Finite Group Databasebeta

← #13138934

Subgroups — #13138934

Order 1728.

510 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 83 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×C8×#272C211yesmaximal, characteristic, complemented
2C4×#6200C211yesmaximal, characteristic
3C2×#88430C211yesmaximal, characteristic, complemented
4#63647⋊C3C211yesmaximal, characteristic, complemented
5#63641⋊C3C211yesmaximal, characteristic, complemented
6#11720323C211yesmaximal, characteristic, complemented
7#11720312C211yesmaximal, characteristic, complemented
8#10590727—31nomaximal
9#10590723—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#6200C411yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C8×#272—22no—
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451C411yescharacteristic
20#88450C411yescharacteristic
21#88448C411yescharacteristic
22#88430—22no—
23C2×C3×D3×C8—31no—
24C2×C3×D3×C8—31no—
25C2×C3×D3×C8—61no—
26C4×#626—31no—
27C4×#625—31no—
28C2×#3712—31no—
29C2×#3711—31no—
30#63647D311yescharacteristic, complemented
31#63641D311yescharacteristic, complemented
32C3×(C2×D3)⋊C8—31no—
33C3×Dic3⋊C8—31no—
34(C2×D3)⋊(C3⋊C8)—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36#63555—31no—
37#63554—31no—
38C22×#272C2×C411yescharacteristic
39C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2×C411yescharacteristic
42C4×#272—21no—
43C4×#272—21no—
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
46C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
47C4×(C3⋊D3)⋊C3—21no—
48C4×(C3⋊D3)⋊C3—21no—
49#6200—22no—
50C8×H(3,𝔽3)—21no—
51#6180D412yescomplemented
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5111—91no—
55#4836—91no—
56#4728—91no—
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×C3×C4×D3—31no—
59C2×C3×C4×D3—31no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—61no—
62C2×#626—31no—
63C2×#625—31no—
64C2×C32×C8—21no—
65C2×C32×C8C2×D311yescharacteristic
66C2×C32×C8C2×D311yescharacteristic
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×C4×Dic3—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—61no—
76C3×D3×C8—62no—
77C3×D3×C8—62no—
78C3×D3×C8—64no—
79#3724—31no—
80Dic3⋊Dic3—31no—
81#3722—31no—
82(C2×D3)⋊Dic3—31no—
83(C2×D3)⋊Dic3—31no—
84(C2×D3)⋊Dic3—31no—
85#3712—62no—
86#3711—62no—
87C22×H(3,𝔽3)C22×C411yescharacteristic
88C2×#272—21no—
89C2×#272—21no—
90C2×#272M1612yescomplemented
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—21no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96#1084—21no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3—21no—
100C22×C3×C8—91no—
101C4×#67—91no—
102C4×#67—91no—
103C2×#208—91no—
104C2×ΓL(1,𝔽25)—91no—
105C2×ΓL(1,𝔽25)—91no—
106C2×D3×C8—31no—
107C2×D3×C8—31no—
108C2×D3×C8—61no—
109C4×D12—91no—
110C3×C4⋊C8—91no—
111C3×C22⋊C8—91no—
112#830—91no—
113(C2×D3)⋊C8—31no—
114(C2×D3)⋊C8—91no—
115(C2×D3)⋊C8—91no—
116(C2×D3)⋊C8—91no—
117#815—91no—
118#815—91no—
119Dic3⋊C8—31no—
120Dic3⋊C8—91no—
121#803—91no—
122C22×C3⋊D3C4×D311yescharacteristic
123C22×C3×D3—31no—
124C22×C3×D3—31no—
125C22×C3×D3—31no—
126C22×C3×D3—61no—
127C2×C32×C4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3C4×D311yescharacteristic
