Finite Group Databasebeta

← #13138977

Subgroups — #13138977

Order 1728.

488 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 79 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
2C2×#3743⋊C3C211yesmaximal, characteristic, complemented
3#11721156C211yesmaximal, characteristic
4#11721107C211yesmaximal, characteristic, complemented
5#63645⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720324C211yesmaximal, characteristic, complemented
8#10590845—31nomaximal
9#10590833—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#6200—21no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15#88528—22no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#3743⋊C3—22no—
20#88451—21no—
21#88450—21no—
22#88448—21no—
23C2×#3743D311yescharacteristic, complemented
24C2×C3×#204—31no—
25C2×C3×#204—31no—
26C2×C3×#204—61no—
27C2×C3×ΓL(1,𝔽25)—31no—
28#63916—31no—
29#63908—31no—
30#63871—31no—
31#63856—31no—
32#63645D311yescharacteristic, complemented
33C3×#818—31no—
34#63566—31no—
35#63565—31no—
36#63563—31no—
37#63562—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C4⋊#272—21no—
43C4⋊#272—21no—
44#6229—21no—
45#6229—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4×(C3⋊D3)⋊C3—21no—
50C4×(C3⋊D3)⋊C3—21no—
51#6200—42no—
52C8×H(3,𝔽3)D412yescomplemented
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5174—91no—
55#5134—91no—
56#4886—91no—
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×C3×D12—31no—
59C2×C3×D12—31no—
60C2×C3×D12—61no—
61C2×C3×C4×D3—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—61no—
65C2×#626—61no—
66C2×#625—61no—
67C2×C32×C8—21no—
68C2×C32×C8C2×D311yescharacteristic
69C2×C32×C8C2×D311yescharacteristic
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73#3743—22no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C2×C3×C3⋊C8—31no—
77C3×#204—64no—
78C3×#204—62no—
79C3×#204—62no—
80C3×ΓL(1,𝔽25)—62no—
81#3724—61no—
82Dic3⋊Dic3—61no—
83#3722—61no—
84(C2×D3)⋊Dic3—61no—
85(C2×D3)⋊Dic3—61no—
86(C2×D3)⋊Dic3—61no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—41no—
89C2×#272—41no—
90C2×#272—41no—
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—21no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96#1084—41no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×SD16—91no—
101C2×C3×M16—91no—
102#946—91no—
103#938—91no—
104#930—91no—
105#924—91no—
106C2×#204—31no—
107C2×#204—31no—
108C2×#204—61no—
109C2×ΓL(1,𝔽25)—31no—
110C2×ΓL(1,𝔽25)—91no—
111#897—91no—
112#895—91no—
113C3×#107—91no—
114#835—91no—
115#832—91no—
116#821—91no—
117#818—31no—
118#818—91no—
119#816—91no—
120#810—91no—
121#809—91no—
122C22×C3⋊D3D1211yescharacteristic
123C22×C3×D3—31no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C22×C3×D3—121no—
127C2×C32×C4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3D1211yescharacteristic
133C4×C3⋊D3—21no—
134C4×C3⋊D3—21no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—121no—
141C3×D12—62no—
142C3×D12—61no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—121no—
147C3×C4×D3—61no—
148C3×C4×D3—61no—
149C3×Dic6—62no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C3×Dic6—121no—
155#626—122no—
156#625—122no—
157C32×C8—22no—
158C32×C8D1212yescomplemented
159C32×C8#6712yescomplemented
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—61no—
162#468—271noSylow
163C2×H(3,𝔽3)#12111yescharacteristic
164C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
165C2×H(3,𝔽3)#13711yescharacteristic
166#272—81no—
167(C3⋊D3)⋊C3—41no—
168C2×C3×Q8—91no—
169C2×C3×D4—91no—
170C22×C3×C4—91no—
171C2×#67—181no—
172C2×#67—181no—
173C2×D12—31no—
174C2×D12—31no—
175C2×D12—61no—
176C2×D12—181no—
177C2×C4×D3—31no—
178C2×C4×D3—91no—
179C2×Dic6—31no—
180C2×Dic6—31no—
181C2×Dic6—61no—
182C3×SD16—182no—
183C3×M16—182no—
184C2×C3×C8—31no—
185C2×C3×C8—31no—
186C2×C3×C8—61no—
187C2×C3×C8—91no—
188C2×C3×C8D3211yescharacteristic
189C3×C4⋊C4—91no—
190C3×C4⋊C4—91no—
191#217—181no—
192#217—181no—
193(C2×D3)⋊C4—181no—
194(C2×D3)⋊C4—181no—
195(C2×D3)⋊C4—181no—
196(C2×D3)⋊C4—181no—
197C4⋊Dic3—31no—
198C4⋊Dic3—31no—
199C4⋊Dic3—91no—
200C4⋊Dic3—91no—
201C4⋊Dic3—181no—
202Dic3⋊C4—181no—
203Dic3⋊C4—181no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—91no—
206#204—62no—
207#204—62no—
208#204—64no—
209ΓL(1,𝔽25)—62no—
210ΓL(1,𝔽25)—182no—
211C22×C32—21no—
212C22×C32C2×D1211yescharacteristic
213C22×C32D3×D411yescharacteristic
214C2×C3⋊D3—21no—
215C2×C3⋊D3—21no—
216C2×C3⋊D3—21no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—122no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C2×C3×D3—241no—
