Finite Group Databasebeta

← #13138871

Subgroups — #13138871

Order 1728.

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

normal subgroups only

This group has 488 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×#88527C211yesmaximal, characteristic, complemented
2#11721159C211yesmaximal, characteristic
3#11721094C211yesmaximal, characteristic, complemented
4C2×#88431C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic, complemented
6#63643⋊C3C211yesmaximal, characteristic, complemented
7#11720315C211yesmaximal, characteristic, complemented
8#10590850—31nomaximal
9#10590828—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#6200—21no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#88527—22no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#88451—21no—
20#88450—21no—
21#88448—21no—
22#88431—22no—
23C2×C3×ΓL(1,𝔽25)—31no—
24C2×C3×ΓL(1,𝔽25)—31no—
25C2×C3×ΓL(1,𝔽25)—61no—
26#63909—31no—
27#63908—31no—
28#63857—31no—
29#63856—31no—
30C2×#3717—31no—
31C2×#3716—31no—
32#63648D311yescharacteristic, complemented
33#63643D311yescharacteristic, complemented
34C3×#821—31no—
35C3×#816—31no—
36#63575—31no—
37#63575—31no—
38C22×#272D411yescharacteristic
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084D411yescharacteristic
42C4×#272—21no—
43C4×#272—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51#6200—42no—
52C8×H(3,𝔽3)—21no—
53#6180D412yescomplemented
54#5134—91no—
55#4812—91no—
56#4780—91no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×#634C2×D311yescharacteristic
59C2×C3×D12—31no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—61no—
63C2×C3×Dic6—31no—
64C2×#626—61no—
65C2×#625—61no—
66C2×C32×C8—21no—
67C2×C32×C8C2×D311yescharacteristic
68C2×C32×C8C2×D311yescharacteristic
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—61no—
76C3×ΓL(1,𝔽25)—62no—
77C3×ΓL(1,𝔽25)—62no—
78C3×ΓL(1,𝔽25)—64no—
79#3724—61no—
80Dic3⋊Dic3—61no—
81#3722—61no—
82(C2×D3)⋊Dic3—61no—
83(C2×D3)⋊Dic3—61no—
84(C2×D3)⋊Dic3—61no—
85#3717—62no—
86#3716—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—21no—
89C2×#272—21no—
90C2×#272—21no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96#1084—21no—
97(C3⋊Dic3)⋊C3—41no—
98(C3⋊Dic3)⋊C3—41no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×M16—91no—
101#946—91no—
102C2×#215—91no—
103#938—91no—
104C2×#214—91no—
105#930—91no—
106#924—91no—
107C2×#204—91no—
108C2×ΓL(1,𝔽25)—31no—
109C2×ΓL(1,𝔽25)—31no—
110C2×ΓL(1,𝔽25)—61no—
111#897—91no—
112#895—91no—
113C3×#103—91no—
114C3×#102—91no—
115#821—31no—
116#821—91no—
117#818—91no—
118#816—31no—
119#816—91no—
120#807—91no—
121#807—91no—
122C22×C3⋊D3—21no—
123C22×C3×D3—31no—
124C22×C3×D3—31no—
125C22×C3×D3—61no—
126C22×C3×D3—61no—
127C2×C32×C4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—21no—
133C4⋊(C3⋊D3)—21no—
134C4⋊(C3⋊D3)—21no—
135#634—21no—
136#634—21no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C3×D12—61no—
144C3×D12—61no—
145C3×C4×D3—62no—
146C3×C4×D3—62no—
147C3×C4×D3—61no—
148C3×C4×D3—61no—
149C3×C4×D3—61no—
150C3×C4×D3—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153#626—122no—
154#625—122no—
155C32×C8—22no—
156C32×C8—21no—
157C32×C8—21no—
158C3×C3⋊C8—32no—
159C3×C3⋊C8—32no—
160C3×C3⋊C8—62no—
161#468—271noSylow
162C2×H(3,𝔽3)#12111yescharacteristic
163C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
164C2×H(3,𝔽3)#13711yescharacteristic
165#272—41no—
166(C3⋊D3)⋊C3—81no—
167C2×C3×Q8—91no—
168C2×C3×D4—91no—
169C22×C3×C4—91no—
170C2×#67—181no—
171C2×#67—181no—
172C2×D12—31no—
173C2×D12—91no—
174C2×D12—181no—
175C2×C4×D3—31no—
176C2×C4×D3—31no—
177C2×C4×D3—61no—
178C2×Dic6—31no—
179C2×Dic6—91no—
180C3×M16—182no—
181C2×C3×C8—31no—
182C2×C3×C8—31no—
183C2×C3×C8—61no—
184C2×C3×C8—91no—
185C2×C3×C8D3211yescharacteristic
186C3×C4⋊C4—91no—
187C3×C4⋊C4—91no—
188#217—181no—
189#217—181no—
190#215—182no—
191#214—182no—
192(C2×D3)⋊C4—181no—
193(C2×D3)⋊C4—181no—
194(C2×D3)⋊C4—181no—
195(C2×D3)⋊C4—181no—
196C4⋊Dic3—31no—
197C4⋊Dic3—31no—
198C4⋊Dic3—91no—
199C4⋊Dic3—91no—
200C4⋊Dic3—181no—
201Dic3⋊C4—181no—
202Dic3⋊C4—181no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—61no—
206#204—182no—
207ΓL(1,𝔽25)—62no—
208ΓL(1,𝔽25)—62no—
209ΓL(1,𝔽25)—64no—
210C22×C32—21no—
211C22×C32D3×D411yescharacteristic
212C22×C32D3×D411yescharacteristic
213C2×C3⋊D3—41no—
214C2×C3⋊D3—41no—
215C2×C3⋊D3—41no—
216C2×C3×D3—62no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—61no—
