Finite Group Databasebeta

← #13139036

Subgroups — #13139036

Order 1728.

603 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 95 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×#88650C211yesmaximal, characteristic
2C2×#3743⋊C3C211yesmaximal, characteristic, complemented
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
4#11720553C211yesmaximal, characteristic, complemented
5#63647⋊C3C211yesmaximal, characteristic, complemented
6#11720323C211yesmaximal, characteristic, complemented
7#11720317C211yesmaximal, characteristic, complemented
8#10590959—31nomaximal
9#10590955—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#6200C411yescharacteristic
14C2×#6200C411yescharacteristic
15C2×#6198C411yescharacteristic
16C2×#6196C411yescharacteristic
17#88650—22no—
18#88650—22no—
19C2×C8×H(3,𝔽3)C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#3743⋊C3—22no—
24((C3⋊D3)⋊C3)⋊C8—22no—
25C2×#3798—31no—
26C2×#3798—31no—
27C2×#3743D311yescharacteristic, complemented
28C2×C3×ΓL(1,𝔽25)—31no—
29C2×#3710—31no—
30C2×D3×C3⋊C8—31no—
31#63647D311yescharacteristic, complemented
32C3×(C2×D3)⋊C8—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×(C2×D3)⋊C8—31no—
35C3×(C2×D3)⋊C8—61no—
36#63557—31no—
37(C2×D3)⋊(C3⋊C8)—31no—
38(C2×D3)⋊(C3⋊C8)—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×#272C2×C411yescharacteristic
41C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
42C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084C2×C411yescharacteristic
45C4×#272—21no—
46C4×#272—21no—
47C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
48C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
49C4×(C3⋊D3)⋊C3—21no—
50C4×(C3⋊D3)⋊C3—21no—
51C4×(C3⋊D3)⋊C3D412yes—
52C4×(C3⋊D3)⋊C3D412yes—
53#6200—22no—
54#6200—22no—
55#6198—22no—
56#6196—22no—
57C8×H(3,𝔽3)—21no—
58#6180—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60(C3⋊(C3⋊C8))⋊C3—21no—
61#5129—91no—
62#4727—91no—
63#4727—91no—
64C2×C4×C3⋊D3C2×D311yescharacteristic
65C2×C4×C3⋊D3C2×D311yescharacteristic
66C2×C3×C4×D3—31no—
67C2×C3×C4×D3—31no—
68C2×C3×C4×D3—31no—
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—61no—
71C2×#627—31no—
72C2×#627—31no—
73C2×#626—31no—
74C2×#626—31no—
75C2×#626—31no—
76C2×#626—31no—
77C2×#625—31no—
78C2×#625—31no—
79#3798—62no—
80#3798—62no—
81#3798—62no—
82#3798—62no—
83C2×C32×C8—21no—
84C2×C32×C8C2×D311yescharacteristic
85C2×C32×C8C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88#3743—22no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—61no—
94C3×ΓL(1,𝔽25)—62no—
95#3710—62no—
96D3×C3⋊C8—62no—
97C22×H(3,𝔽3)C22×C411yescharacteristic
98C2×#272—21no—
99C2×#272—21no—
100C2×#272—21no—
101C2×(C3⋊D3)⋊C3—21no—
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3—21no—
104C2×(C3⋊D3)⋊C3—21no—
