Finite Group Databasebeta

← #13139066

Subgroups — #13139066

Order 1728.

609 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 95 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×#88648C211yesmaximal, characteristic
2C2×#3743⋊C3C211yesmaximal, characteristic, complemented
3#11721129C211yesmaximal, characteristic, complemented
4#11720549C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590957—31nomaximal
9#10590671—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#6198—21no—
14C2×((C3⋊D3)⋊C3)⋊C4—21no—
15C2×((C3⋊D3)⋊C3)⋊C4—21no—
16#88648—22no—
17#88648C414yescomplemented
18C2×C8×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3743⋊C3—22no—
23#88448—21no—
24C2×#3799—31no—
25C2×#3796—31no—
26C2×#3743D311yescharacteristic, complemented
27C2×C3×ΓL(1,𝔽25)—31no—
28#63900—31no—
29D3×C4⋊Dic3—31no—
30#63648D311yescharacteristic, complemented
31C3×#821—31no—
32C3×#816—31no—
33C3×#816—31no—
34C3×#816—61no—
35#63566—31no—
36#63565—31no—
37#63564—31no—
38#63562—31no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C2×#1084—21no—
44#6229—21no—
45#6229C2×C412yescomplemented
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
50C4×(C3⋊D3)⋊C3C2×C412yes—
51C4×(C3⋊D3)⋊C3D412yes—
52#6198—24no—
53((C3⋊D3)⋊C3)⋊C4—24no—
54((C3⋊D3)⋊C3)⋊C4—42no—
55C8×H(3,𝔽3)—21no—
56(C3⋊(C3⋊C8))⋊C3—21no—
57#5133—91no—
58#4803—91no—
59#4771—91no—
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×C4×D3—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67C2×#626—61no—
68C2×#626—61no—
69C2×#624—61no—
70C2×#624—61no—
71C2×D3×Dic3—61no—
72C2×D3×Dic3—61no—
73#3799—34no—
74#3799—62no—
75#3796—34no—
76#3796—62no—
77C2×C32×C8—21no—
78C2×C32×C8C2×D311yescharacteristic
79C2×C32×C8C2×D311yescharacteristic
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82#3743—22no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—61no—
87C2×C3×C3⋊C8—31no—
88C3×ΓL(1,𝔽25)—62no—
89#3724—61no—
90Dic3⋊Dic3—61no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—21no—
94C2×(C3⋊D3)⋊C3—22no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3C22⋊C412yes—
98C4×H(3,𝔽3)C22×C411yescharacteristic
99C4×H(3,𝔽3)C2×D411yescharacteristic
100#1084—22no—
101#1084—41no—
102(C3⋊Dic3)⋊C3—41no—
103(C3⋊Dic3)⋊C3C22⋊C412yes—
104C2×#239—91no—
105C2×#237—91no—
106C2×#235—91no—
107C2×C3×M16—91no—
108C2×C4⋊Dic3—91no—
109C2×ΓL(1,𝔽25)—31no—
110C2×ΓL(1,𝔽25)—91no—
111D3×C4⋊C4—91no—
112D3×C4⋊C4—91no—
113C3×#103—91no—
114C3×#102—91no—
115#835—91no—
116#832—91no—
117#821—31no—
118#821—91no—
119#821—91no—
120#816—31no—
121#816—31no—
122#816—61no—
123#816—91no—
124#810—91no—
125#809—91no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3D1211yescharacteristic
128C22×C3×D3—31no—
129C22×C3×D3—61no—
130C2×C32×C4—21no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3D1211yescharacteristic
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)C4×D312yescomplemented
137C4×C3⋊D3C4×D312yes—
138C4×C3⋊D3#6712yes—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—121no—
146C2×C3×Dic3—121no—
147C3×D12—32no—
148C3×D12—61no—
149C3×C4×D3—32no—
150C3×C4×D3—32no—
151C3×Dic6—32no—
152C3×Dic6—32no—
153C3×Dic6—62no—
