Finite Group Databasebeta

← #13141085

Subgroups — #13141085

Order 1728.

426 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 77 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
1#11721871C211yesmaximal, characteristic
2#11721689C211yesmaximal, characteristic, complemented
3#11721064C211yesmaximal, characteristic, complemented
4#63664⋊C3C211yesmaximal, characteristic, complemented
5#11720328C211yesmaximal, characteristic, complemented
6#11720316C211yesmaximal, characteristic, complemented
7#11720312C211yesmaximal, characteristic, complemented
8#10591676—31nomaximal
9#10591674—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11#88740C2211yescharacteristic
12#88550C411yescharacteristic, complemented
13#88549C411yescharacteristic, complemented
14#88546C411yescharacteristic, complemented
15#88545C411yescharacteristic, complemented
16C2×#6180C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18#3748⋊C3—21no—
19#3748⋊C3C2211yescharacteristic
20#3748⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88435—21no—
24#88434—21no—
25#64157D311yescharacteristic, complemented
26C3×#950—31no—
27C3×#950—31no—
28C3×#950—61no—
29C3×#948—31no—
30#63815—31no—
31#63814—31no—
32#63664D311yescharacteristic, complemented
33C3×#837—31no—
34#63577—31no—
35#63577—31no—
36#63569—31no—
37#63568—31no—
38#63555—31no—
39#63554—31no—
40Q8×H(3,𝔽3)C2×C411yescharacteristic
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)C2×C411yescharacteristic
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#6235—21no—
46C4⋊#272C2×C411yescharacteristic
47C4×#272—21no—
48#6229C2×C411yescharacteristic
49C2×(C3⋊Dic3)⋊C3—21no—
50#6180C2×C411yescharacteristic
51#6180C2×C411yescharacteristic
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
56#5197—91no—
57#5145—91no—
58#4823—91no—
59C32×#41—21no—
60C32×#41C2×D311yescharacteristic
61C32×#41C2×D311yescharacteristic
62C3×#235—31no—
63C3×#235—31no—
64C3×#235—61no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66#3748—21no—
67#3748C2×D311yescharacteristic
68#3748C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—21no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#216—31no—
72C3×#216—31no—
73C3×#216—61no—
74C3×#215—31no—
75C3×#215—31no—
76C3×#215—61no—
77C3×#214—31no—
78C3×#214—31no—
79C3×#214—61no—
80C3×#213—31no—
81C3×#213—31no—
82C3×#213—61no—
83C3×C4×Dic3—31no—
84C3×#208—31no—
85C3×#208—31no—
86C3×#208—61no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—61no—
91C2×C3×C3⋊C8—61no—
92#3711—61no—
93#3711—61no—
94D3×C3⋊C8—61no—
95D3×C3⋊C8—61no—
96C22×H(3,𝔽3)—21no—
97C22×H(3,𝔽3)#4111yescharacteristic
98C2×#272—21no—
99C4×H(3,𝔽3)—21no—
100C4×H(3,𝔽3)C22×C411yescharacteristic
101C4×H(3,𝔽3)C2×D411yescharacteristic
102#1084—21no—
103(C3⋊Dic3)⋊C3—41no—
104C3×#135—91no—
105C3×#131—91no—
106#950—31no—
107#950—31no—
108#950—61no—
109#948—31no—
110#948—91no—
111#948—91no—
112#907—91no—
113#901—91no—
114C3×#104—91no—
115#837—31no—
116#837—91no—
117#837—91no—
118#825—91no—
119#822—91no—
120#822—91no—
121#819—91no—
122#815—91no—
123#815—91no—
124#805—91no—
125#803—91no—
126C32×Q8—21no—
