Finite Group Databasebeta

← #13140574

Subgroups — #13140574

Order 1728.

514 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721848C211yesmaximal, characteristic
2C2×#3754⋊C3C211yesmaximal, characteristic, complemented
3#11721094C211yesmaximal, characteristic, complemented
4C2×#88435C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591502—31nomaximal
9#10591462—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12C2×C4×#272C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14#88552—21no—
15C4⋊#1084C2211yescharacteristic
16#3754⋊C3—22no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88451—21no—
21#88448—21no—
22#88435—22no—
23C2×#3754D311yescharacteristic, complemented
24C3×#938—31no—
25C3×#938—31no—
26C3×#938—61no—
27C2×C3×#214—31no—
28#63857—31no—
29#63856—31no—
30C2×#3711—31no—
31C2×#3711—31no—
32#63659D311yescharacteristic, complemented
33C3×#832—31no—
34#63576—31no—
35#63575—31no—
36#63567—31no—
37#63565—31no—
38C23×H(3,𝔽3)—21no—
39C22×#272D411yescharacteristic
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43#6235—42no—
44C2×#1084—21no—
45C2×#1084D411yescharacteristic
46C4×#272—21no—
47C4×#272—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54#5174—91no—
55#5042—91no—
56#4780—91no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×C3×#67—61no—
62C2×C3×#67—61no—
63C2×C3×#67—62no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—31no—
66C2×C3×C4×D3—61no—
67C2×C3×Dic6—31no—
68#3754—22no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×#217—61no—
73C3×#217—61no—
74C3×#217—62no—
75C3×#214—62no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—61no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82#3724—61no—
83Dic3⋊Dic3—61no—
84(C2×D3)⋊Dic3—61no—
85(C2×D3)⋊Dic3—61no—
86#3711—62no—
87#3711—62no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×#272—21no—
93C2×#272—21no—
94C2×#272—21no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97#1084—21no—
98#1084—41no—
99(C3⋊Dic3)⋊C3—41no—
100(C3⋊Dic3)⋊C3—41no—
101C2×C3×SD16—91no—
102C3×#121—91no—
103#946—91no—
104#938—31no—
105#938—31no—
106#938—61no—
107C2×#214—31no—
108C2×#214—91no—
109#924—91no—
110C2×#208—91no—
111C2×ΓL(1,𝔽25)—91no—
112C2×ΓL(1,𝔽25)—91no—
113#897—91no—
114C3×#102—91no—
115#835—91no—
116#832—31no—
117#832—91no—
118#818—91no—
119#816—91no—
120#810—91no—
121#807—91no—
122#807—91no—
123C23×C32—22no—
124C23×C32—21no—
125C23×C32—21no—
126C22×C3×D3—31no—
127C22×C3×D3—31no—
128C22×C3×D3—61no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—41no—
134C32×D4—41no—
135C2×C32×C4—21no—
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3—21no—
140#634—21no—
141#634—21no—
142C3×#67—124no—
143C3×#67—122no—
144C3×#67—122no—
145C2×C3×Dic3—31no—
146C2×C3×Dic3—31no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C3×C4×D3—61no—
154C3×C4×D3—61no—
155C3×C4×D3—61no—
156C3×C4×D3—61no—
157C3×C4×D3—121no—
158C3×C4×D3—121no—
159C3×Dic6—61no—
160C3×Dic6—61no—
161C3⋊(C3⋊C8)—21no—
162C3⋊(C3⋊C8)#6712yescomplemented
163C3×C3⋊C8—32no—
164C3×C3⋊C8—61no—