221C2×C3×D3—61no—
222C2×C3×D3—61no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C32×C4—21no—
229C32×C4—21no—
230C32×C4#23511yescharacteristic
231C32×C4#23511yescharacteristic
232C32×C4D3×D411yescharacteristic
233C32×C4D3×D411yescharacteristic
234C3⋊Dic3—41no—
235C3⋊Dic3—41no—
236C3⋊Dic3—41no—
237C3×Dic3—62no—
238C3×Dic3—61no—
239C3×Dic3—61no—
240C3×Dic3—121no—
241C3×Dic3—121no—
242C3×Dic3—121no—
243C2×SD16—271no—
244C2×M16—91no—
245C22⋊Q8—271no—
246#121—271no—
247#107—271no—
248#103—91no—
249#102—91no—
250H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
251C23×C3—91no—
252C23×C3—181no—
253C22×D3—31no—
254C22×D3—31no—
255C22×D3—61no—
256C22×D3—61no—
257C22×D3—181no—
258C3×Q8—181no—
259C3×Q8—181no—
260C3×D4—181no—
261C3×D4—181no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—61no—
265C2×C3×C4—91no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4C2×D3211yescharacteristic
272#67—362no—
273#67—362no—
274C2×Dic3—31no—
275C2×Dic3—31no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—181no—
281C2×Dic3—181no—
282C2×Dic3—181no—
283D12—61no—
284D12—61no—
285D12—181no—
286D12—181no—
287D12—362no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—61no—
291C4×D3—61no—
292C4×D3—62no—
293C4×D3—62no—
294Dic6—61no—
295Dic6—61no—
296Dic6—181no—
297Dic6—181no—
298C3×C8—21no—
299C3×C8—61no—
300C3×C8—61no—
301C3×C8—62no—
302C3×C8—92no—
303C3⋊C8—32no—
304C3⋊C8—32no—
305C3⋊C8—62no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32#88411yescharacteristic
310C2×C32#88411yescharacteristic
311C2×C32#91111yescharacteristic
312C2×C32#91411yescharacteristic
313C2×C32#91511yescharacteristic
314C2×C32#91811yescharacteristic
315C3⋊D3—81no—
316C3×D3—122no—
317C3×D3—121no—
318C3×D3—121no—
319C3×D3—241no—
320C2×Q8—91no—
321C2×D4—91no—
322C2×D4—541no—
323C22×C4—91no—
324SD16—542no—
325M16—182no—
326C2×C8—91no—
327C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
328C4⋊C4—91no—
329C4⋊C4—91no—
330C4⋊C4—541no—
331C22⋊C4—541no—
332C22⋊C4—541no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—61no—
336C22×C3—181no—
337C22×C3—181no—
338C22×C3—181no—
339C22×C3—361no—
340C22×C3—361no—
341C22×C3—361no—
342C22×C3#380211yescharacteristic
343C2×D3—61no—
344C2×D3—61no—
345C2×D3—61no—
346C2×D3—61no—
347C2×D3—61no—
348C2×D3—61no—
349C2×D3—62no—
350C2×D3—121no—
351C2×D3—121no—
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—361no—
356C2×D3—361no—
357C2×D3—361no—
358C3×C4—31no—
359C3×C4—31no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—61no—
363C3×C4—61no—
364C3×C4—181no—
365C3×C4—361no—
366C3×C4—361no—
367C3×C4—361no—
368C3×C4#379811yescharacteristic
369C3×C4#381111yescharacteristic
370Dic3—61no—
371Dic3—61no—
372Dic3—62no—
373Dic3—121no—
374Dic3—121no—
375Dic3—121no—
376Dic3—361no—
377Dic3—361no—
378Dic3—361no—
379C32—21no—
380C32#478011yescharacteristic, complemented
381C32#481211yescharacteristic, complemented
382C23—91no—
383C23—181no—
384Q8—181no—
385Q8—181no—
386D4—181no—
387D4—181no—
388D4—1082no—
389C2×C4—91no—
390C2×C4—181no—
391C2×C4—181no—
392C2×C4—181no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
396C8—21no—
397C8—92no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—61no—
405C2×C3—61no—
406C2×C3—61no—
407C2×C3—361no—
408C2×C3—721no—
409C2×C3#6379811yescharacteristic
410C2×C3#6381111yescharacteristic
411C2×C3#6391211yescharacteristic
412D3—121no—
413D3—121no—
414D3—122no—
415D3—241no—
416D3—721no—
417C22—181no—
418C22—181no—
419C22—181no—
420C22—361no—
421C22—361no—
422C22—361no—
423C22#8865511yescharacteristic
424C4—181no—
425C4—361no—
426C4—361no—
427C4—361no—
428C4#8865011yescharacteristic
429C4#8867711yescharacteristic
430C3—31no—
431C3—31no—
432C3—61no—
433C3#1059084211yescharacteristic
434C2—361no—
435C2—721no—
436C2#1172083911yescharacteristic
437C2#1172084511yescharacteristic
438C2#1172114211yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radical#11721159C2
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×#88527C2semidirect
3. #11721094C2semidirect
4. C2×#88431C2semidirect
5. #63648⋊C3C2semidirect
6. #63643⋊C3C2semidirect
7. #11720315C2semidirect
32. #63648D3semidirect
33. #63643D3semidirect
53. #6180D4semidirect
250. H(3,𝔽3)#468semidirect
380. C32#4780semidirect
381. C32#4812semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
#117211592
#117210942
C2×#884312
#63648⋊C32
#63643⋊C32
#117203152
#105908503
#105908283