Finite Group Databasebeta

← #13139027

Subgroups — #13139027

Order 1728.

757 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 121 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×C4×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic
2C2×C8×(C3⋊D3)⋊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#10590935—31nomaximal
9D3×(C2×D3)⋊C8—31nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2C411yescharacteristic
11C2×C4×#272C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#6199C411yescharacteristic
15C2×((C3⋊D3)⋊C3)⋊C4C411yescharacteristic
16C2×((C3⋊D3)⋊C3)⋊C4C411yescharacteristic
17C4×((C3⋊D3)⋊C3)⋊C2—22no—
18C4×((C3⋊D3)⋊C3)⋊C2—22no—
19C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
20C2×#6180C2211yescharacteristic, complemented
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
23C8×(C3⋊D3)⋊C3—22no—
24((C3⋊D3)⋊C3)⋊C8—22no—
25C2×C4×D32—31no—
26C2×C4×D32—31no—
27C2×C8×C3⋊D3D311yescharacteristic, complemented
28C2×C3×D3×C8—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×((C3⋊D3)⋊C3)⋊C2—22no—
44C2×((C3⋊D3)⋊C3)⋊C2—22no—
45C2×((C3⋊D3)⋊C3)⋊C2C814yescomplemented
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×#1084C2×C411yescharacteristic
48C4×#272—21no—
49C4×#272—21no—
50C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
51C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
52C4×(C3⋊D3)⋊C3—21no—
53C4×(C3⋊D3)⋊C3—21no—
54C4×(C3⋊D3)⋊C3D412yes—
55C4×(C3⋊D3)⋊C3D412yes—
56#6199—22no—
57((C3⋊D3)⋊C3)⋊C4—22no—
58((C3⋊D3)⋊C3)⋊C4—22no—
59C8×H(3,𝔽3)—21no—
60#6180—21no—
61(C3⋊(C3⋊C8))⋊C3—21no—
62(C3⋊(C3⋊C8))⋊C3—21no—
63C2×(C2×D3)⋊C8—91no—
64D3×C22⋊C8—91no—
65D3×C22⋊C8—91no—
66C22×D32—31no—
67C22×D32—31no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—61no—
75C2×#624—31no—
76C2×#624—31no—
77C2×D3×Dic3—31no—
78C2×D3×Dic3—31no—
79C2×D3×Dic3—31no—
80C2×D3×Dic3—31no—
81C4×D32—62no—
82C4×D32—62no—
83C4×D32—62no—
84C4×D32—62no—
85C2×C32×C8—21no—
86C2×C32×C8C2×D311yescharacteristic, complemented
87C2×C32×C8C2×D311yescharacteristic, complemented
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
90C8×C3⋊D3—22no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—61no—
96C3×D3×C8—62no—
97#3710—62no—
98D3×C3⋊C8—62no—
99C22×H(3,𝔽3)C22×C411yescharacteristic
100C2×#272—21no—
101C2×#272—21no—
102C2×#272C2×C812yescomplemented
103C2×(C3⋊D3)⋊C3—21no—
104C2×(C3⋊D3)⋊C3—21no—
105C2×(C3⋊D3)⋊C3C22⋊C412yes—
106C2×(C3⋊D3)⋊C3C2×C812yes—
107C2×(C3⋊D3)⋊C3C2×C812yescomplemented
108C2×(C3⋊D3)⋊C3M1612yes—
109((C3⋊D3)⋊C3)⋊C2—28no—
110C4×H(3,𝔽3)C2×D411yescharacteristic
111C4×H(3,𝔽3)C2×D411yescharacteristic
112#1084—21no—
113(C3⋊Dic3)⋊C3—21no—
114(C3⋊Dic3)⋊C3C22⋊C412yes—
115C22×C4×D3—91no—
116C22×C4×D3—91no—
117C22×C4×D3—91no—
118C22×C3×C8—91no—
119C22×C3⋊C8—91no—
120C2×D3×C8—31no—
121C2×D3×C8—91no—
122C2×D3×C8—91no—
123C2×D3×C8—91no—
124C3×C22⋊C8—91no—
125C3×C22⋊C8—91no—
126#830—91no—
127#830—91no—
128(C2×D3)⋊C8—31no—
129(C2×D3)⋊C8—31no—
130(C2×D3)⋊C8—31no—
131(C2×D3)⋊C8—61no—
