Finite Group Databasebeta

← #13139197

Subgroups — #13139197

Order 1728.

697 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 697 conjugacy classes of subgroups — too many to draw. Only its 89 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×#88654C211yesmaximal, characteristic
2M16×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#11721053C211yesmaximal, characteristic, complemented
4#11720560C211yesmaximal, characteristic, complemented
5#63650⋊C3C211yesmaximal, characteristic, complemented
6#11720325C211yesmaximal, characteristic, complemented
7#11720320C211yesmaximal, characteristic, complemented
8#10591001—31nomaximal
9D3×#823—31nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2C412yes—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#6200C412yes—
15#88654—21no—
16#88654—21no—
17#88654—21no—
18#88654—21no—
19M16×H(3,𝔽3)C2211yescharacteristic, complemented
20#88538C2211yescharacteristic, complemented
21#3748⋊C3C2211yescharacteristic, complemented
22#3748⋊C3C2211yescharacteristic, complemented
23#3743⋊C3—21no—
24C8×(C3⋊D3)⋊C3—21no—
25#88430—21no—
26((C3⋊D3)⋊C3)⋊C8—21no—
27C2×#3802—31no—
28C2×D3×D12—31no—
29M16×C3⋊D3D311yescharacteristic, complemented
30C3×D3×M16—31no—
31#63818—31no—
32D3×#208—31no—
33#63650D311yescharacteristic, complemented
34C3×#823—31no—
35C3×#823—31no—
36C3×#823—31no—
37C3×#823—61no—
38#63559—31no—
39#63558—31no—
40#63558—31no—
41#63558—31no—
42C22×#272C2×C412yes—
43C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
44C22×(C3⋊D3)⋊C3C2×C412yes—
45C2×((C3⋊D3)⋊C3)⋊C2—22no—
46C2×((C3⋊D3)⋊C3)⋊C2—22no—
47C2×((C3⋊D3)⋊C3)⋊C2—44no—
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C4⋊#272—21no—
50C4⋊#272—21no—
51C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)—21no—
54C4×(C3⋊D3)⋊C3D411yescharacteristic
55C4×(C3⋊D3)⋊C3D411yescharacteristic
56C4×(C3⋊D3)⋊C3D411yescharacteristic
57C4×(C3⋊D3)⋊C3D411yescharacteristic
58#6200—22no—
59#6200—22no—
60C8×H(3,𝔽3)—21no—
61#6180—21no—
62(C3⋊(C3⋊C8))⋊C3—21no—
63(C3⋊(C3⋊C8))⋊C3—21no—
64C2×#823—91no—
65D3×#100—91no—
66D3×#100—91no—
67C22×D32—32no—
68C22×D32—32no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×C4×D3—31no—
76C2×#626—32no—
77C2×#625—32no—
78#3802—61no—
79#3802—61no—
80#3802—61no—
81#3802—61no—
82D3×D12—61no—
83D3×D12—61no—
84D3×D12—61no—
85D3×D12—61no—
86C32×M16—21no—
87C32×M16C2×D311yescharacteristic, complemented
88C32×M16C2×D311yescharacteristic, complemented
89#3748C2×D311yescharacteristic, complemented
90#3748C2×D311yescharacteristic, complemented
91#3743—21no—
92C8×C3⋊D3—21no—
93C3×#208—31no—
94C3×#208—31no—
95C3×#208—31no—
96C3×#208—31no—
97C3×#208—61no—
98C3×ΓL(1,𝔽25)—61no—
99C3×D3×C8—61no—
100#3712—61no—
101#3711—61no—
102#3710—61no—
103D3×C3⋊C8—61no—
104C22×H(3,𝔽3)C22×C411yescharacteristic
105C2×#272—22no—
106C2×#272—42no—
107C2×(C3⋊D3)⋊C3—22no—