127C32×Q8C4×D311yescharacteristic
128C32×Q8C4×D311yescharacteristic
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—41no—
133C32×D4C4×D311yescharacteristic
134C32×D4C4×D311yescharacteristic
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—41no—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3—21no—
142C3×#67—61no—
143C3×#67—61no—
144C3×#67—121no—
145C2×C3×Dic3—61no—
146C3×D12—31no—
147C3×D12—31no—
148C3×D12—61no—
149C3×C4×D3—61no—
150C3×C4×D3—61no—
151C3×C4×D3—121no—
152C3×Dic6—31no—
153C3×Dic6—31no—
154C3×Dic6—61no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)—21no—
157C3⋊(C3⋊C8)#6711yescharacteristic
158C3⋊(C3⋊C8)#6711yescharacteristic
159C3×C3⋊C8—31no—
160C3×C3⋊C8—31no—
161C3×C3⋊C8—31no—
162C3×C3⋊C8—31no—
163C3×C3⋊C8—31no—
164C3×C3⋊C8—31no—
165C3×C3⋊C8—61no—
166C3×C3⋊C8—61no—
167C3×C3⋊C8—61no—
168C3×C3⋊C8—61no—
169#444—271noSylow
170C2×H(3,𝔽3)—21no—
171C2×H(3,𝔽3)—41no—
172C2×H(3,𝔽3)C4×D411yescharacteristic
173#272—41no—
174C3×#41—31no—
175C3×#41—31no—
176C3×#41—61no—
177C3×#41—91no—
178C3×#41D3211yescharacteristic
179#235—31no—
180#235—31no—
181#235—61no—
182C3×Q16—91no—
183C3×SD16—91no—
184C3×SD16—91no—
185C3×D8—91no—
186C3×M16—91no—
187C3×M16—91no—
188C3×M16—181no—
189C2×C3×C8—91no—
190C2×C3×C8—91no—
191C2×C3×C8—181no—
192C3×C42—91no—
193#216—31no—
194#216—31no—
195#216—61no—
196#215—31no—
197#215—31no—
198#215—61no—
199#214—31no—
200#214—31no—
201#214—61no—
202#213—31no—
203#213—31no—
204#213—61no—
205C4×Dic3—31no—
206C4×Dic3—91no—
207#208—31no—
208#208—31no—
209#208—61no—
210#208—91no—
211#208—91no—
212#208—181no—
213#208—181no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—31no—
216C2×C3⋊C8—31no—
217C2×C3⋊C8—61no—
218C2×C3⋊C8—61no—
219C2×C3⋊C8—91no—
220C2×C3⋊C8—181no—
221C2×C3⋊C8—181no—
222ΓL(1,𝔽25)—181no—
223ΓL(1,𝔽25)—181no—
224D3×C8—181no—
225D3×C8—181no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—41no—
230C22×C32#23511yescharacteristic
231C22×C32#23711yescharacteristic
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C2×C3×D3—121no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—41no—
240C32×C4C2×C4×D311yescharacteristic
241C32×C4C2×C4×D311yescharacteristic
242C32×C4D3×D411yescharacteristic
243C32×C4C2×#6711yescharacteristic
244C3⋊Dic3—41no—
245C3×Dic3—61no—
246C3×Dic3—61no—
247C3×Dic3—121no—
248C3×Dic3—121no—
249#135—91no—
250#131—91no—
251#131—271no—
252#108—271no—
253#104—91no—
254#104—271no—
255#97—271no—
256H(3,𝔽3)#44411yescharacteristic, complemented, Sylow
257C3×Q8—31no—
258C3×Q8—31no—
259C3×Q8—61no—
260C3×Q8—91no—
261C3×Q8#62411yescharacteristic
262C3×D4—21no—
263C3×D4—31no—
264C3×D4—31no—
265C3×D4—61no—
266C3×D4—61no—
267C3×D4—61no—
268C3×D4—91no—
269C3×D4—121no—
270C3×D4—181no—
271C3×D4#62411yescharacteristic
272C2×C3×C4—21no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—121no—
279C2×C3×C4—181no—
280C2×C3×C4—181no—
281C2×C3×C4C2×D3211yescharacteristic
282#67—61no—
283#67—61no—
284#67—121no—
285C2×Dic3—61no—
286C2×Dic3—181no—
287D12—31no—
288D12—31no—
289D12—61no—
290C4×D3—61no—
291C4×D3—61no—
292C4×D3—121no—
293Dic6—31no—
294Dic6—31no—