132(C2×D3)⋊C8—91no—
133(C2×D3)⋊C8—91no—
134(C2×D3)⋊C8—91no—
135(C2×D3)⋊C8—91no—
136(C2×D3)⋊C8—91no—
137C22×C3⋊D3C4×D311yescharacteristic
138C22×C3⋊D3C4×D311yescharacteristic
139C22×C3×D3—31no—
140C22×C3×D3—31no—
141C22×C3×D3—31no—
142C22×C3×D3—31no—
143C22×C3×D3—61no—
144C2×D32—34no—
145C2×D32—34no—
146C2×D32—62no—
147C2×D32—62no—
148C2×D32—62no—
149C2×D32—62no—
150C2×C32×C4—21no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C3⋊Dic3C4×D311yescharacteristic
154C2×C3⋊Dic3C4×D311yescharacteristic
155C4×C3⋊D3—21no—
156C4×C3⋊D3—21no—
157C4×C3⋊D3D1212yes—
158C4×C3⋊D3#6712yes—
159C2×C3×Dic3—31no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—31no—
163C2×C3×Dic3—61no—
164C3×C4×D3—32no—
165C3×C4×D3—32no—
166C3×C4×D3—61no—
167C3×C4×D3—61no—
168C3×C4×D3—61no—
169C3×C4×D3—61no—
170C3×C4×D3—61no—
171C3×C4×D3—61no—
172C3×C4×D3—121no—
173C3×C4×D3—121no—
174#624—62no—
175#624—62no—
176D3×Dic3—62no—
177D3×Dic3—62no—
178D3×Dic3—62no—
179D3×Dic3—62no—
180C32×C8—21no—
181C32×C8—21no—
182C32×C8—41no—
183C3⋊(C3⋊C8)—21no—
184C3⋊(C3⋊C8)—21no—
185C3×C3⋊C8—61no—
186C3×C3⋊C8—61no—
187C3×C3⋊C8—61no—
188C3×C3⋊C8—61no—
189C3×C3⋊C8—121no—
190C2×C22⋊C8—271noSylow
191C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
192C2×H(3,𝔽3)C22×C811yescharacteristic
193C2×H(3,𝔽3)C2×M1611yescharacteristic
194#272—22no—
195(C3⋊D3)⋊C3—22no—
196(C3⋊D3)⋊C3C22⋊C814yescomplemented
197C23×D3—91no—
198C23×D3—91no—
199C23×D3—91no—
200C22×C3×C4—91no—
201C22×C3×C4—91no—
202C22×C3×C4—91no—
203C22×Dic3—91no—
204C22×Dic3—91no—
205C22×Dic3—91no—
206C2×C4×D3—31no—
207C2×C4×D3—31no—
208C2×C4×D3—31no—
209C2×C4×D3—31no—
210C2×C4×D3—61no—
211C2×C4×D3—91no—
212C2×C4×D3—91no—
213C2×C4×D3—91no—
214C2×C4×D3—91no—
215C2×C4×D3—91no—
216C2×C4×D3—91no—
217C2×C4×D3—91no—
218C2×C4×D3—91no—
219C2×C4×D3—182no—
220C2×C4×D3—182no—
221C2×C4×D3—182no—
222C2×C4×D3—182no—
223C2×C4×D3—182no—
224C2×C4×D3—182no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—61no—
228C2×C3×C8—91no—
229C2×C3×C8—91no—
230C2×C3×C8—91no—
231C2×C3×C8—182no—
232C2×C3×C8D3211yescharacteristic
233C2×C3⋊C8—31no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—31no—
236C2×C3⋊C8—31no—
237C2×C3⋊C8—61no—
238C2×C3⋊C8—91no—
239C2×C3⋊C8—91no—
240C2×C3⋊C8—182no—
241D3×C8—62no—
242D3×C8—182no—
243D3×C8—182no—
244D3×C8—182no—
245C22×C32—21no—
246C22×C32C2×C4×D311yescharacteristic
247C22×C32C2×C4×D311yescharacteristic
248C2×C3⋊D3—21no—
249C2×C3⋊D3—21no—
250C2×C3⋊D3D3×C812yes—
251C2×C3⋊D3D3×C812yescomplemented
252C2×C3⋊D3ΓL(1,𝔽25)12yes—
253C2×C3⋊D3(C2×D3)⋊C412yes—
254C2×C3×D3—32no—
255C2×C3×D3—32no—
256C2×C3×D3—32no—
257C2×C3×D3—32no—
258C2×C3×D3—32no—
259C2×C3×D3—32no—
260C2×C3×D3—62no—
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C2×C3×D3—61no—
264C2×C3×D3—61no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—121no—
268C2×C3×D3—121no—
269D32—68no—
270D32—68no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4C2×D1211yescharacteristic
274C32×C4D3×D411yescharacteristic