108C2×(C3⋊D3)⋊C3—22no—
109C2×(C3⋊D3)⋊C3—42no—
110C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
111C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
112((C3⋊D3)⋊C3)⋊C2—44no—
113((C3⋊D3)⋊C3)⋊C2—44no—
114C4×H(3,𝔽3)C2×D411yescharacteristic
115C4×H(3,𝔽3)C2×D411yescharacteristic
116(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
117(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
118C2×D3×D4—91no—
119C2×D3×D4—91no—
120C22×D12—91no—
121C2×C3×M16—91no—
122C2×#208—91no—
123D3×M16—31no—
124D3×M16—91no—
125D3×M16—91no—
126D3×M16—91no—
127C3×#100—91no—
128C3×#100—91no—
129#833—91no—
130#833—91no—
131#823—31no—
132#823—31no—
133#823—31no—
134#823—61no—
135#823—91no—
136#823—91no—
137#823—91no—
138#823—91no—
139#823—91no—
140C22×C3⋊D3C4×D311yescharacteristic
141C22×C3⋊D3C4×D312yes—
142C22×C3×D3—31no—
143C22×C3×D3—32no—
144C22×C3×D3—32no—
145C22×C3×D3—32no—
146C22×C3×D3—62no—
147C2×D32—62no—
148C2×D32—62no—
149C2×D32—62no—
150C2×D32—62no—
151C2×D32—124no—
152C2×D32—124no—
153C2×C32×C4—21no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3C4×D311yescharacteristic
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)—21no—
159C4×C3⋊D3D1211yescharacteristic
160C4×C3⋊D3D1211yescharacteristic
161C4×C3⋊D3#6711yescharacteristic
162C4×C3⋊D3#6711yescharacteristic
163C2×C3×Dic3—31no—
164C3×D12—61no—
165C3×D12—61no—
166C3×D12—61no—
167C3×D12—61no—
168C3×D12—61no—
169C3×D12—61no—
170C3×D12—121no—
171C3×D12—121no—
172C3×C4×D3—31no—
173C3×C4×D3—31no—
174C3×C4×D3—31no—
175C3×C4×D3—31no—
176#626—62no—
177#626—62no—
178#625—62no—
179#625—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×#100—271noSylow
191C2×H(3,𝔽3)—21no—
192C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
193#272—42no—
194(C3⋊D3)⋊C3—21no—
195(C3⋊D3)⋊C3—42no—
196(C3⋊D3)⋊C3#10012yescomplemented
197C23×D3—92no—
198C23×D3—92no—
199C23×D3—92no—
200C2×C3×D4—91no—
201C2×C3×D4—91no—
202C22×C3×C4—91no—
203C2×#67—92no—
204C2×#67—92no—
205D3×D4—181no—
206D3×D4—181no—
207D3×D4—181no—
208D3×D4—181no—
209D3×D4—181no—
210D3×D4—181no—
211D3×D4—181no—
212D3×D4—181no—
213C2×D12—31no—
214C2×D12—31no—
215C2×D12—31no—
216C2×D12—61no—
217C2×D12—91no—
218C2×D12—92no—
219C2×D12—181no—
220C2×D12—181no—
221C2×D12—181no—
222C2×D12—181no—
223C2×C4×D3—31no—
224C2×C4×D3—91no—
225C3×M16—31no—
226C3×M16—31no—
227C3×M16—61no—
228C3×M16—91no—
229C3×M16—91no—
230C3×M16—91no—
231C3×M16—181no—
232C3×M16D3211yescharacteristic
233C2×C3×C8—181no—
234#208—31no—
235#208—31no—
236#208—31no—
237#208—31no—
238#208—61no—
239#208—91no—
240#208—91no—
241#208—181no—
242C2×C3⋊C8—181no—
243ΓL(1,𝔽25)—61no—
244ΓL(1,𝔽25)—181no—
245ΓL(1,𝔽25)—181no—
246ΓL(1,𝔽25)—181no—
247D3×C8—61no—
248D3×C8—181no—
249D3×C8—181no—
250D3×C8—181no—
251C22×C32—21no—
252C22×C32C2×C4×D311yescharacteristic
253C22×C32C2×C4×D311yescharacteristic