105C2×(C3⋊D3)⋊C3—21no—
106C2×(C3⋊D3)⋊C3C22⋊C412yes—
107C4×H(3,𝔽3)C2×D411yescharacteristic
108C4×H(3,𝔽3)C2×D411yescharacteristic
109#1084—21no—
110(C3⋊Dic3)⋊C3—21no—
111(C3⋊Dic3)⋊C3C22⋊C412yes—
112C2×#235—91no—
113C2×#235—91no—
114C2×#235—91no—
115C2×C3×M16—91no—
116C22×C3⋊C8—91no—
117C2×ΓL(1,𝔽25)—31no—
118C2×ΓL(1,𝔽25)—91no—
119C2×D3×C8—91no—
120C2×D3×C8—91no—
121C3×C22⋊C8—91no—
122C3×C22⋊C8—91no—
123#830—91no—
124#830—91no—
125(C2×D3)⋊C8—31no—
126(C2×D3)⋊C8—31no—
127(C2×D3)⋊C8—31no—
128(C2×D3)⋊C8—61no—
129(C2×D3)⋊C8—91no—
130(C2×D3)⋊C8—91no—
131(C2×D3)⋊C8—91no—
132(C2×D3)⋊C8—91no—
133(C2×D3)⋊C8—91no—
134C22×C3⋊D3C4×D311yescharacteristic
135C22×C3⋊D3C4×D311yescharacteristic
136C22×C3×D3—31no—
137C22×C3×D3—31no—
138C22×C3×D3—31no—
139C22×C3×D3—31no—
140C22×C3×D3—61no—
141C2×C32×C4—21no—
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C3⋊Dic3C4×D311yescharacteristic
145C2×C3⋊Dic3C4×D311yescharacteristic
146C4×C3⋊D3—21no—
147C4×C3⋊D3—21no—
148C4×C3⋊D3D1212yes—
149C4×C3⋊D3#6712yes—
150C2×C3×Dic3—31no—
151C2×C3×Dic3—31no—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—31no—
154C2×C3×Dic3—61no—
155C3×C4×D3—32no—
156C3×C4×D3—32no—
157C3×C4×D3—61no—
158C3×C4×D3—61no—
159C3×C4×D3—61no—
160C3×C4×D3—61no—
161C3×C4×D3—61no—
162C3×C4×D3—61no—
163C3×C4×D3—121no—
164C3×C4×D3—121no—
165#627—62no—
166#627—62no—
167#626—62no—
168#626—62no—
169#626—62no—
170#626—62no—
171#625—62no—
172#625—62no—
173C32×C8—21no—
174C32×C8—21no—
175C32×C8—41no—
176C3⋊(C3⋊C8)—21no—
177C3⋊(C3⋊C8)—21no—
178C3×C3⋊C8—61no—
179C3×C3⋊C8—61no—
180C3×C3⋊C8—61no—
181C3×C3⋊C8—61no—
182C3×C3⋊C8—121no—
183#408—271noSylow
184C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
185C2×H(3,𝔽3)#13111yescharacteristic
186C2×H(3,𝔽3)#13111yescharacteristic
187#272—41no—
188(C3⋊D3)⋊C3—22no—
189(C3⋊D3)⋊C3—41no—
190C22×C3×C4—91no—
191C22×C3×C4—91no—
192C22×C3×C4—91no—
193C2×#67—91no—
194C2×#67—91no—
195C2×#67—91no—
196C2×#67—91no—
197C2×#67—91no—
198C2×#67—91no—
199#235—182no—
200#235—182no—
201#235—182no—
202#235—182no—
203#235—182no—
204#235—182no—
205C2×D12—91no—
206C2×D12—91no—
207C2×D12—91no—
208C2×C4×D3—31no—
209C2×C4×D3—31no—
210C2×C4×D3—31no—
211C2×C4×D3—31no—
212C2×C4×D3—61no—
213C2×C4×D3—91no—
214C2×C4×D3—91no—
215C2×Dic6—91no—
216C2×Dic6—91no—
217C2×Dic6—91no—
218C3×M16—182no—
219C2×C3×C8—31no—
220C2×C3×C8—31no—
221C2×C3×C8—61no—
222C2×C3×C8—91no—
223C2×C3×C8—91no—
224C2×C3×C8—91no—
225C2×C3×C8D3211yescharacteristic
226C2×C3⋊C8—31no—
227C2×C3⋊C8—31no—
228C2×C3⋊C8—31no—
229C2×C3⋊C8—31no—
230C2×C3⋊C8—61no—
231C2×C3⋊C8—91no—
232C2×C3⋊C8—91no—
233C2×C3⋊C8—182no—
234ΓL(1,𝔽25)—62no—
235ΓL(1,𝔽25)—182no—
236D3×C8—182no—
237D3×C8—182no—
238C22×C32—21no—
239C22×C32C2×C4×D311yescharacteristic
240C22×C32C2×C4×D311yescharacteristic
