Finite Group Databasebeta

← #13139488

Subgroups — #13139488

Order 1728.

638 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic
2C2×#88414C211yesmaximal, characteristic, complemented
3C2×(C3⋊D9)⋊C8C211yesmaximal, characteristic, complemented
4C9×#830C211yesmaximal, characteristic, complemented
5C3×#63389C211yesmaximal, characteristic, complemented
6#11720282C211yesmaximal, characteristic, complemented
7#11720282C211yesmaximal, characteristic, complemented
8#10591153—31nomaximal
9#10588409—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×#6233C411yescharacteristic
12C2×#6233C411yescharacteristic
13#88736—22no—
14#88736—22no—
15C2×C4⋊(C3⋊D9)C411yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C4×C3⋊D9C2211yescharacteristic
18C2×#6226C411yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23#88414—22no—
24(C3⋊D9)⋊C8—22no—
25C2×#3828—31no—
26C2×#3712—31no—
27C2×#3710—31no—
28C2×#3697—31no—
29C2×#3697—61no—
30C3×#830—31no—
31C3×#830D311yescharacteristic
32#63557—31no—
33#63557—31no—
34C2×#3663—31no—
35C2×C8×D9—31no—
36C9×C22⋊C8—31no—
37#63389D311yescharacteristic, complemented
38(C2×D9)⋊C8—31no—
39(C2×D9)⋊C8—31no—
40C23×C3×C9C2×C411yescharacteristic
41C22×C3⋊D9C2×C411yescharacteristic
42C22×C3⋊D9C2×C411yescharacteristic
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2×C411yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47#6233—22no—
48#6233—22no—
49C2×C3⋊Dic9C2×C411yescharacteristic
50C2×C3⋊Dic9C2×C411yescharacteristic
51C4⋊(C3⋊D9)—22no—
52C4×C3⋊D9—21no—
53C4×C3⋊D9—21no—
54C4×C3⋊D9D412yes—
55C4×C3⋊D9D412yes—
56#6226—22no—
57C9×C3⋊C8—21no—
58C9×C3⋊C8—21no—
59C3×C9⋊C8—21no—
60C3×C9⋊C8—21no—
61#4727—91no—
62#4727—91no—
63C22×C32×C4C2×D311yescharacteristic
64C2×#638—31no—
65C2×#638—31no—
66#3828—62no—
67#3828—62no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4×C3⋊D3—31no—
70C2×C4×C3⋊D3—31no—
71C2×#634—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75C2×C3×C3⋊C8C2×D311yescharacteristic
76#3712—62no—
77#3710—62no—
78C22×C4×C9—21no—
79C22×C4×C9C2×D311yescharacteristic
80C2×#611—31no—
81C2×#611—31no—
82C2×#611—61no—
83C2×#611—61no—
84#3697—62no—
85#3697—62no—
86#3697—64no—
87#3697—64no—
88C2×D36—31no—
89C2×D36—61no—
90C2×C4×D9—31no—
91C2×C4×D9—31no—
92C2×C4×D9—61no—
93C2×C4×D9—61no—
94C2×Dic18—31no—
95C2×Dic18—61no—
96C2×C8×C9—31no—
97C2×C8×C9—31no—
98C2×C9⋊C8C2×D311yescharacteristic
99C2×C9⋊C8C2×D311yescharacteristic
100#3663—62no—
101C8×D9—62no—
102C22×C3×C9—21no—
103C22×C3×C9—21no—
104C22×C3×C9—21no—
105C22×C3×C9C22×C411yescharacteristic
106C2×C3⋊D9—21no—
107C2×C3⋊D9—21no—
108C2×C3⋊D9—21no—
109C2×C3⋊D9—21no—
110C2×C3⋊D9—21no—
111C2×C3⋊D9C22⋊C412yes—
112C3×C4×C9—21no—
113C3×C4×C9C2×D411yescharacteristic
114C3×C4×C9C2×D411yescharacteristic
115C3⋊Dic9—21no—
116C3⋊Dic9C22⋊C412yes—
117C2×#235—91no—
118C2×#235—91no—
119C2×#235—181no—
120C2×ΓL(1,𝔽25)—91no—
121C2×ΓL(1,𝔽25)—91no—
122C2×D3×C8—91no—
123C2×D3×C8—91no—
124C3×C22⋊C8—31no—
125C3×C22⋊C8—91no—
126#830—31no—
127#830D911yescharacteristic, complemented
128(C2×D3)⋊C8—91no—
129(C2×D3)⋊C8—91no—
130(C2×D3)⋊C8—91no—
131(C2×D3)⋊C8—91no—
132C23×C32C4×D311yescharacteristic
133C22×C3⋊D3—31no—
134C22×C3⋊D3—31no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4C4×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139#638—62no—
140#638—62no—
141C2×C3⋊Dic3—31no—
142C2×C3⋊Dic3—31no—
143C4⋊(C3⋊D3)—62no—
144C4×C3⋊D3—32no—
145C4×C3⋊D3—32no—
146C4×C3⋊D3—61no—
147C4×C3⋊D3—61no—
148#634—62no—
149C23×C9—21no—