154C3×Dic6—61no—
155C3×Dic6—61no—
156C3×Dic6—121no—
157#626—64no—
158#626—64no—
159#624—64no—
160#624—122no—
161D3×Dic3—64no—
162D3×Dic3—122no—
163C32×C8—21no—
164C32×C8—21no—
165C32×C8—41no—
166C3⋊(C3⋊C8)—21no—
167C3×C3⋊C8—61no—
168#417—271noSylow
169C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
170C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
171C2×H(3,𝔽3)#13711yescharacteristic
172(C3⋊D3)⋊C3—22no—
173(C3⋊D3)⋊C3—42no—
174C2×C3×Q8—91no—
175C2×C3×D4—91no—
176C22×C3×C4—91no—
177C2×#67—181no—
178C2×#67—181no—
179C22×Dic3—181no—
180C22×Dic3—181no—
181#239—94no—
182#239—182no—
183#237—94no—
184#237—182no—
185#235—94no—
186#235—182no—
187C2×D12—31no—
188C2×D12—91no—
189C2×D12—181no—
190C2×C4×D3—31no—
191C2×C4×D3—91no—
192C2×C4×D3—181no—
193C2×C4×D3—181no—
194C2×C4×D3—181no—
195C2×C4×D3—181no—
196C2×Dic6—31no—
197C2×Dic6—31no—
198C2×Dic6—61no—
199C3×M16—182no—
200C2×C3×C8—31no—
201C2×C3×C8—31no—
202C2×C3×C8—61no—
203C2×C3×C8—91no—
204C2×C3×C8D3211yescharacteristic
205C3×C4⋊C4—91no—
206C3×C4⋊C4—91no—
207C4⋊Dic3—31no—
208C4⋊Dic3—31no—
209C4⋊Dic3—31no—
210C4⋊Dic3—61no—
211C4⋊Dic3—91no—
212C4⋊Dic3—181no—
213Dic3⋊C4—181no—
214Dic3⋊C4—181no—
215C2×C3⋊C8—31no—
216C2×C3⋊C8—91no—
217ΓL(1,𝔽25)—62no—
218ΓL(1,𝔽25)—182no—
219C22×C32—21no—
220C22×C32C2×D1211yescharacteristic
221C22×C32D3×D411yescharacteristic
222C2×C3⋊D3—21no—
223C2×C3⋊D3—21no—
224C2×C3⋊D3—22no—
225C2×C3⋊D3—41no—
226C2×C3⋊D3—41no—
227C2×C3⋊D3(C2×D3)⋊C412yes—
228C2×C3×D3—32no—
229C2×C3×D3—61no—
230C2×C3×D3—61no—
231C2×C3×D3—62no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4C2×C4×D311yescharacteristic
237C32×C4C2×C4×D311yescharacteristic
238C32×C4D3×D411yescharacteristic
239C32×C4C2×#6711yescharacteristic
240C3⋊Dic3—41no—
241C3⋊Dic3(C2×D3)⋊C412yes—
242C3×Dic3—32no—
243C3×Dic3—62no—
244C3×Dic3—62no—
245C3×Dic3—122no—
246C3×Dic3—121no—
247C3×Dic3—121no—
248C3×Dic3—121no—
249C3×Dic3—241no—
250C2×#41—271no—
251C2×M16—91no—
252C2×C4⋊C4—271no—
253#103—91no—
254#103—271no—
255#102—91no—
256#102—271no—
257H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
258C23×C3—91no—
259C23×C3—181no—
260C22×D3—31no—
261C22×D3—61no—
262C22×D3—91no—
263C22×D3—181no—
264C3×Q8—92no—
265C3×Q8—181no—
266C3×D4—92no—
267C3×D4—181no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—61no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—91no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4C2×D3211yescharacteristic
278#67—184no—
279#67—184no—
280C2×Dic3—31no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—91no—
287C2×Dic3—121no—
288C2×Dic3—121no—
289C2×Dic3—181no—
290C2×Dic3—184no—
291C2×Dic3—362no—
292D12—32no—
293D12—61no—
294D12—92no—
295D12—181no—
296D12—184no—
297C4×D3—32no—
298C4×D3—32no—
299C4×D3—92no—
300C4×D3—92no—
301C4×D3—184no—
302C4×D3—184no—
303C4×D3—362no—
304C4×D3—362no—
305Dic6—32no—
306Dic6—32no—
307Dic6—62no—
308Dic6—61no—
309Dic6—61no—
310Dic6—121no—
311C3×C8—21no—
312C3×C8—61no—
313C3×C8—61no—
314C3×C8—121no—
315C3×C8—181no—
316C3⋊C8—61no—
317C3⋊C8—181no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32D3×C22⋊C411yescharacteristic