241C2×C3⋊D3—21no—
242C2×C3⋊D3—21no—
243C2×C3⋊D3—21no—
244C2×C3⋊D3—21no—
245C2×C3⋊D3—21no—
246C2×C3⋊D3(C2×D3)⋊C412yes—
247C2×C3×D3—32no—
248C2×C3×D3—61no—
249C2×C3×D3—61no—
250C2×C3×D3—62no—
251C2×C3×D3—61no—
252C2×C3×D3—61no—
253C2×C3×D3—61no—
254C2×C3×D3—61no—
255C2×C3×D3—61no—
256C2×C3×D3—61no—
257C2×C3×D3—61no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—121no—
261C2×C3×D3—121no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4C2×D1211yescharacteristic
265C32×C4D3×D411yescharacteristic
266C32×C4D3×D411yescharacteristic
267C32×C4C2×#6711yescharacteristic
268C3⋊Dic3—21no—
269C3⋊Dic3(C2×D3)⋊C412yes—
270C3×Dic3—32no—
271C3×Dic3—61no—
272C3×Dic3—61no—
273C3×Dic3—61no—
274C3×Dic3—121no—
275C2×#41—271no—
276C2×M16—91no—
277C22×C8—271no—
278C22⋊C8—91no—
279C22⋊C8—91no—
280C22⋊C8—271no—
281C22⋊C8—271no—
282H(3,𝔽3)#40811yescharacteristic, complemented, Sylow
283C23×C3—91no—
284C23×C3—91no—
285C23×C3—91no—
286C22×D3—31no—
287C22×D3—31no—
288C22×D3—31no—
289C22×D3—31no—
290C22×D3—61no—
291C22×D3—91no—
292C22×D3—91no—
293C2×C3×C4—31no—
294C2×C3×C4—31no—
295C2×C3×C4—61no—
296C2×C3×C4—92no—
297C2×C3×C4—92no—
298C2×C3×C4—91no—
299C2×C3×C4—91no—
300C2×C3×C4—91no—
301C2×C3×C4—181no—
302C2×C3×C4—181no—
303C2×C3×C4—181no—
304C2×C3×C4—181no—
305C2×C3×C4C2×D3211yescharacteristic
306#67—182no—
307#67—182no—
308#67—182no—
309#67—182no—
310#67—182no—
311#67—182no—
312C2×Dic3—31no—
313C2×Dic3—31no—
314C2×Dic3—31no—
315C2×Dic3—31no—
316C2×Dic3—61no—
317C2×Dic3—91no—
318C2×Dic3—91no—
319D12—182no—
320D12—182no—
321D12—182no—
322C4×D3—32no—
323C4×D3—32no—
324C4×D3—61no—
325C4×D3—61no—
326C4×D3—61no—
327C4×D3—61no—
328C4×D3—61no—
329C4×D3—61no—
330C4×D3—92no—
331C4×D3—92no—
332C4×D3—121no—
333C4×D3—121no—
334C4×D3—181no—
335C4×D3—181no—
336Dic6—182no—
337Dic6—182no—
338Dic6—182no—
339C3×C8—21no—
340C3×C8—61no—
341C3×C8—61no—
342C3×C8—121no—
343C3×C8—181no—
344C3×C8—181no—
345C3×C8—181no—
346C3⋊C8—61no—
347C3⋊C8—61no—
348C3⋊C8—61no—
349C3⋊C8—61no—
350C3⋊C8—121no—
351C3⋊C8—181no—
352C3⋊C8—181no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32D3×C22⋊C411yescharacteristic
357C2×C32#90111yescharacteristic
358C2×C32#90711yescharacteristic
359C2×C32#90711yescharacteristic
360C2×C32#90711yescharacteristic
361C2×C32C2×(C2×D3)⋊C411yescharacteristic
362C3⋊D3—22no—
363C3⋊D3—41no—
364C3×D3—62no—
365C3×D3—122no—
366C3×D3—121no—
367C3×D3—121no—
368C3×D3—121no—
369#41—542no—
370#41—542no—
371C2×Q8—271no—
372C2×D4—271no—
373C2×D4—271no—
374C2×D4—271no—
375C22×C4—91no—
376C22×C4—91no—
377C22×C4—91no—
378M16—182no—
379C2×C8—91no—
380C2×C8—91no—
381C2×C8—91no—
382C2×C8—542no—
383C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
384C22×C3—31no—
385C22×C3—31no—
386C22×C3—61no—
387C22×C3—92no—
388C22×C3—181no—
389C22×C3—181no—