275C32×C4D3×D411yescharacteristic
276C32×C4C2×#6711yescharacteristic
277C3⋊Dic3—21no—
278C3⋊Dic3(C2×D3)⋊C412yes—
279C3×Dic3—32no—
280C3×Dic3—61no—
281C3×Dic3—61no—
282C3×Dic3—61no—
283C3×Dic3—121no—
284C23×C4—271no—
285C22×C8—91no—
286C22×C8—271no—
287C22⋊C8—91no—
288C22⋊C8—91no—
289C22⋊C8—271no—
290C22⋊C8—271no—
291H(3,𝔽3)C2×C22⋊C811yescharacteristic, complemented, Sylow
292C23×C3—91no—
293C23×C3—91no—
294C23×C3—91no—
295C22×D3—31no—
296C22×D3—31no—
297C22×D3—31no—
298C22×D3—31no—
299C22×D3—61no—
300C22×D3—91no—
301C22×D3—91no—
302C22×D3—94no—
303C22×D3—94no—
304C22×D3—94no—
305C22×D3—182no—
306C22×D3—182no—
307C22×D3—182no—
308C22×D3—182no—
309C22×D3—182no—
310C22×D3—182no—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—61no—
314C2×C3×C4—92no—
315C2×C3×C4—92no—
316C2×C3×C4—91no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4—181no—
320C2×C3×C4—181no—
321C2×C3×C4—181no—
322C2×C3×C4—181no—
323C2×C3×C4C2×D3211yescharacteristic
324C2×Dic3—31no—
325C2×Dic3—31no—
326C2×Dic3—31no—
327C2×Dic3—31no—
328C2×Dic3—61no—
329C2×Dic3—91no—
330C2×Dic3—91no—
331C2×Dic3—182no—
332C2×Dic3—182no—
333C2×Dic3—182no—
334C4×D3—32no—
335C4×D3—32no—
336C4×D3—61no—
337C4×D3—61no—
338C4×D3—61no—
339C4×D3—61no—
340C4×D3—61no—
341C4×D3—61no—
342C4×D3—92no—
343C4×D3—92no—
344C4×D3—121no—
345C4×D3—121no—
346C4×D3—181no—
347C4×D3—181no—
348C4×D3—182no—
349C4×D3—182no—
350C4×D3—182no—
351C4×D3—182no—
352C4×D3—182no—
353C4×D3—182no—
354C3×C8—21no—
355C3×C8—61no—
356C3×C8—61no—
357C3×C8—121no—
358C3×C8—181no—
359C3×C8—181no—
360C3×C8—181no—
361C3⋊C8—61no—
362C3⋊C8—61no—
363C3⋊C8—61no—
364C3⋊C8—61no—
365C3⋊C8—121no—
366C3⋊C8—181no—
367C3⋊C8—181no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32D3×C22⋊C411yescharacteristic
372C2×C32C2×D3×C811yescharacteristic
373C2×C32C2×D3×C811yescharacteristic
374C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
375C2×C32D3×M1611yescharacteristic
376C2×C32C2×(C2×D3)⋊C411yescharacteristic
377C3⋊D3—22no—
378C3⋊D3(C2×D3)⋊C814yescomplemented
379C3×D3—34no—
380C3×D3—62no—
381C3×D3—62no—
382C3×D3—62no—
383C3×D3—122no—
384C24—271no—
385C22×C4—91no—
386C22×C4—91no—
387C22×C4—91no—
388C22×C4—271no—
389C22×C4—271no—
390C22×C4—271no—
391C22×C4—542no—
392C22×C4—542no—
393C2×C8—91no—
394C2×C8—91no—
395C2×C8—91no—
396C2×C8—182no—
397C2×C8—542no—
398C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
399C22×C3—31no—
400C22×C3—31no—
401C22×C3—61no—
402C22×C3—92no—
403C22×C3—92no—
404C22×C3—92no—
405C22×C3—92no—
406C22×C3—92no—
407C22×C3—181no—
408C22×C3—181no—
409C22×C3—181no—
410C22×C3—181no—
411C22×C3C4×D3211yescharacteristic
412C2×D3—32no—
413C2×D3—32no—
414C2×D3—32no—
415C2×D3—32no—
416C2×D3—32no—
417C2×D3—32no—
418C2×D3—62no—
419C2×D3—61no—
420C2×D3—61no—
421C2×D3—61no—
422C2×D3—61no—
423C2×D3—61no—
424C2×D3—61no—
425C2×D3—92no—
426C2×D3—92no—
427C2×D3—92no—
428C2×D3—92no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—181no—
432C2×D3—181no—
433C2×D3—188no—
434C2×D3—188no—
435C2×D3—188no—
