Finite Group Databasebeta

← #13140031

Subgroups — #13140031

Order 1728.

510 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C22×#6180C211yesmaximal, characteristic, complemented
2C4×#638⋊C3C211yesmaximal, characteristic
3C2×#88430C211yesmaximal, characteristic, complemented
4#63657⋊C3C211yesmaximal, characteristic, complemented
5#11720317C211yesmaximal, characteristic, complemented
6#11720312C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10591157—31nomaximal
9#10591155—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3C411yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#6180—22no—
14C2×#6180C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3757⋊C3C411yescharacteristic
17#3752⋊C3C411yescharacteristic
18#3750⋊C3C411yescharacteristic
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#88430—22no—
23C4×#638D311yescharacteristic
24C3×C4×#67—31no—
25C22×C3×C3⋊C8—31no—
26C22×C3×C3⋊C8—31no—
27C22×C3×C3⋊C8—61no—
28C2×#3712—31no—
29C2×#3711—31no—
30#63657D311yescharacteristic, complemented
31C3×#830—31no—
32#63572—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63557—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36#63555—31no—
37#63554—31no—
38C23×H(3,𝔽3)C2×C411yescharacteristic
39C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2×C411yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#638⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
46C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
47C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
48C4×(C3⋊D3)⋊C3—21no—
49C4×(C3⋊D3)⋊C3—21no—
50#6180—21no—
51#6180D412yescomplemented
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5111—91no—
55#5011—91no—
56#4730—91no—
57C22×C32×C4—21no—
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C2×D311yescharacteristic
60C2×#638Dic311yescharacteristic
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×C3×#67—31no—
63C2×C3×C4×D3—31no—
64#3757Dic311yescharacteristic
65#3752Dic311yescharacteristic
66#3750Dic311yescharacteristic
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×#217—31no—
71C3×(C2×D3)⋊C4—31no—
72C3×Dic3⋊C4—31no—
73C3×C4×Dic3—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—62no—
81C2×C3×C3⋊C8—62no—
82C2×C3×C3⋊C8—61no—
83C2×C3×C3⋊C8—61no—
84C2×C3×C3⋊C8—64no—
85#3712—62no—
86#3711—62no—
87C22×H(3,𝔽3)—21no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)M1612yescomplemented
90C22×H(3,𝔽3)C22×C411yescharacteristic
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—21no—
94C4×H(3,𝔽3)—21no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3—21no—
100C22×C3×C8—91no—
101C3×C4×D4—91no—
102C4×#67—31no—
103C4×#67—91no—
104C2×#208—91no—
105C22×C3⋊C8—31no—
106C22×C3⋊C8—31no—
107C22×C3⋊C8—61no—
108C2×ΓL(1,𝔽25)—91no—
109C2×ΓL(1,𝔽25)—91no—
110C3×C22⋊C8—91no—
111#830—31no—
112#830—91no—
113#830—91no—
114(C2×D3)⋊C8—91no—
115(C2×D3)⋊C8—91no—
116#815—91no—
117#815—91no—
118Dic3⋊C8—91no—
119Dic3⋊C8—91no—
120C4⋊(C3⋊C8)—91no—
121#803—91no—
122C23×C32—21no—
123C23×C32C4×D311yescharacteristic
124C23×C32C2×Dic311yescharacteristic
125C22×C3⋊D3C2×Dic311yescharacteristic
126C22×C3×D3—31no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4C4×D311yescharacteristic