131C2×C3⋊Dic3C4×D311yescharacteristic
132C2×C3⋊Dic3C4×D311yescharacteristic
133C4×C3⋊D3—21no—
134C4×C3⋊D3—21no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—61no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143C3×C4×D3—61no—
144C3×C4×D3—61no—
145C3×C4×D3—61no—
146C3×C4×D3—61no—
147C3×C4×D3—61no—
148C3×C4×D3—61no—
149#626—62no—
150#625—62no—
151C32×C8—22no—
152C32×C8—21no—
153C32×C8—21no—
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—32no—
157C3×C3⋊C8—32no—
158C3×C3⋊C8—62no—
159C3×C3⋊C8—61no—
160C3×C3⋊C8—61no—
161#435—271noSylow
162C2×H(3,𝔽3)C4×D411yescharacteristic
163C2×H(3,𝔽3)C2×M1611yescharacteristic
164C2×H(3,𝔽3)#13111yescharacteristic
165#272—22no—
166(C3⋊D3)⋊C3—41no—
167C22×C3×C4—91no—
168C22×C3×C4—91no—
169C2×#67—91no—
170C2×#67—91no—
171C2×D12—91no—
172C2×C4×D3—31no—
173C2×C4×D3—31no—
174C2×C4×D3—31no—
175C2×C4×D3—61no—
176C2×C4×D3—91no—
177C2×C3×C8—31no—
178C2×C3×C8—31no—
179C2×C3×C8—61no—
180C2×C3×C8—91no—
181C2×C3×C8—91no—
182C2×C3×C8—91no—
183C2×C3×C8—182no—
184C2×C3×C8D3211yescharacteristic
185C3×C42—91no—
186#217—91no—
187#217—91no—
188(C2×D3)⋊C4—91no—
189(C2×D3)⋊C4—91no—
190(C2×D3)⋊C4—91no—
191(C2×D3)⋊C4—91no—
192C4⋊Dic3—91no—
193Dic3⋊C4—91no—
194Dic3⋊C4—91no—
195C4×Dic3—31no—
196C4×Dic3—91no—
197#208—182no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—61no—
203C2×C3⋊C8—91no—
204C2×C3⋊C8—91no—
205ΓL(1,𝔽25)—182no—
206ΓL(1,𝔽25)—182no—
207D3×C8—62no—
208D3×C8—62no—
209D3×C8—64no—
210C22×C32—21no—
211C22×C32C2×C4×D311yescharacteristic
212C22×C32C2×C4×D311yescharacteristic
213C2×C3⋊D3—21no—
214C2×C3⋊D3—21no—
215C2×C3⋊D3—21no—
216C2×C3×D3—32no—
217C2×C3×D3—32no—
218C2×C3×D3—62no—
219C2×C3×D3—62no—
220C2×C3×D3—62no—
221C2×C3×D3—61no—
222C2×C3×D3—61no—
223C2×C3×D3—61no—
224C2×C3×D3—61no—
225C2×C3×D3—61no—
226C2×C3×D3—61no—
227C2×C3×D3—61no—
228C32×C4—21no—
229C32×C4—21no—
230C32×C4D3×D411yescharacteristic
231C32×C4D3×D411yescharacteristic
232C32×C4#23711yescharacteristic
233C32×C4#23911yescharacteristic
234C3⋊Dic3—21no—
235C3⋊Dic3—21no—
236C3⋊Dic3—21no—
237C3×Dic3—62no—
238C3×Dic3—61no—
239C3×Dic3—61no—
240C3×Dic3—61no—
241C3×Dic3—61no—
242C3×Dic3—61no—
243C2×M16—271no—
244C22×C8—91no—
245C4×D4—271no—
246C4⋊C8—91no—
247C22⋊C8—91no—
248C22⋊C8—271no—
249#97—271no—
250H(3,𝔽3)#43511yescharacteristic, complemented, Sylow
251C23×C3—91no—
252C23×C3—91no—
253C22×D3—31no—
254C22×D3—31no—
255C22×D3—31no—
256C22×D3—61no—
257C22×D3—91no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—61no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4—91no—
264C2×C3×C4—91no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4C2×D3211yescharacteristic
270#67—182no—
271#67—182no—
272C2×Dic3—31no—
273C2×Dic3—31no—
274C2×Dic3—31no—
275C2×Dic3—31no—
276C2×Dic3—31no—
277C2×Dic3—61no—
278C2×Dic3—91no—
279C2×Dic3—91no—
280C2×Dic3—91no—
281D12—182no—
282C4×D3—61no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—61no—
286C4×D3—62no—
287C4×D3—62no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—181no—
291C4×D3—181no—
292C3×C8—21no—
293C3×C8—61no—
294C3×C8—61no—
295C3×C8—62no—
296C3×C8—92no—
297C3×C8—181no—
298C3×C8—181no—
299C3⋊C8—32no—
300C3⋊C8—32no—
301C3⋊C8—62no—
302C3⋊C8—61no—
