Finite Group Databasebeta

← #13140600

Subgroups — #13140600

Order 1728.

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

normal subgroups only

This group has 510 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
1C2×#88545C211yesmaximal, characteristic
2#64145⋊C3C211yesmaximal, characteristic, complemented
3#11721149C211yesmaximal, characteristic, complemented
4C2×#88437C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720313C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10591434—31nomaximal
9#10591431—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#6199—21no—
14#88545—22no—
15C2×#6180C2211yescharacteristic
16#3757⋊C3—21no—
17#3757⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88451—21no—
22#88437—22no—
23#64145D311yescharacteristic, complemented
24C3×#936—31no—
25C2×C3×#213—31no—
26C2×C3×#213—31no—
27C2×C3×#213—61no—
28#63898—31no—
29Dic3×D12—31no—
30C2×#3716—31no—
31C2×#3715—31no—
32#63659D311yescharacteristic, complemented
33C3×#832—31no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63567—31no—
37#63566—31no—
38C23×H(3,𝔽3)—21no—
39C22×#272—21no—
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C4⋊#272—21no—
44C4⋊#272D412yes—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49#634⋊C3—21no—
50#634⋊C3—21no—
51#6199—42no—
52#6180—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5153—91no—
55#5057—91no—
56#4787—91no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—61no—
64C2×C3×Dic6—31no—
65C2×D3×Dic3—61no—
66C2×D3×Dic3—61no—
67#3757—21no—
68#3757—21no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×#217—61no—
73C3×#217—61no—
74C3×#213—62no—
75C3×#213—62no—
76C3×#213—64no—
77C3×C4⋊Dic3—31no—
78C3×C4×Dic3—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—61no—
83(C2×D3)⋊Dic3—61no—
84(C2×D3)⋊Dic3—61no—
85#3716—62no—
86#3715—62no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×#272—22no—
92C2×#272—41no—
93C2×#272—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96(C3⋊Dic3)⋊C3—21no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×D8—91no—
101C3×#124—91no—
102C2×#215—91no—
103#936—31no—
104#936—91no—
105D4×Dic3—91no—
106C2×#214—91no—
107C2×#213—31no—
108C2×#213—31no—
109C2×#213—61no—
110C2×#204—91no—
111#893—91no—
112C4×D12—91no—
113C3×#102—91no—
114#835—91no—
115#832—31no—
116#832—91no—
117#816—91no—
118Dic3⋊C8—91no—
119Dic3⋊C8—91no—
120#810—91no—
121C4⋊(C3⋊C8)—91no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—41no—
125C22×C3×D3—61no—
126C22×C3×D3—61no—
127C22×C3×D3—121no—
128C32×D4—22no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—41no—
134C2×C32×C4—21no—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3#6711yescharacteristic
140C2×C3⋊Dic3#6711yescharacteristic
141#634—21no—
142#634—21no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C3×D12—32no—
148C3×D12—32no—
149C3×D12—62no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—121no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155D3×Dic3—122no—
156D3×Dic3—122no—
157C3⋊(C3⋊C8)—21no—
158C3×C3⋊C8—62no—
159C3×C3⋊C8—61no—
160C3×C3⋊C8—61no—
161C3×C3⋊C8—61no—
162#463—271noSylow
163C2×H(3,𝔽3)—81no—
164C2×H(3,𝔽3)#12111yescharacteristic
165C2×H(3,𝔽3)#13511yescharacteristic
166C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
167#272—42no—
168C2×C3×Q8—91no—
169C2×C3×D4—31no—
170C2×C3×D4—31no—
171C2×C3×D4—61no—
172C2×C3×D4—91no—
173C2×C3×D4D3211yescharacteristic
174C22×Dic3—181no—
175C2×D12—31no—
176C2×D12—31no—
177C2×D12—61no—
178C2×C4×D3—181no—
179C2×C4×D3—181no—
180C2×Dic6—31no—
181C2×Dic6—91no—
182C3×D8—182no—
183C2×C3×C8—91no—
184C2×C3×C8—91no—
185C3×C4⋊C4—91no—
186C3×C22⋊C4—181no—
187C3×C22⋊C4—181no—
188C3×C42—91no—
189#217—61no—
190#217—61no—
191#217—181no—
192#217—181no—
193#217—181no—
194#215—182no—
195#214—182no—
196#213—62no—
197#213—62no—
198#213—64no—
199(C2×D3)⋊C4—181no—
200(C2×D3)⋊C4—181no—
201C4⋊Dic3—31no—
202C4⋊Dic3—91no—
203C4×Dic3—31no—
204C4×Dic3—91no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—61no—
209C2×C3⋊C8—91no—
210#204—182no—
211C22×C32—21no—
212C22×C32—42no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—81no—
220C22×C32—81no—
221C22×C32D3×D411yescharacteristic
222C22×C32C2×#6711yescharacteristic
223C2×C3×D3—62no—
224C2×C3×D3—62no—
225C2×C3×D3—122no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—241no—
231C2×C3×D3—241no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4#23511yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4D3×D411yescharacteristic