322C2×C32#90811yescharacteristic
323C2×C32#90911yescharacteristic
324C2×C32#91111yescharacteristic
325C2×C32#91511yescharacteristic
326C2×C32C2×(C2×D3)⋊C411yescharacteristic
327C3⋊D3—22no—
328C3⋊D3—42no—
329C3×D3—62no—
330C3×D3—122no—
331#41—274no—
332#41—542no—
333C2×Q8—91no—
334C2×D4—91no—
335C2×D4—541no—
336C22×C4—91no—
337C22×C4—541no—
338C22×C4—541no—
339M16—182no—
340C2×C8—91no—
341C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C4⋊C4—91no—
343C4⋊C4—91no—
344C4⋊C4—541no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—61no—
348C22×C3—92no—
349C22×C3—181no—
350C22×C3—181no—
351C22×C3—182no—
352C22×C3—361no—
353C22×C3—361no—
354C22×C3D3×D1211yescharacteristic
355C2×D3—32no—
356C2×D3—61no—
357C2×D3—61no—
358C2×D3—62no—
359C2×D3—92no—
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—181no—
363C2×D3—181no—
364C2×D3—182no—
365C2×D3—361no—
366C2×D3—361no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—92no—
374C3×C4—182no—
375C3×C4—361no—
376C3×C4—361no—
377C3×C4C4×D3211yescharacteristic
378C3×C4D3×#6711yescharacteristic
379Dic3—32no—
380Dic3—62no—
381Dic3—62no—
382Dic3—92no—
383Dic3—122no—
384Dic3—121no—
385Dic3—121no—
386Dic3—121no—
387Dic3—241no—
388Dic3—361no—
389C32—21no—
390C32#477111yescharacteristic, complemented
391C32#513311yescharacteristic, complemented
392C23—91no—
393C23—181no—
394Q8—92no—
395Q8—181no—
396D4—92no—
397D4—181no—
398D4—544no—
399C2×C4—91no—
400C2×C4—92no—
401C2×C4—92no—
402C2×C4—181no—
403C2×C4—181no—
404C2×C4—181no—
405C2×C4—544no—
406C2×C4—1082no—
407C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
408C8—21no—
409C8—181no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—31no—
416C2×C3—61no—
417C2×C3—61no—
418C2×C3—61no—
419C2×C3—182no—
420C2×C3—362no—
421C2×C3#6379511yescharacteristic
422C2×C3#6380011yescharacteristic
423C2×C3D3×(C2×D3)⋊C411yescharacteristic
424D3—62no—
425D3—122no—
426D3—182no—
427D3—362no—
428C22—92no—
429C22—181no—
430C22—181no—
431C22—182no—
432C22—361no—
433C22—361no—
434C22#8865411yescharacteristic
435C4—92no—
436C4—182no—
437C4—361no—
438C4—361no—
439C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
440C4#8867611yescharacteristic
441C3—31no—
442C3—31no—
443C3—61no—
444C3#1059067011yescharacteristic
445C2—182no—
446C2—362no—
447C2#1172084911yescharacteristic
448C2#1172085211yescharacteristic
449C2#1172116011yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×#88648C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#417
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#3743⋊C3C2semidirect
3. #11721129C2semidirect
4. #11720549C2semidirect
5. #63648⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720308C2semidirect
17. #88648C4semidirect
26. C2×#3743D3semidirect
30. #63648D3semidirect
45. #6229C2×C4semidirect
49. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
136. C4⋊(C3⋊D3)C4×D3semidirect
257. H(3,𝔽3)#417semidirect
390. C32#4771semidirect
391. C32#5133semidirect

Maximal subgroups

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

subgroupindex
C2×#886482
C2×#3743⋊C32
#117211292
#117205492
#63648⋊C32
#117203262
#117203082
#105909573
#105906713