295Dic6—61no—
296C3×C8—91no—
297C3×C8—91no—
298C3×C8—91no—
299C3×C8—91no—
300C3×C8—181no—
301C3×C8—181no—
302C3⋊C8—31no—
303C3⋊C8—31no—
304C3⋊C8—31no—
305C3⋊C8—31no—
306C3⋊C8—31no—
307C3⋊C8—31no—
308C3⋊C8—61no—
309C3⋊C8—61no—
310C3⋊C8—61no—
311C3⋊C8—61no—
312C3⋊C8—91no—
313C3⋊C8—91no—
314C3⋊C8—181no—
315C3⋊C8—181no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—41no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—81no—
323C2×C32#67⋊C411yescharacteristic
324C2×C32C4×#6711yescharacteristic
325C3×D3—121no—
326C3×D3—121no—
327C3×D3—241no—
328#41—91no—
329#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
330Q16—91no—
331SD16—91no—
332SD16—91no—
333D8—91no—
334M16—91no—
335M16—91no—
336M16—181no—
337M16—541no—
338C2×C8—91no—
339C2×C8—91no—
340C2×C8—181no—
341C2×C8—541no—
342C42—91no—
343C22×C3—21no—
344C22×C3—31no—
345C22×C3—31no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—121no—
350C22×C3—181no—
351C22×C3#380411yescharacteristic
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—121no—
355C3×C4—21no—
356C3×C4—31no—
357C3×C4—31no—
358C3×C4—31no—
359C3×C4—31no—
360C3×C4—61no—
361C3×C4—61no—
362C3×C4—61no—
363C3×C4—61no—
364C3×C4—121no—
365C3×C4—181no—
366C3×C4—361no—
367C3×C4C2×#62411yescharacteristic
368C3×C4D3×#6711yescharacteristic
369Dic3—61no—
370Dic3—61no—
371Dic3—121no—
372Dic3—121no—
373Dic3—361no—
374C32—21no—
375C32#482311yescharacteristic, complemented
376C32#514511yescharacteristic, complemented
377Q8—91no—
378Q8#619911yescharacteristic
379D4—21no—
380D4—91no—
381D4—181no—
382D4#619911yescharacteristic
383C2×C4—21no—
384C2×C4—181no—
385C2×C4—181no—
386C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
387C8—91no—
388C8—91no—
389C8—91no—
390C8—91no—
391C8—181no—
392C8—181no—
393C2×C3—21no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—41no—
397C2×C3—61no—
398C2×C3—61no—
399C2×C3—61no—
400C2×C3—121no—
401C2×C3—121no—
402C2×C3—121no—
403C2×C3—241no—
404C2×C3—361no—
405C2×C3#6395211yescharacteristic
406D3—121no—
407D3—121no—
408D3—241no—
409C22—21no—
410C22—181no—
411C22#8867111yescharacteristic
412C4—21no—
413C4—181no—
414C4—361no—
415C4C2×#619911yescharacteristic
416C4#8867611yescharacteristic
417C3—31no—
418C3—31no—
419C3—61no—
420C3#1059166311yescharacteristic
421C2—21no—
422C2—41no—
423C2—361no—
424C2#1172129311yescharacteristic
subgroupG/H
centerC4#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3#63952
prime radical#11721871C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#444
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #11721689C2semidirect
3. #11721064C2semidirect
4. #63664⋊C3C2semidirect
5. #11720328C2semidirect
6. #11720316C2semidirect
7. #11720312C2semidirect
12. #88550C4semidirect
13. #88549C4semidirect
14. #88546C4semidirect
15. #88545C4semidirect
25. #64157D3semidirect
32. #63664D3semidirect
256. H(3,𝔽3)#444semidirect
375. C32#4823semidirect
376. C32#5145semidirect

Maximal subgroups

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

subgroupindex
#117218712
#117216892
#117210642
#63664⋊C32
#117203282
#117203162
#117203122
#105916763
#105916743