—
380D4—21no—
381D4—21no—
382D4—181no—
383D4—181no—
384D4—181no—
385D4—181no—
386D4—364no—
387C2×C4—21no—
388C2×C4—92no—
389C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
390C8—92no—
391C8#638⋊C312yescomplemented
392C2×C3—32no—
393C2×C3—32no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—62no—
397C2×C3—61no—
398C2×C3—81no—
399C2×C3—241no—
400C2×C3—241no—
401C2×C3—242no—
402C2×C3—721no—
403C2×C3—721no—
404C2×C3C3×#91412yes—
405C2×C3C3×#94011yescharacteristic
406D3—241no—
407D3—241no—
408D3—721no—
409D3—721no—
410C22—42no—
411C22—41no—
412C22—362no—
413C22—362no—
414C22—361no—
415C22—361no—
416C22D4×(C3⋊D3)⋊C311yescharacteristic
417C4—41no—
418C4—182no—
419C4D4×(C3⋊D3)⋊C311yescharacteristic
420C4C2×#638⋊C311yescharacteristic
421C3—31no—
422C3—31no—
423C3—61no—
424C3C3×#517611yescharacteristic
425C2—81no—
426C2—721no—
427C2—721no—
428C2#64123⋊C312yes—
429C2#64149⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×SD16×H(3,𝔽3)C2
socleC22×C3C3×D3×D4
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#496
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×SD16×H(3,𝔽3)C2semidirect
2. #64156⋊C3C2semidirect
3. C2×#3755⋊C3C2semidirect
4. #64149⋊C3C2semidirect
5. C2×#3753⋊C3C2semidirect
6. C2×#3745⋊C3C2semidirect
7. #63642⋊C3C2semidirect
8. #10592875C3semidirect
24. C2×C32×SD16D3semidirect
25. C2×C32×SD16C2×C3semidirect
26. #64156C2×C3semidirect
27. C2×#3755C2×C3semidirect
28. #64149C2×C3semidirect
29. C2×#3753C2×C3semidirect
30. C2×#3745C2×C3semidirect
36. #63642C2×C3semidirect
53. C8×H(3,𝔽3)D4semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
159. C32×C8#67semidirect
160. C32×C8C3×D4semidirect
161. C3⋊(C3⋊C8)C3×D4semidirect
241. C2×SD16(C3⋊D3)⋊C3semidirect
247. H(3,𝔽3)#496semidirect
373. C32#5176semidirect
374. C32C3×#496semidirect
391. C8#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×H(3,𝔽3)2
#64156⋊C32
C2×#3755⋊C32
#64149⋊C32
C2×#3753⋊C32
C2×#3745⋊C32
#63642⋊C32
#105928753
C3×#51763