254C2×C3⋊D3—22no—
255C2×C3⋊D3—22no—
256C2×C3⋊D3—42no—
257C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
258C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
259C2×C3×D3—31no—
260C2×C3×D3—31no—
261C2×C3×D3—62no—
262C2×C3×D3—62no—
263C2×C3×D3—62no—
264C2×C3×D3—62no—
265C2×C3×D3—122no—
266C2×C3×D3—122no—
267C2×C3×D3—122no—
268C2×C3×D3—122no—
269C2×C3×D3—242no—
270D32—124no—
271D32—124no—
272D32—124no—
273D32—124no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4C2×D1211yescharacteristic
277C32×C4D3×D411yescharacteristic
278C32×C4D3×D411yescharacteristic
279C32×C4C2×#6711yescharacteristic
280C3⋊Dic3(C2×D3)⋊C411yescharacteristic
281C3⋊Dic3(C2×D3)⋊C411yescharacteristic
282C3×Dic3—31no—
283C3×Dic3—31no—
284C22×D4—271no—
285C2×M16—91no—
286C2×M16—271no—
287#100—91no—
288#100—91no—
289#100—271no—
290#100—271no—
291H(3,𝔽3)C2×#10011yescharacteristic, complemented, Sylow
292C23×C3—91no—
293C23×C3—92no—
294C23×C3—92no—
295C22×D3—31no—
296C22×D3—32no—
297C22×D3—32no—
298C22×D3—32no—
299C22×D3—62no—
300C22×D3—91no—
301C22×D3—92no—
302C22×D3—182no—
303C22×D3—182no—
304C22×D3—182no—
305C22×D3—182no—
306C22×D3—182no—
307C22×D3—182no—
308C22×D3—364no—
309C22×D3—364no—
310C22×D3—364no—
311C3×D4—181no—
312C3×D4—181no—
313C3×D4—181no—
314C3×D4—181no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—61no—
318C2×C3×C4—91no—
319C2×C3×C4—91no—
320C2×C3×C4—91no—
321C2×C3×C4—91no—
322C2×C3×C4—91no—
323C2×C3×C4C2×D3211yescharacteristic
324#67—182no—
325#67—182no—
326#67—182no—
327#67—182no—
328C2×Dic3—31no—
329C2×Dic3—91no—
330D12—61no—
331D12—61no—
332D12—61no—
333D12—61no—
334D12—61no—
335D12—61no—
336D12—121no—
337D12—121no—
338D12—181no—
339D12—181no—
340D12—182no—
341D12—182no—
342C4×D3—31no—
343C4×D3—31no—
344C4×D3—31no—
345C4×D3—31no—
346C4×D3—91no—
347C4×D3—91no—
348C4×D3—91no—
349C4×D3—91no—
350C3×C8—21no—
351C3×C8—61no—
352C3×C8—61no—
353C3×C8—121no—
354C3×C8—181no—
355C3×C8—181no—
356C3×C8—181no—
357C3⋊C8—61no—
358C3⋊C8—61no—
359C3⋊C8—61no—
360C3⋊C8—61no—
361C3⋊C8—121no—
362C3⋊C8—181no—
363C3⋊C8—181no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—41no—
368C2×C32D3×C22⋊C411yescharacteristic
369C2×C32C2×(C2×D3)⋊C411yescharacteristic
370C3⋊D3—21no—
371C3⋊D3—42no—
372C3⋊D3#82312yescomplemented
373C3×D3—32no—
374C3×D3—61no—
375C3×D3—122no—
376C3×D3—122no—
377C3×D3—122no—
378C3×D3—242no—
379C24—272no—
380C2×D4—91no—
381C2×D4—91no—
382C2×D4—272no—
383C2×D4—541no—
384C2×D4—541no—
385C2×D4—541no—
386C2×D4—541no—
387C22×C4—91no—
388M16—91no—
389M16—91no—
390M16—91no—
391M16—181no—
392M16—541no—
393M16((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
394C2×C8—181no—
395C2×C8—541no—
396C22×C3—31no—
397C22×C3—31no—
398C22×C3—61no—
399C22×C3—91no—
400C22×C3—91no—
401C22×C3—182no—
402C22×C3—182no—
403C22×C3—182no—
404C22×C3—362no—
405C22×C3—362no—