136C2×C32×C4C2×Dic311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139#638—22no—
140C2×C3⋊Dic3C2×Dic311yescharacteristic
141C2×C3⋊Dic3C2×Dic311yescharacteristic
142C2×C3⋊Dic3C2×Dic311yescharacteristic
143C4×C3⋊D3—21no—
144C4×C3⋊D3—21no—
145C3×#67—62no—
146C2×C3×Dic3—31no—
147C2×C3×Dic3—31no—
148C2×C3×Dic3—31no—
149C3×C4×D3—61no—
150C3×C4×D3—61no—
151C3⋊(C3⋊C8)—21no—
152C3⋊(C3⋊C8)—21no—
153C3×C3⋊C8—32no—
154C3×C3⋊C8—32no—
155C3×C3⋊C8—62no—
156C3×C3⋊C8—62no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—61no—
159C3×C3⋊C8—61no—
160C3×C3⋊C8—61no—
161#435—271noSylow
162C2×H(3,𝔽3)—22no—
163C2×H(3,𝔽3)C4×D411yescharacteristic
164C2×H(3,𝔽3)C2×M1611yescharacteristic
165C2×H(3,𝔽3)#13111yescharacteristic
166(C3⋊D3)⋊C3—41no—
167C2×C3×D4—91no—
168C22×C3×C4—31no—
169C22×C3×C4—31no—
170C22×C3×C4—61no—
171C22×C3×C4—91no—
172C22×C3×C4D3211yescharacteristic
173C2×#67—31no—
174C2×#67—91no—
175C2×C4×D3—31no—
176C2×C4×D3—91no—
177C2×C3×C8—91no—
178C2×C3×C8—91no—
179C2×C3×C8—91no—
180C2×C3×C8—91no—
181C2×C3×C8—182no—
182C3×C4⋊C4—91no—
183C3×C22⋊C4—91no—
184C3×C22⋊C4—91no—
185C3×C42—91no—
186#217—31no—
187#217—91no—
188(C2×D3)⋊C4—31no—
189(C2×D3)⋊C4—91no—
190Dic3⋊C4—31no—
191Dic3⋊C4—91no—
192C4×Dic3—31no—
193C4×Dic3—91no—
194#208—182no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—31no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—61no—
202C2×C3⋊C8—61no—
203C2×C3⋊C8—62no—
204C2×C3⋊C8—62no—
205C2×C3⋊C8—64no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—91no—
208ΓL(1,𝔽25)—182no—
209ΓL(1,𝔽25)—182no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—22no—
217C22×C32—21no—
218C22×C32ΓL(1,𝔽25)12yescomplemented
219C22×C32#20812yescomplemented
220C22×C32C2×C4×D311yescharacteristic
221C22×C32C22×Dic311yescharacteristic
222C2×C3⋊D3—21no—
223C2×C3⋊D3—21no—
224C2×C3⋊D3—21no—
225C2×C3×D3—61no—
226C2×C3×D3—61no—
227C2×C3×D3—61no—
228C32×C4—22no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4D3×D411yescharacteristic
234C32×C4D3×D411yescharacteristic
235C32×C4#23711yescharacteristic
236C32×C4#23711yescharacteristic
237C3⋊Dic3—21no—
238C3⋊Dic3—21no—
239C3⋊Dic3—21no—
240C3×Dic3—61no—
241C3×Dic3—61no—
242C3×Dic3—61no—
243C2×M16—271no—
244C22×C8—91no—
245C4×D4—91no—
246C4⋊C8—271no—
247C22⋊C8—91no—
248C22⋊C8—271no—
249#97—271no—
250H(3,𝔽3)#43511yescharacteristic, complemented, Sylow
251C23×C3—31no—
252C23×C3—31no—
253C23×C3—61no—
254C23×C3—91no—
255C23×C3D3×Dic311yescharacteristic
256C22×D3—31no—
257C22×D3—91no—
258C3×D4—182no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—91no—
274C2×C3×C4—91no—
275C2×C3×C4—91no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4D3×Dic311yescharacteristic
279C2×C3×C4C2×D3211yescharacteristic
280#67—62no—
281#67—182no—
282C2×Dic3—31no—
283C2×Dic3—31no—
284C2×Dic3—31no—
285C2×Dic3—91no—
286C2×Dic3—91no—
287C2×Dic3—91no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—181no—
291C4×D3—181no—
292C3×C8—92no—
293C3×C8—181no—
294C3×C8—181no—
295C3×C8—181no—
296C3⋊C8—32no—
297C3⋊C8—32no—
298C3⋊C8—62no—
299C3⋊C8—61no—
300C3⋊C8—61no—
301C3⋊C8—62no—
302C3⋊C8—61no—
303C3⋊C8—61no—
304C3⋊C8—181no—
305C3⋊C8—181no—
306C2×C32—24no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32#67⋊C411yescharacteristic