237C32×C4#23711yescharacteristic
238C3⋊Dic3—21no—
239C3⋊Dic3—21no—
240C3⋊Dic3—41no—
241C3⋊Dic3—41no—
242C3×Dic3—61no—
243C3×Dic3—61no—
244C3×Dic3—121no—
245C3×Dic3—121no—
246C2×SD16—271no—
247C2×D8—91no—
248#124—91no—
249C4×D4—271no—
250C4⋊C8—271no—
251#103—271no—
252#102—91no—
253H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
254C23×C3—21no—
255C23×C3—61no—
256C23×C3—61no—
257C23×C3—121no—
258C23×C3—181no—
259C22×D3—61no—
260C22×D3—61no—
261C22×D3—121no—
262C3×Q8—181no—
263C3×Q8—181no—
264C3×D4—21no—
265C3×D4—21no—
266C3×D4—61no—
267C3×D4—61no—
268C3×D4—61no—
269C3×D4—61no—
270C3×D4—62no—
271C3×D4—92no—
272C3×D4—121no—
273C3×D4—181no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—61no—
277C2×C3×C4—91no—
278C2×C3×C4—91no—
279C2×C3×C4—181no—
280C2×C3×C4—181no—
281C2×C3×C4C2×D3211yescharacteristic
282C2×Dic3—31no—
283C2×Dic3—31no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—91no—
287C2×Dic3—91no—
288C2×Dic3—181no—
289C2×Dic3—181no—
290C2×Dic3—362no—
291D12—32no—
292D12—32no—
293D12—62no—
294D12—61no—
295D12—61no—
296D12—121no—
297C4×D3—362no—
298C4×D3—362no—
299Dic6—61no—
300Dic6—61no—
301Dic6—181no—
302Dic6—181no—
303C3×C8—181no—
304C3×C8—181no—
305C3⋊C8—61no—
306C3⋊C8—61no—
307C3⋊C8—62no—
308C3⋊C8—61no—
309C3⋊C8—181no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—82no—
314C2×C32—81no—
315C2×C32—81no—
316C2×C32#88411yescharacteristic
317C2×C32#91111yescharacteristic
318C2×C32#91611yescharacteristic
319C2×C32#93811yescharacteristic
320C2×C32#94911yescharacteristic
321C2×C32#95011yescharacteristic
322C3×D3—122no—
323C3×D3—122no—
324C3×D3—242no—
325C2×Q8—91no—
326C2×D4—91no—
327C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
328C22×C4—541no—
329SD16—542no—
330D8—182no—
331C2×C8—91no—
332C2×C8—91no—
333C4⋊C4—91no—
334C22⋊C4—181no—
335C22⋊C4—181no—
336C22⋊C4—541no—
337C42—91no—
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—61no—
344C22×C3—121no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—121no—
348C22×C3—121no—
349C22×C3—121no—
350C22×C3—122no—
351C22×C3—182no—
352C22×C3—241no—
353C22×C3—241no—
354C22×C3—361no—
355C22×C3—361no—
356C22×C3D3×#6711yescharacteristic
357C2×D3—62no—
358C2×D3—62no—
359C2×D3—122no—
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—121no—
364C2×D3—241no—
365C2×D3—241no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—61no—
371C3×C4—61no—
372C3×C4—181no—
373C3×C4—181no—
374C3×C4—361no—
375C3×C4—361no—
376C3×C4#380411yescharacteristic
377C3×C4#381111yescharacteristic
378Dic3—61no—
379Dic3—61no—
380Dic3—121no—
381Dic3—121no—
382Dic3—181no—
383Dic3—181no—
384Dic3—361no—
385Dic3—361no—
386C32—21no—
387C32#478711yescharacteristic, complemented
388C32#505711yescharacteristic, complemented
389C23—21no—
390C23—181no—
391Q8—181no—
392Q8—181no—
393D4—21no—
394D4—21no—
395D4—92no—
396D4—181no—
397C2×C4—91no—
398C2×C4—91no—
399C2×C4—181no—
400C2×C4—181no—
401C2×C4—1082no—
402C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
403C8—181no—
404C8—181no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—61no—
412C2×C3—61no—
413C2×C3—61no—
414C2×C3—81no—
415C2×C3—241no—
416C2×C3—241no—
417C2×C3—242no—
418C2×C3—362no—
419C2×C3#6392511yescharacteristic
420C2×C3#6393511yescharacteristic
421C2×C3#6395811yescharacteristic
422D3—122no—
423D3—122no—
424D3—242no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—182no—
429C22—361no—
430C22—361no—
431C22#8867611yescharacteristic
432C4—181no—
433C4—181no—
434C4—361no—
435C4—361no—
436C4#8867111yescharacteristic
437C4#8867711yescharacteristic
438C3—31no—
439C3—31no—
440C3—61no—
441C3#1059143211yescharacteristic
442C2—81no—
443C2—362no—
444C2#1172119511yescharacteristic
445C2#1172119711yescharacteristic
446C2#1172130311yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#88545C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#463
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. #64145⋊C3C2semidirect
3. #11721149C2semidirect
4. C2×#88437C2semidirect
5. #63659⋊C3C2semidirect
6. #11720313C2semidirect
7. #11720311C2semidirect
23. #64145D3semidirect
32. #63659D3semidirect
253. H(3,𝔽3)#463semidirect
387. C32#4787semidirect
388. C32#5057semidirect

Maximal subgroups

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

subgroupindex
C2×#885452
#64145⋊C32
#117211492
C2×#884372
#63659⋊C32
#117203132
#117203112
#105914343
#105914313