406C22×C3C4×D3211yescharacteristic
407C2×D3—31no—
408C2×D3—31no—
409C2×D3—62no—
410C2×D3—62no—
411C2×D3—62no—
412C2×D3—62no—
413C2×D3—91no—
414C2×D3—91no—
415C2×D3—122no—
416C2×D3—122no—
417C2×D3—122no—
418C2×D3—122no—
419C2×D3—182no—
420C2×D3—182no—
421C2×D3—242no—
422C2×D3—362no—
423C2×D3—364no—
424C2×D3—364no—
425C2×D3—364no—
426C2×D3—364no—
427C2×D3—364no—
428C2×D3—364no—
429C3×C4—31no—
430C3×C4—31no—
431C3×C4—31no—
432C3×C4—31no—
433C3×C4—61no—
434C3×C4—61no—
435C3×C4—91no—
436C3×C4—91no—
437C3×C4D3×D1211yescharacteristic
438C3×C4D3×#6711yescharacteristic
439Dic3—31no—
440Dic3—31no—
441Dic3—91no—
442Dic3—91no—
443C32—21no—
444C32D3×#10011yescharacteristic, complemented
445C32C2×#82311yescharacteristic, complemented
446C23—91no—
447C23—92no—
448C23—92no—
449C23—542no—
450C23—542no—
451C23—1084no—
452D4—181no—
453D4—181no—
454D4—181no—
455D4—181no—
456D4—542no—
457D4—542no—
458C2×C4—91no—
459C2×C4—91no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
464C8—21no—
465C8—181no—
466C8—181no—
467C8—181no—
468C2×C3—21no—
469C2×C3—31no—
470C2×C3—31no—
471C2×C3—61no—
472C2×C3—61no—
473C2×C3—61no—
474C2×C3—92no—
475C2×C3—121no—
476C2×C3—181no—
477C2×C3—362no—
478C2×C3—362no—
479C2×C3D3×(C2×D3)⋊C411yescharacteristic
480D3—32no—
481D3—61no—
482D3—92no—
483D3—122no—
484D3—122no—
485D3—122no—
486D3—181no—
487D3—242no—
488D3—362no—
489C22—91no—
490C22—91no—
491C22—182no—
492C22—182no—
493C22—182no—
494C22—362no—
495C22—362no—
496C22—1084no—
497C22—1084no—
498C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
499C4—91no—
500C4—91no—
501C4#8865411yescharacteristic
502C4#8867611yescharacteristic
503C3—31no—
504C3—31no—
505C3—61no—
506C3D3×#82311yescharacteristic
507C2—21no—
508C2—92no—
509C2—181no—
510C2—362no—
511C2—362no—
512C2#1172116011yescharacteristic
subgroupG/H
centerC2#11721160
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingM16×H(3,𝔽3)C22
socleC2×C3D3×(C2×D3)⋊C4
prime radicalC2×#88654C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterM16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#100
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. M16×(C3⋊D3)⋊C3C2semidirect
3. #11721053C2semidirect
4. #11720560C2semidirect
5. #63650⋊C3C2semidirect
6. #11720325C2semidirect
7. #11720320C2semidirect
19. M16×H(3,𝔽3)C22semidirect
20. #88538C22semidirect
21. #3748⋊C3C22semidirect
22. #3748⋊C3C22semidirect
29. M16×C3⋊D3D3semidirect
33. #63650D3semidirect
87. C32×M16C2×D3semidirect
88. C32×M16C2×D3semidirect
89. #3748C2×D3semidirect
90. #3748C2×D3semidirect
196. (C3⋊D3)⋊C3#100semidirect
291. H(3,𝔽3)C2×#100semidirect
372. C3⋊D3#823semidirect
393. M16((C3⋊D3)⋊C3)⋊C2semidirect
444. C32D3×#100semidirect
445. C32C2×#823semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
M16×(C3⋊D3)⋊C32
#117210532
#117205602
#63650⋊C32
#117203252
#117203202
#105910013
D3×#8233