Finite Group Databasebeta

← #13138992

Subgroups — #13138992

Order 1728.

432 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 69 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×#88528C211yesmaximal, characteristic, complemented
2#64110⋊C3C211yesmaximal, characteristic, complemented
3#11721074C211yesmaximal, characteristic, complemented
4#11721068C211yesmaximal, characteristic
5#63646⋊C3C211yesmaximal, characteristic, complemented
6#11720327C211yesmaximal, characteristic, complemented
7#11720325C211yesmaximal, characteristic, complemented
8#10590928—31nomaximal
9#10590925—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12#3828⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14#88528—21no—
15#88528—21no—
16#3748⋊C3C2211yescharacteristic
17#3748⋊C3C2211yescharacteristic
18#3748⋊C3C2211yescharacteristic
19#3743⋊C3—21no—
20C8×(C3⋊D3)⋊C3—21no—
21#88439—21no—
22#88438—21no—
23#88437—21no—
24#88436—21no—
25#64110D311yescharacteristic, complemented
26C2×C3×#204—31no—
27C2×C3×#204—31no—
28C2×C3×#204—61no—
29C3×#901—31no—
30#63833—31no—
31#63830—31no—
32#63827—31no—
33#63823—31no—
34#63646D311yescharacteristic, complemented
35C3×#819—31no—
36#63560—31no—
37#63560—31no—
38#63558—31no—
39#63558—31no—
40C22×#272—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C4⋊#272—21no—
44C4⋊#272—21no—
45#6229—21no—
46#6229—21no—
47#638⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
49C4×(C3⋊D3)⋊C3—21no—
50#634⋊C3D411yescharacteristic
51C8×H(3,𝔽3)D411yescharacteristic
52C8×H(3,𝔽3)D411yescharacteristic
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5177—91no—
57#5141—91no—
58#4898—91no—
59#3828C2×D311yescharacteristic
60C3×#235—31no—
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—61no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67C2×C32×C8—21no—
68C2×C32×C8C2×D311yescharacteristic
69C2×C32×C8C2×D311yescharacteristic
70#3748C2×D311yescharacteristic
71#3748C2×D311yescharacteristic
72#3748C2×D311yescharacteristic
73#3743—21no—
74C8×C3⋊D3—21no—
75C3×#208—31no—
76C3×#208—31no—
77C3×#208—31no—
78C3×#204—62no—
79C3×#204—62no—
80C3×#204—61no—
81C3×#204—61no—
82C3×#204—61no—
83C3×#204—61no—
84C3×ΓL(1,𝔽25)—61no—
85C3×D3×C8—61no—
86#3719—61no—
87#3718—61no—
88#3717—61no—
89#3716—61no—
90#3715—61no—
91#3715—61no—
92#3714—61no—
93#3713—61no—
94C22×H(3,𝔽3)#4111yescharacteristic
95C2×#272—41no—
96C2×#272—81no—
97C2×(C3⋊D3)⋊C3—21no—
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)C2×D411yescharacteristic
100#1084—41no—
101(C3⋊Dic3)⋊C3—21no—
102C2×C3×SD16—91no—
103C3×#131—91no—
104#951—91no—
105#949—91no—
106#942—91no—
107#932—91no—
108#909—91no—
109#908—91no—
110C2×#204—31no—
111C2×#204—31no—
112C2×#204—61no—
113#901—31no—
114#901—91no—
115C3×#108—91no—
116#836—91no—
117#833—91no—
118#824—91no—
119#824—91no—
120#823—91no—
121#823—91no—
122#819—31no—
123#819—91no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C22×C3×D3—121no—
127C2×C32×C4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130#638—21no—
131C4⋊(C3⋊D3)D1211yescharacteristic
132C4×C3⋊D3—21no—
133#634D1211yescharacteristic
134C3×#67—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—121no—
138C3×D12—31no—
139C3×D12—62no—
140C3×D12—61no—
141C3×D12—61no—
142C3×D12—61no—
143C3×D12—61no—
144C3×D12—121no—
145C3×C4×D3—61no—
146C3×Dic6—31no—
147C3×Dic6—62no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—121no—
153C32×C8—21no—
154C32×C8—21no—
155C32×C8D1211yescharacteristic
156C32×C8D1211yescharacteristic
157C32×C8#6711yescharacteristic
158C32×C8#6711yescharacteristic
159C3⋊(C3⋊C8)—21no—
160C3⋊(C3⋊C8)—21no—
161C3⋊(C3⋊C8)—21no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—61no—
165#471—271noSylow
166C2×H(3,𝔽3)—21no—
167C2×H(3,𝔽3)#12111yescharacteristic
168#272—81no—
169(C3⋊D3)⋊C3—41no—
170C3×#41—91no—
171C2×C3×Q8—91no—
172C2×C3×D4—91no—
173#235—31no—
174#235—91no—
175C2×D12—31no—
176C2×D12—31no—
177C2×D12—61no—
178C2×Dic6—31no—
179C2×Dic6—31no—
180C2×Dic6—61no—
181C3×SD16—181no—
182C3×SD16—181no—
183C3×M16—91no—
184C3×M16—91no—
185C3×M16—91no—
186C3×M16—181no—
187C2×C3×C8—31no—
188C2×C3×C8—31no—
189C2×C3×C8—61no—
190C2×C3×C8—181no—
191C2×C3×C8D3211yescharacteristic
192#216—181no—
193#216—181no—
194#215—181no—
195#215—181no—
196#214—181no—
197#214—181no—
198#213—181no—
199#213—181no—
200#208—31no—
201#208—31no—
202#208—31no—
203#208—91no—
204#208—91no—
205#208—91no—