313C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
314C2×C32#90711yescharacteristic
315C2×C32C2×#20811yescharacteristic
316C2×C32D4×Dic311yescharacteristic
317C2×C32#94811yescharacteristic
318C3⋊D3—41no—
319C3×D3—121no—
320C2×D4—91no—
321C22×C4—91no—
322C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
323M16—542no—
324C2×C8—91no—
325C2×C8—91no—
326C2×C8—91no—
327C2×C8—91no—
328C2×C8—182no—
329C4⋊C4—91no—
330C22⋊C4—91no—
331C22⋊C4—91no—
332C42—91no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—32no—
338C22×C3—32no—
339C22×C3—61no—
340C22×C3—61no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3—62no—
344C22×C3—62no—
345C22×C3—61no—
346C22×C3—62no—
347C22×C3—181no—
348C22×C3—181no—
349C22×C3—181no—
350C22×C3#371112yes—
351C22×C3C2×D3×Dic311yescharacteristic
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—61no—
355C2×D3—181no—
356C2×D3—181no—
357C2×D3—181no—
358C3×C4—21no—
359C3×C4—31no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—61no—
364C3×C4—61no—
365C3×C4—62no—
366C3×C4—61no—
367C3×C4—61no—
368C3×C4—181no—
369C3×C4—181no—
370C3×C4—181no—
371C3×C4#380511yescharacteristic
372C3×C4#381111yescharacteristic
373Dic3—61no—
374Dic3—61no—
375Dic3—61no—
376Dic3—181no—
377Dic3—181no—
378Dic3—181no—
379C32—21no—
380C32#473011yescharacteristic, complemented
381C32#501111yescharacteristic, complemented
382C23—91no—
383C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
384D4—182no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—91no—
388C2×C4—91no—
389C2×C4—91no—
390C2×C4—181no—
391C2×C4—181no—
392C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
393C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
394C8—92no—
395C8—181no—
396C8—181no—
397C8—181no—
398C2×C3—22no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—62no—
406C2×C3—62no—
407C2×C3—64no—
408C2×C3—61no—
409C2×C3—61no—
410C2×C3—61no—
411C2×C3—361no—
412C2×C3#6381511yescharacteristic
413C2×C3C2×#371111yescharacteristic
414C2×C3#67⋊Dic311yescharacteristic
415D3—121no—
416D3—361no—
417C22—21no—
418C22—21no—
419C22—181no—
420C22—181no—
421C22—181no—
422C22#8843012yescomplemented
423C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
424C4—21no—
425C4—181no—
426C4—181no—
427C4—181no—
428C4#8867211yescharacteristic
429C4#8867711yescharacteristic
430C3—31no—
431C3—31no—
432C3—61no—
433C3#1059115811yescharacteristic
434C2—22no—
435C2—361no—
436C2#1172105511yescharacteristic
437C2C2×#8843011yescharacteristic
438C2#1172131511yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC2×#638⋊C3C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#435
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6180C2semidirect
3. C2×#88430C2semidirect
4. #63657⋊C3C2semidirect
5. #11720317C2semidirect
6. #11720312C2semidirect
7. #11720311C2semidirect
30. #63657D3semidirect
51. #6180D4semidirect
89. C22×H(3,𝔽3)M16semidirect
218. C22×C32ΓL(1,𝔽25)semidirect
219. C22×C32#208semidirect
250. H(3,𝔽3)#435semidirect
380. C32#4730semidirect
381. C32#5011semidirect
422. C22#88430semidirect

Maximal subgroups

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

subgroupindex
C22×#61802
C4×#638⋊C32
C2×#884302
#63657⋊C32
#117203172
#117203122
#117203112
#105911573
#105911553