390C22×C3—181no—
391C22×C3—181no—
392C22×C3—181no—
393C22×C3—181no—
394C22×C3—181no—
395C22×C3—181no—
396C22×C3C4×D3211yescharacteristic
397C2×D3—32no—
398C2×D3—61no—
399C2×D3—61no—
400C2×D3—61no—
401C2×D3—61no—
402C2×D3—61no—
403C2×D3—61no—
404C2×D3—61no—
405C2×D3—61no—
406C2×D3—62no—
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—61no—
410C2×D3—92no—
411C2×D3—121no—
412C2×D3—121no—
413C2×D3—181no—
414C2×D3—181no—
415C2×D3—181no—
416C2×D3—181no—
417C2×D3—181no—
418C3×C4—31no—
419C3×C4—31no—
420C3×C4—31no—
421C3×C4—31no—
422C3×C4—61no—
423C3×C4—61no—
424C3×C4—92no—
425C3×C4—181no—
426C3×C4—181no—
427C3×C4D3×D1211yescharacteristic
428C3×C4D3×#6711yescharacteristic
429Dic3—32no—
430Dic3—61no—
431Dic3—61no—
432Dic3—61no—
433Dic3—92no—
434Dic3—121no—
435Dic3—181no—
436C32—21no—
437C32#472711yescharacteristic, complemented
438C32#512911yescharacteristic, complemented
439C23—91no—
440C23—91no—
441C23—91no—
442Q8—542no—
443D4—542no—
444D4—542no—
445D4—542no—
446C2×C4—91no—
447C2×C4—92no—
448C2×C4—92no—
449C2×C4—91no—
450C2×C4—91no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—181no—
455C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
456C8—21no—
457C8—181no—
458C8—181no—
459C8—181no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—31no—
465C2×C3—31no—
466C2×C3—61no—
467C2×C3—61no—
468C2×C3—61no—
469C2×C3—182no—
470C2×C3—361no—
471C2×C3—361no—
472C2×C3#6380411yescharacteristic
473C2×C3#6380511yescharacteristic
474C2×C3D3×(C2×D3)⋊C411yescharacteristic
475D3—62no—
476D3—121no—
477D3—121no—
478D3—122no—
479D3—121no—
480D3—182no—
481D3—361no—
482C22—92no—
483C22—181no—
484C22—181no—
485C22—181no—
486C22—181no—
487C22—181no—
488C22—181no—
489C22—181no—
490C22—181no—
491C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
492C4—92no—
493C4—181no—
494C4—181no—
495C4#8865411yescharacteristic
496C4#8867611yescharacteristic
497C3—31no—
498C3—31no—
499C3—61no—
500C3#1059095511yescharacteristic
501C2—182no—
502C2—361no—
503C2—361no—
504C2#1172084111yescharacteristic
505C2#1172084211yescharacteristic
506C2#1172116011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC2×#88650C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#408
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#3743⋊C3C2semidirect
3. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
4. #11720553C2semidirect
5. #63647⋊C3C2semidirect
6. #11720323C2semidirect
7. #11720317C2semidirect
27. C2×#3743D3semidirect
31. #63647D3semidirect
282. H(3,𝔽3)#408semidirect
437. C32#4727semidirect
438. C32#5129semidirect

Maximal subgroups

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

subgroupindex
C2×#886502
C2×#3743⋊C32
C2×((C3⋊D3)⋊C3)⋊C82
#117205532
#63647⋊C32
#117203232
#117203172
#105909593
#105909553