228C2×C3×D3—241no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4C2×D1211yescharacteristic
232C32×C4#23511yescharacteristic
233C32×C4#23911yescharacteristic
234C32×C4C2×#6711yescharacteristic
235C3⋊Dic3—21no—
236C3⋊Dic3—41no—
237C3⋊Dic3—41no—
238C3×Dic3—61no—
239C3×Dic3—122no—
240C3×Dic3—121no—
241C3×Dic3—121no—
242C3×Dic3—121no—
243C3×Dic3—121no—
244C2×SD16—91no—
245C2×M16—91no—
246C22⋊Q8—271no—
247#121—271no—
248#107—91no—
249#103—271no—
250#102—271no—
251H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
252C23×C3—91no—
253C23×C3—181no—
254C22×D3—31no—
255C22×D3—61no—
256C22×D3—61no—
257C22×D3—91no—
258C22×D3—121no—
259C3×Q8—181no—
260C3×Q8—181no—
261C3×D4—181no—
262C3×D4—181no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—61no—
266C2×C3×C4—91no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4C2×D3211yescharacteristic
273#67—362no—
274#67—362no—
275C2×Dic3—31no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—91no—
281C2×Dic3—121no—
282C2×Dic3—181no—
283C2×Dic3—181no—
284D12—61no—
285D12—61no—
286D12—61no—
287D12—61no—
288D12—62no—
289D12—121no—
290D12—362no—
291C4×D3—61no—
292C4×D3—61no—
293C4×D3—181no—
294C4×D3—181no—
295Dic6—61no—
296Dic6—61no—
297Dic6—61no—
298Dic6—61no—
299Dic6—62no—
300Dic6—121no—
301C3×C8—32no—
302C3×C8—32no—
303C3×C8—62no—
304C3×C8—181no—
305C3×C8#62612yes—
306C3⋊C8—61no—
307C3⋊C8—181no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32#89511yescharacteristic
312C2×C32#90811yescharacteristic
313C2×C32#90911yescharacteristic
314C2×C32#91111yescharacteristic
315C2×C32#91811yescharacteristic
316C2×C32#93011yescharacteristic
317C3⋊D3—41no—
318C3×D3—121no—
319C3×D3—242no—
320C3×D3—241no—
321C3×D3—241no—
322C2×Q8—91no—
323C2×D4—91no—
324C2×D4—541no—
325C22×C4—91no—
326SD16—182no—
327M16—182no—
328C2×C8—91no—
329C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
330C4⋊C4—91no—
331C4⋊C4—91no—
332C4⋊C4—541no—
333C22⋊C4—541no—
334C22⋊C4—541no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—61no—
338C22×C3—181no—
339C22×C3—181no—
340C22×C3—181no—
341C22×C3—361no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3D3×D1211yescharacteristic
345C2×D3—61no—
346C2×D3—61no—
347C2×D3—61no—
348C2×D3—121no—
349C2×D3—121no—
350C2×D3—121no—
351C2×D3—121no—
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—122no—
355C2×D3—181no—
356C2×D3—181no—
357C2×D3—181no—
358C2×D3—241no—
359C2×D3—241no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—61no—
365C3×C4—61no—
366C3×C4—181no—
367C3×C4—361no—
368C3×C4—361no—
369C3×C4—361no—
370C3×C4#379911yescharacteristic
371C3×C4C2×#62611yescharacteristic
372Dic3—61no—
373Dic3—121no—
374Dic3—121no—
375Dic3—122no—
376Dic3—121no—
377Dic3—121no—
378Dic3—181no—
379Dic3—361no—
380Dic3—361no—
381C32—21no—
382C32#488611yescharacteristic, complemented
383C32#513411yescharacteristic, complemented
384C23—91no—
385C23—181no—
386Q8—181no—
387Q8—181no—
388D4—181no—
389D4—181no—
390D4—1082no—
391C2×C4—91no—
392C2×C4—181no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
398C8—181no—
399C8#620012yescomplemented
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—61no—
407C2×C3—61no—
408C2×C3—61no—
409C2×C3—361no—
410C2×C3—721no—
411C2×C3#6379511yescharacteristic
412C2×C3#6380111yescharacteristic
413C2×C3#6390911yescharacteristic
414D3—121no—
415D3—241no—
416D3—241no—
417D3—242no—
418D3—361no—
419C22—181no—
420C22—181no—
421C22—181no—
422C22—361no—
423C22—361no—
424C22—361no—
425C22#8865411yescharacteristic
426C4—181no—
427C4—361no—
428C4—361no—
429C4—361no—
430C4#8865211yescharacteristic
431C4C2×#620011yescharacteristic
432C3—31no—
433C3—31no—
434C3—61no—
435C3#1059083811yescharacteristic
436C2—361no—
437C2—721no—
438C2#1172084911yescharacteristic
439C2#1172085311yescharacteristic
440C2#1172115611yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radical#11721156C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#468
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88528C2semidirect
2. C2×#3743⋊C3C2semidirect
4. #11721107C2semidirect
5. #63645⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720324C2semidirect
23. C2×#3743D3semidirect
32. #63645D3semidirect
52. C8×H(3,𝔽3)D4semidirect
158. C32×C8D12semidirect
159. C32×C8#67semidirect
251. H(3,𝔽3)#468semidirect
382. C32#4886semidirect
383. C32#5134semidirect
399. C8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#885282
C2×#3743⋊C32
#117211562
#117211072
#63645⋊C32
#117203262
#117203242
#105908453
#105908333