150C23×C9C4×D311yescharacteristic
151C22×D9—31no—
152C22×D9—31no—
153C22×D9—61no—
154C22×D9—61no—
155C3×C3⋊C8—21no—
156C3×C3⋊C8—21no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—61no—
159C2×C4×C9—22no—
160C2×C4×C9—22no—
161C2×C4×C9—21no—
162C2×C4×C9—21no—
163C2×C4×C9—21no—
164C2×C4×C9—21no—
165C2×C4×C9C4×D311yescharacteristic
166C2×C4×C9C22×D311yescharacteristic
167#611—62no—
168#611—62no—
169#611—64no—
170#611—64no—
171C2×Dic9—31no—
172C2×Dic9—31no—
173C2×Dic9—61no—
174C2×Dic9—61no—
175D36—62no—
176D36—64no—
177C4×D9—32no—
178C4×D9—32no—
179C4×D9—62no—
180C4×D9—62no—
181C4×D9—62no—
182C4×D9—62no—
183C4×D9—61no—
184C4×D9—61no—
185Dic18—62no—
186Dic18—64no—
187C8×C9—61no—
188C8×C9—61no—
189C9⋊C8—21no—
190C9⋊C8—21no—
191#408—271noSylow
192C2×C3×C9—41no—
193C2×C3×C9C2×C22⋊C411yescharacteristic
194C2×C3×C9#13111yescharacteristic
195C2×C3×C9#13111yescharacteristic
196C3⋊D9—22no—
197C3⋊D9—41no—
198C22×C3×C4—21no—
199C22×C3×C4C2×D911yescharacteristic
200C22×C3×C4D3211yescharacteristic
201C2×#67—91no—
202C2×#67—91no—
203C2×#67—91no—
204C2×#67—91no—
205C2×#67—181no—
206C2×#67—181no—
207#235—182no—
208#235—182no—
209#235—182no—
210#235—182no—
211#235—184no—
212#235—184no—
213C2×D12—91no—
214C2×D12—91no—
215C2×D12—181no—
216C2×C4×D3—91no—
217C2×C4×D3—91no—
218C2×C4×D3—91no—
219C2×C4×D3—91no—
220C2×C4×D3—181no—
221C2×C4×D3—181no—
222C2×Dic6—91no—
223C2×Dic6—91no—
224C2×Dic6—181no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—91no—
228C2×C3×C8—91no—
229C2×C3⋊C8—31no—
230C2×C3⋊C8—31no—
231C2×C3⋊C8C2×D911yescharacteristic
232C2×C3⋊C8C2×D911yescharacteristic
233ΓL(1,𝔽25)—182no—
234ΓL(1,𝔽25)—182no—
235D3×C8—182no—
236D3×C8—182no—
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32C2×C4×D311yescharacteristic
241C2×C3⋊D3—32no—
242C2×C3⋊D3—61no—
243C2×C3⋊D3—61no—
244C2×C3⋊D3—61no—
245C2×C3⋊D3—61no—
246C2×C3⋊D3—61no—
247C32×C4—21no—
248C32×C4D3×D411yescharacteristic
249C32×C4D3×D411yescharacteristic
250C3⋊Dic3—32no—
251C3⋊Dic3—61no—
252C22×C9—22no—
253C22×C9—21no—
254C22×C9—21no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—41no—
258C22×C9—41no—
259C22×C9C2×C4×D311yescharacteristic
260C2×D9—32no—
261C2×D9—62no—
262C2×D9—61no—
263C2×D9—61no—
264C2×D9—62no—
265C2×D9—61no—
266C2×D9—61no—
267C2×D9—61no—
268C2×D9—121no—
269C2×D9—121no—
270C2×D9—121no—
271C2×D9—121no—
272C4×C9—22no—
273C4×C9—21no—
274C4×C9—21no—
275C4×C9—21no—
276C4×C9D3×D411yescharacteristic
277C4×C9D3×D411yescharacteristic
278Dic9—32no—
279Dic9—62no—
280Dic9—62no—
281Dic9—61no—
282C2×#41—271no—
283C2×M16—271no—
284C22×C8—271no—
285C22⋊C8—31no—
286C22⋊C8—91no—
287C22⋊C8—271no—
288C22⋊C8—271no—
289C3×C9#40811yescharacteristic, complemented, Sylow
290C23×C3—21no—
291C23×C3C4×D911yescharacteristic
292C23×C3#62411yescharacteristic
293C22×D3—91no—
294C22×D3—91no—
295C22×D3—91no—
296C22×D3—91no—
297C22×D3—181no—
298C22×D3—181no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4C4×D911yescharacteristic
308C2×C3×C4C22×D911yescharacteristic
309C2×C3×C4#62411yescharacteristic
310C2×C3×C4C2×D3211yescharacteristic
311#67—182no—
312#67—182no—
313#67—182no—
314#67—182no—
315#67—184no—
316#67—184no—
317C2×Dic3—91no—
318C2×Dic3—91no—
319C2×Dic3—91no—
320C2×Dic3—91no—
321C2×Dic3—181no—
322C2×Dic3—181no—
323D12—182no—
324D12—182no—
325D12—184no—
326C4×D3—92no—
327C4×D3—92no—
328C4×D3—92no—
329C4×D3—92no—
330C4×D3—182no—
331C4×D3—182no—
332C4×D3—182no—
333C4×D3—182no—
334C4×D3—181no—
335C4×D3—181no—
336C4×D3—181no—