165#468—271noSylow
166C2×H(3,𝔽3)—81no—
167C2×H(3,𝔽3)#12111yescharacteristic
168C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
169C2×H(3,𝔽3)#13711yescharacteristic
170#272—41no—
171C2×C3×Q8—91no—
172C2×C3×D4—31no—
173C2×C3×D4—31no—
174C2×C3×D4—61no—
175C2×C3×D4—181no—
176C2×C3×D4D3211yescharacteristic
177C22×C3×C4—91no—
178C2×#67—61no—
179C2×#67—61no—
180C2×#67—62no—
181C2×C4×D3—31no—
182C2×C4×D3—31no—
183C2×C4×D3—61no—
184C2×Dic6—31no—
185C2×Dic6—91no—
186C3×SD16—182no—
187C2×C3×C8—91no—
188C2×C3×C8—91no—
189C3×C4⋊C4—91no—
190C3×C4⋊C4—91no—
191C3×C22⋊C4—181no—
192#217—61no—
193#217—61no—
194#217—62no—
195#217—181no—
196#214—62no—
197#214—182no—
198(C2×D3)⋊C4—181no—
199(C2×D3)⋊C4—181no—
200C4⋊Dic3—31no—
201C4⋊Dic3—31no—
202C4⋊Dic3—31no—
203C4⋊Dic3—61no—
204C4⋊Dic3—91no—
205C4⋊Dic3—181no—
206Dic3⋊C4—181no—
207Dic3⋊C4—181no—
208#208—182no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—91no—
212C2×C3⋊C8—91no—
213ΓL(1,𝔽25)—182no—
214ΓL(1,𝔽25)—182no—
215C22×C32—21no—
216C22×C32—42no—
217C22×C32—42no—
218C22×C32—42no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32D3×D411yescharacteristic
226C22×C32D3×D411yescharacteristic
227C2×C3×D3—62no—
228C2×C3×D3—61no—
229C2×C3×D3—61no—
230C2×C3×D3—61no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4#23511yescharacteristic
239C32×C4D3×D411yescharacteristic
240C32×C4#23711yescharacteristic
241C32×C4C2×#6711yescharacteristic
242C3⋊Dic3—41no—
243C3⋊Dic3—41no—
244C3×Dic3—61no—
245C3×Dic3—61no—
246C3×Dic3—121no—
247C3×Dic3—122no—
248C3×Dic3—121no—
249C3×Dic3—121no—
250C3×Dic3—121no—
251C3×Dic3—121no—
252C2×SD16—91no—
253C2×M16—271no—
254C22⋊Q8—271no—
255#121—91no—
256#107—271no—
257#103—271no—
258#102—91no—
259H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
260C23×C3—21no—
261C23×C3—61no—
262C23×C3—61no—
263C23×C3—62no—
264C23×C3—91no—
265C22×D3—31no—
266C22×D3—31no—
267C22×D3—61no—
268C3×Q8—181no—
269C3×Q8—181no—
270C3×D4—21no—
271C3×D4—21no—
272C3×D4—61no—
273C3×D4—61no—
274C3×D4—61no—
275C3×D4—61no—
276C3×D4—121no—
277C3×D4—121no—
278C3×D4—362no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—61no—
282C2×C3×C4—91no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4—181no—
287C2×C3×C4—181no—
288C2×C3×C4C2×D3211yescharacteristic
289#67—122no—
290#67—122no—
291#67—124no—
292C2×Dic3—31no—
293C2×Dic3—31no—
294C2×Dic3—61no—
295C2×Dic3—61no—
296C2×Dic3—61no—
297C2×Dic3—62no—
298C2×Dic3—61no—
299C2×Dic3—61no—
300C2×Dic3—181no—
301C2×Dic3—181no—
302C4×D3—61no—
303C4×D3—61no—
304C4×D3—61no—
305C4×D3—61no—
306C4×D3—121no—
307C4×D3—121no—
308Dic6—61no—
309Dic6—61no—
310Dic6—181no—
311Dic6—181no—
312C3×C8—92no—
313C3×C8—181no—
314C3⋊C8—32no—
315C3⋊C8—61no—
316C3⋊C8—92no—
317C3⋊C8—181no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32—82no—
322C2×C32—81no—
323C2×C32—81no—
324C2×C32#88411yescharacteristic
325C2×C32#91111yescharacteristic
326C2×C32#91411yescharacteristic
327C2×C32#91511yescharacteristic
328C2×C32#91511yescharacteristic
329C2×C32#93811yescharacteristic
330C3×D3—122no—