436C3×C4—31no—
437C3×C4—31no—
438C3×C4—31no—
439C3×C4—31no—
440C3×C4—61no—
441C3×C4—61no—
442C3×C4—92no—
443C3×C4—181no—
444C3×C4—181no—
445C3×C4D3×D1211yescharacteristic
446C3×C4D3×#6711yescharacteristic
447Dic3—32no—
448Dic3—61no—
449Dic3—61no—
450Dic3—61no—
451Dic3—92no—
452Dic3—121no—
453Dic3—181no—
454C32—21no—
455C32D3×C22⋊C811yescharacteristic, complemented
456C32C2×(C2×D3)⋊C811yescharacteristic, complemented
457C23—91no—
458C23—91no—
459C23—91no—
460C23—274no—
461C23—542no—
462C23—542no—
463C2×C4—91no—
464C2×C4—92no—
465C2×C4—92no—
466C2×C4—91no—
467C2×C4—91no—
468C2×C4—181no—
469C2×C4—181no—
470C2×C4—181no—
471C2×C4—181no—
472C2×C4—542no—
473C2×C4—542no—
474C2×C4—542no—
475C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
476C8—21no—
477C8—181no—
478C8—181no—
479C8—181no—
480C2×C3—31no—
481C2×C3—31no—
482C2×C3—31no—
483C2×C3—31no—
484C2×C3—31no—
485C2×C3—31no—
486C2×C3—61no—
487C2×C3—61no—
488C2×C3—61no—
489C2×C3—94no—
490C2×C3—182no—
491C2×C3—182no—
492C2×C3D32×C811yescharacteristic
493C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
494C2×C3D3×(C2×D3)⋊C411yescharacteristic
495D3—34no—
496D3—62no—
497D3—62no—
498D3—62no—
499D3—94no—
500D3—122no—
501D3—182no—
502C22—92no—
503C22—92no—
504C22—92no—
505C22—92no—
506C22—92no—
507C22—181no—
508C22—181no—
509C22—181no—
510C22—181no—
511C22—548no—
512C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
513C4—92no—
514C4—181no—
515C4—181no—
516C4#8865411yescharacteristic
517C4#8867611yescharacteristic
518C3—31no—
519C3—31no—
520C3—61no—
521C3D3×(C2×D3)⋊C811yescharacteristic
522C2—94no—
523C2—182no—
524C2—182no—
525C2C8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
526C2#1172083511yescharacteristic
527C2#1172116011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×((C3⋊D3)⋊C3)⋊C2C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C22⋊C8
LCSG ▷ C2×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×C8×(C3⋊D3)⋊C3C2semidirect
3. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
4. #11720553C2semidirect
5. #63647⋊C3C2semidirect
6. #11720323C2semidirect
7. #11720317C2semidirect
19. C2×C8×H(3,𝔽3)C22semidirect
20. C2×#6180C22semidirect
21. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
22. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
27. C2×C8×C3⋊D3D3semidirect
31. #63647D3semidirect
45. C2×((C3⋊D3)⋊C3)⋊C2C8semidirect
86. C2×C32×C8C2×D3semidirect
87. C2×C32×C8C2×D3semidirect
88. C2×C3⋊(C3⋊C8)C2×D3semidirect
89. C2×C3⋊(C3⋊C8)C2×D3semidirect
102. C2×#272C2×C8semidirect
107. C2×(C3⋊D3)⋊C3C2×C8semidirect
196. (C3⋊D3)⋊C3C22⋊C8semidirect
251. C2×C3⋊D3D3×C8semidirect
291. H(3,𝔽3)C2×C22⋊C8semidirect
378. C3⋊D3(C2×D3)⋊C8semidirect
398. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
455. C32D3×C22⋊C8semidirect
456. C32C2×(C2×D3)⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×((C3⋊D3)⋊C3)⋊C22
C2×C8×(C3⋊D3)⋊C32
C2×((C3⋊D3)⋊C3)⋊C82
#117205532
#63647⋊C32
#117203232
#117203172
#105909353
D3×(C2×D3)⋊C83