337C4×D3—181no—
338Dic6—182no—
339Dic6—182no—
340Dic6—184no—
341C3×C8—61no—
342C3×C8—61no—
343C3×C8—181no—
344C3×C8—181no—
345C3⋊C8—21no—
346C3⋊C8—21no—
347C3⋊C8—61no—
348C3⋊C8—61no—
349C2×C32—41no—
350C2×C32D3×C22⋊C411yescharacteristic
351C2×C32#90111yescharacteristic
352C2×C32#90711yescharacteristic
353C3⋊D3—62no—
354C3⋊D3—121no—
355C2×C9—21no—
356C2×C9—21no—
357C2×C9—21no—
358C2×C9—42no—
359C2×C9—41no—
360C2×C9D3×C22⋊C411yescharacteristic
361C2×C9#90111yescharacteristic
362C2×C9#90711yescharacteristic
363D9—62no—
364D9—122no—
365D9—122no—
366D9—121no—
367#41—542no—
368#41—542no—
369C2×Q8—271no—
370C2×D4—271no—
371C2×D4—271no—
372C2×D4—271no—
373C22×C4—271no—
374C22×C4—271no—
375C22×C4D3×D911yescharacteristic
376M16—542no—
377C2×C8—31no—
378C2×C8—31no—
379C2×C8—91no—
380C2×C8—91no—
381C2×C8—542no—
382C22×C3—22no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3C2×C4×D911yescharacteristic
393C22×C3C2×#62411yescharacteristic
394C2×D3—92no—
395C2×D3—92no—
396C2×D3—182no—
397C2×D3—181no—
398C2×D3—181no—
399C2×D3—181no—
400C2×D3—181no—
401C2×D3—182no—
402C2×D3—181no—
403C2×D3—181no—
404C2×D3—181no—
405C2×D3—181no—
406C2×D3—181no—
407C2×D3—181no—
408C2×D3—361no—
409C2×D3—361no—
410C2×D3—361no—
411C2×D3—361no—
412C3×C4—22no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4D4×D911yescharacteristic
418C3×C4D4×D911yescharacteristic
419C3×C4#381111yescharacteristic
420C3×C4#381111yescharacteristic
421Dic3—92no—
422Dic3—92no—
423Dic3—182no—
424Dic3—182no—
425Dic3—181no—
426Dic3—181no—
427C32#472711yescharacteristic
428C9—21no—
429C9#472711yescharacteristic, complemented
430C23—271no—
431C23—271no—
432C23(C3⋊D9)⋊C411yescharacteristic
433Q8—542no—
434D4—542no—
435D4—542no—
436D4—542no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—271no—
440C2×C4—272no—
441C2×C4—272no—
442C2×C4—271no—
443C2×C4—541no—
444C2×C4—541no—
445C2×C4(C3⋊D9)⋊C411yescharacteristic
446C2×C4C2×D3×D911yescharacteristic
447C8—61no—
448C8—61no—
449C8—181no—
450C8—181no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—42no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3C22⋊C4×D911yescharacteristic
458C2×C3#6346411yescharacteristic
459C2×C3#6346911yescharacteristic
460C2×C3#6381711yescharacteristic
461C2×C3#6381711yescharacteristic
462C2×C3#6397411yescharacteristic
463D3—182no—
464D3—182no—
465D3—362no—
466D3—362no—
467D3—361no—
468D3—361no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—272no—
473C22—541no—
474C22—541no—
475C22—541no—
476C22—541no—
477C22—541no—
478C22C2×(C3⋊D9)⋊C411yescharacteristic
479C4—21no—
480C4—272no—
481C4—541no—
482C4#8866811yescharacteristic
483C4#8866811yescharacteristic
484C3—21no—
485C3#1058840911yescharacteristic, complemented
486C3#1059115311yescharacteristic
487C2—41no—
488C2—542no—
489C2—1081no—
490C2#1172086111yescharacteristic
491C2#1172086211yescharacteristic
492C2#1172127411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×C4×D3
prime radicalC2×#88736C2
cosocleC2×C3×C4×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#408
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88414C2semidirect
3. C2×(C3⋊D9)⋊C8C2semidirect
4. C9×#830C2semidirect
5. C3×#63389C2semidirect
6. #11720282C2semidirect
7. #11720282C2semidirect
37. #63389D3semidirect
127. #830D9semidirect
289. C3×C9#408semidirect
429. C9#4727semidirect
485. C3#10588409semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C2×#884142
C2×(C3⋊D9)⋊C82
C9×#8302
C3×#633892
#117202822
#105911533
#105884093