331C3×D3—121no—
332C3×D3—121no—
333C2×Q8—91no—
334C2×D4—181no—
335C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
336C22×C4—91no—
337SD16—182no—
338M16—542no—
339C2×C8—91no—
340C2×C8—91no—
341C4⋊C4—91no—
342C4⋊C4—91no—
343C4⋊C4—541no—
344C22⋊C4—181no—
345C22⋊C4—541no—
346C22×C3—31no—
347C22×C3—31no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—61no—
352C22×C3—121no—
353C22×C3—121no—
354C22×C3—121no—
355C22×C3—121no—
356C22×C3—121no—
357C22×C3—121no—
358C22×C3—122no—
359C22×C3—122no—
360C22×C3—122no—
361C22×C3—181no—
362C22×C3—181no—
363C22×C3—181no—
364C22×C3#381111yescharacteristic
365C2×D3—61no—
366C2×D3—61no—
367C2×D3—61no—
368C2×D3—61no—
369C2×D3—61no—
370C2×D3—61no—
371C2×D3—62no—
372C2×D3—121no—
373C2×D3—121no—
374C3×C4—31no—
375C3×C4—31no—
376C3×C4—31no—
377C3×C4—31no—
378C3×C4—61no—
379C3×C4—61no—
380C3×C4—181no—
381C3×C4—361no—
382C3×C4—361no—
383C3×C4—361no—
384C3×C4#380411yescharacteristic
385C3×C4D3×#6711yescharacteristic
386Dic3—61no—
387Dic3—61no—
388Dic3—121no—
389Dic3—121no—
390Dic3—121no—
391Dic3—122no—
392Dic3—121no—
393Dic3—121no—
394Dic3—361no—
395Dic3—361no—
396C32—21no—
397C32#478011yescharacteristic, complemented
398C32#517411yescharacteristic, complemented
399C23—21no—
400C23—91no—
401Q8—181no—
402Q8—181no—
403D4—21no—
404D4—21no—
405D4—362no—
406C2×C4—91no—
407C2×C4—181no—
408C2×C4—181no—
409C2×C4—181no—
410C2×C4—181no—
411C2×C4—181no—
412C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
413C8—92no—
414C8—181no—
415C2×C3—31no—
416C2×C3—31no—
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—61no—
422C2×C3—61no—
423C2×C3—61no—
424C2×C3—81no—
425C2×C3—241no—
426C2×C3—241no—
427C2×C3—242no—
428C2×C3—361no—
429C2×C3#6393111yescharacteristic
430C2×C3#6393711yescharacteristic
431C2×C3#6395811yescharacteristic
432D3—121no—
433D3—121no—
434D3—122no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—181no—
439C22—181no—
440C22—181no—
441C22#8867711yescharacteristic
442C4—181no—
443C4—361no—
444C4—361no—
445C4—361no—
446C4#8867111yescharacteristic
447C4#8867611yescharacteristic
448C3—31no—
449C3—31no—
450C3—61no—
451C3#1059146411yescharacteristic
452C2—81no—
453C2—361no—
454C2#1172118911yescharacteristic
455C2#1172119311yescharacteristic
456C2#1172130311yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721848C2
hypercenterC2×D4((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×D4

Complements

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

NH ≅ G/Ntype
2. C2×#3754⋊C3C2semidirect
3. #11721094C2semidirect
4. C2×#88435C2semidirect
5. #63659⋊C3C2semidirect
6. #11720309C2semidirect
7. #11720306C2semidirect
23. C2×#3754D3semidirect
32. #63659D3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
162. C3⋊(C3⋊C8)#67semidirect
259. H(3,𝔽3)#468semidirect
397. C32#4780semidirect
398. C32#5174semidirect

Maximal subgroups

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

subgroupindex
#117218482
C2×#3754⋊C32
#117210942
C2×#884352
#63659⋊C32
#117203092
#117203062
#105915023
#105914623