303C3⋊C8—61no—
304C3⋊C8—181no—
305C3⋊C8—181no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32#67⋊C411yescharacteristic
310C2×C32#89311yescharacteristic
311C2×C32#90111yescharacteristic
312C2×C32#90111yescharacteristic
313C2×C32D3×M1611yescharacteristic
314C2×C32D3×M1611yescharacteristic
315C3⋊D3—41no—
316C3×D3—64no—
317C3×D3—62no—
318C3×D3—62no—
319C3×D3—121no—
320C2×D4—271no—
321C22×C4—91no—
322C22×C4—91no—
323M16—542no—
324C2×C8—91no—
325C2×C8—91no—
326C2×C8—91no—
327C2×C8—182no—
328C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C4⋊C4—271no—
330C22⋊C4—271no—
331C22⋊C4—271no—
332C42—91no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—61no—
336C22×C3—92no—
337C22×C3—181no—
338C22×C3—181no—
339C22×C3—181no—
340C22×C3—181no—
341C22×C3—181no—
342C22×C3C4×D3211yescharacteristic
343C2×D3—32no—
344C2×D3—32no—
345C2×D3—62no—
346C2×D3—61no—
347C2×D3—61no—
348C2×D3—61no—
349C2×D3—61no—
350C2×D3—62no—
351C2×D3—62no—
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—61no—
355C2×D3—181no—
356C2×D3—181no—
357C2×D3—181no—
358C3×C4—31no—
359C3×C4—31no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—61no—
363C3×C4—61no—
364C3×C4—181no—
365C3×C4—181no—
366C3×C4—181no—
367C3×C4—181no—
368C3×C4#379611yescharacteristic
369C3×C4#381111yescharacteristic
370Dic3—61no—
371Dic3—61no—
372Dic3—62no—
373Dic3—61no—
374Dic3—61no—
375Dic3—61no—
376Dic3—181no—
377Dic3—181no—
378Dic3—181no—
379C32—21no—
380C32#472811yescharacteristic, complemented
381C32#483611yescharacteristic, complemented
382C23—91no—
383C23—91no—
384D4—542no—
385C2×C4—91no—
386C2×C4—91no—
387C2×C4—91no—
388C2×C4—91no—
389C2×C4—181no—
390C2×C4—181no—
391C2×C4—181no—
392C2×C4—181no—
393C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
394C8—21no—
395C8—92no—
396C8—181no—
397C8—181no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—61no—
405C2×C3—61no—
406C2×C3—61no—
407C2×C3—182no—
408C2×C3—361no—
409C2×C3#6379111yescharacteristic
410C2×C3#6380311yescharacteristic
411C2×C3#6388111yescharacteristic
412D3—62no—
413D3—62no—
414D3—64no—
415D3—121no—
416D3—361no—
417C22—92no—
418C22—181no—
419C22—181no—
420C22—181no—
421C22—181no—
422C22—181no—
423C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
424C4—181no—
425C4—181no—
426C4—181no—
427C4—181no—
428C4#8864911yescharacteristic
429C4#8867711yescharacteristic
430C3—31no—
431C3—31no—
432C3—61no—
433C3#1059073711yescharacteristic
434C2—182no—
435C2—361no—
436C2#1172083611yescharacteristic
437C2#1172084011yescharacteristic
438C2#1172112411yescharacteristic
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×#6200C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#435
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
1. C2×C8×#272C2semidirect
3. C2×#88430C2semidirect
4. #63647⋊C3C2semidirect
5. #63641⋊C3C2semidirect
6. #11720323C2semidirect
7. #11720312C2semidirect
30. #63647D3semidirect
31. #63641D3semidirect
51. #6180D4semidirect
90. C2×#272M16semidirect
250. H(3,𝔽3)#435semidirect
380. C32#4728semidirect
381. C32#4836semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#2722
C4×#62002
C2×#884302
#63647⋊C32
#63641⋊C32
#117203232
#117203122
#105907273
#105907233