206Dic12—181no—
207D24—181no—
208#204—61no—
209#204—61no—
210#204—61no—
211#204—61no—
212#204—62no—
213#204—62no—
214#204—181no—
215#204—181no—
216ΓL(1,𝔽25)—61no—
217ΓL(1,𝔽25)—181no—
218D3×C8—61no—
219D3×C8—181no—
220C22×C32—21no—
221C22×C32#23511yescharacteristic
222C22×C32#23911yescharacteristic
223C2×C3⋊D3—21no—
224C2×C3×D3—61no—
225C2×C3×D3—122no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—242no—
229C2×C3×D3—241no—
230C2×C3×D3—241no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4C2×D1211yescharacteristic
234C32×C4C2×D1211yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4C2×#6711yescharacteristic
237C3⋊Dic3—21no—
238C3×Dic3—61no—
239C3×Dic3—122no—
240C3×Dic3—121no—
241C3×Dic3—121no—
242#137—271no—
243AGL(1,ℤ/8)—271no—
244C2×SD16—91no—
245#131—91no—
246#108—91no—
247#101—271no—
248#100—271no—
249H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
250C23×C3—181no—
251C22×D3—61no—
252C22×D3—61no—
253C22×D3—121no—
254C3×Q8—91no—
255C3×Q8—181no—
256C3×Q8—181no—
257C3×D4—91no—
258C3×D4—181no—
259C3×D4—181no—
260C3×D4—181no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—61no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4C2×D3211yescharacteristic
267#67—61no—
268#67—181no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—121no—
272D12—31no—
273D12—61no—
274D12—61no—
275D12—61no—
276D12—61no—
277D12—62no—
278D12—91no—
279D12—121no—
280C4×D3—61no—
281C4×D3—181no—
282Dic6—31no—
283Dic6—61no—
284Dic6—61no—
285Dic6—61no—
286Dic6—61no—
287Dic6—62no—
288Dic6—91no—
289Dic6—121no—
290C3×C8—31no—
291C3×C8—31no—
292C3×C8—31no—
293C3×C8—31no—
294C3×C8—61no—
295C3×C8—61no—
296C3×C8—181no—
297C3×C8—181no—
298C3×C8—181no—
299C3×C8#62611yescharacteristic
300C3×C8#62611yescharacteristic
301C3⋊C8—61no—
302C3⋊C8—61no—
303C3⋊C8—61no—
304C3⋊C8—181no—
305C3⋊C8—181no—
306C3⋊C8—181no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32#89511yescharacteristic
312C2×C32#93011yescharacteristic
313C3⋊D3—41no—
314C3×D3—121no—
315C3×D3—242no—
316C3×D3—241no—
317C3×D3—241no—
318#41—91no—
319C2×Q8—91no—
320C2×D4—91no—
321Q16—541no—
322SD16—181no—
323SD16—181no—
324SD16—541no—
325SD16—541no—
326D8—541no—
327M16—91no—
328M16—91no—
329M16—91no—
330M16—181no—
331C2×C8—181no—
332C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—61no—
336C22×C3—181no—
337C22×C3—361no—
338C22×C3—721no—
339C22×C3#379911yescharacteristic
340C2×D3—61no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—122no—
344C2×D3—181no—
345C2×D3—242no—
346C2×D3—241no—
347C2×D3—241no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—181no—
355C3×C4—361no—
356C3×C4D3×D1211yescharacteristic
357C3×C4C2×#62611yescharacteristic
358Dic3—61no—
359Dic3—121no—
360Dic3—121no—
361Dic3—122no—
362Dic3—181no—
363C32—21no—
364C32#489811yescharacteristic, complemented
365C32#514111yescharacteristic, complemented
366C23—181no—
367Q8—91no—
368Q8—181no—
369Q8—181no—
370D4—91no—
371D4—181no—
372D4—181no—
373D4—181no—
374C2×C4—181no—
375C2×C4—181no—
376C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
377C8—181no—
378C8—181no—
379C8—181no—
380C8#620011yescharacteristic
381C8#620011yescharacteristic
382C2×C3—21no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—61no—
386C2×C3—61no—
387C2×C3—62no—
388C2×C3—61no—
389C2×C3—361no—
390C2×C3—721no—
391C2×C3#6390911yescharacteristic
392D3—121no—
393D3—241no—
394D3—241no—
395D3—242no—
396D3—361no—
397C22—181no—
398C22—361no—
399C22—721no—
400C22#8865211yescharacteristic
401C4—181no—
402C4—361no—
403C4#8865411yescharacteristic
404C4C2×#620011yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3#1059093111yescharacteristic
409C2—21no—
410C2—361no—
411C2—721no—
412C2#1172115611yescharacteristic
subgroupG/H
centerC2#11721156
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3#63909
prime radical#11721068C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#471
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88528C2semidirect
2. #64110⋊C3C2semidirect
3. #11721074C2semidirect
5. #63646⋊C3C2semidirect
6. #11720327C2semidirect
7. #11720325C2semidirect
25. #64110D3semidirect
34. #63646D3semidirect
249. H(3,𝔽3)#471semidirect
364. C32#4898semidirect
365. C32#5141semidirect

Maximal subgroups

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

subgroupindex
C2×#885282
#64110⋊C32
#117210742
#117210682
#63646⋊C32
#117203272
#117203252
#105909283
#105909253