Finite Group Databasebeta

← #13138965

Subgroups — #13138965

Order 1728.

510 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 71 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×#88529C211yesmaximal, characteristic
2#11721149C211yesmaximal, characteristic, complemented
3#11721114C211yesmaximal, characteristic, complemented
4C2×#88437C211yesmaximal, characteristic, complemented
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#11720324C211yesmaximal, characteristic, complemented
8#10590755—31nomaximal
9#10590751—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#6199—21no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15#88529—22no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88451—21no—
21#88451—21no—
22#88437—22no—
23C2×C3×D24—31no—
24C2×C3×D24—31no—
25C2×C3×D24—61no—
26#63898—31no—
27Dic3×D12—31no—
28#63863—31no—
29#63860—31no—
30C2×#3716—31no—
31C2×#3715—31no—
32#63643D311yescharacteristic, complemented
33#63641D311yescharacteristic, complemented
34C3×#816—31no—
35C3×Dic3⋊C8—31no—
36#63563—31no—
37#63562—31no—
38C22×#272—21no—
39C22×#272—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C4⋊#272—21no—
42C4⋊#272—21no—
43C4⋊#272—21no—
44C4⋊#272D412yes—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49#634⋊C3—21no—
50#634⋊C3—21no—
51#6199—42no—
52C8×H(3,𝔽3)—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5153—91no—
55#4844—91no—
56#4819—91no—
57C2×#634C2×D311yescharacteristic
58C2×C3×D12—31no—
59C2×C3×D12—31no—
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—61no—
63C2×C3×D12—61no—
64C2×C3×Dic6—31no—
65C2×D3×Dic3—61no—
66C2×D3×Dic3—61no—
67C2×C32×C8—21no—
68C2×C32×C8C2×D311yescharacteristic
69C2×C32×C8C2×D311yescharacteristic
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C4×C3⋊Dic3C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×C4⋊Dic3—31no—
74C3×C4×Dic3—31no—
75C2×C3×C3⋊C8—31no—
76C3×D24—62no—
77C3×D24—62no—
78C3×D24—64no—
79(C2×D3)⋊Dic3—61no—
80(C2×D3)⋊Dic3—61no—
81(C2×D3)⋊Dic3—61no—
82(C2×D3)⋊Dic3—61no—
83(C2×D3)⋊Dic3—61no—
84(C2×D3)⋊Dic3—61no—
85#3716—62no—
86#3715—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—22no—
89C2×#272—41no—
90C2×#272—41no—
91C2×#272—41no—
92C2×#272—41no—
93C2×#272—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96(C3⋊Dic3)⋊C3—21no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×D8—91no—
101#947—91no—
102C2×#215—91no—
103#936—91no—
104D4×Dic3—91no—
105C2×#214—91no—
106C2×D24—31no—
107C2×D24—31no—
108C2×D24—61no—
109C2×#204—91no—
110#893—91no—
111#875—91no—
112C4×D12—91no—
113C3×C4⋊C8—91no—
114C3×#103—91no—
115#832—91no—
116#821—91no—
117#816—31no—
118#816—91no—
119Dic3⋊C8—31no—
120Dic3⋊C8—91no—
121#809—91no—
122C22×C3×D3—61no—
123C22×C3×D3—61no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C22×C3×D3—121no—
127C22×C3×D3—121no—
128C2×C32×C4—21no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3D1211yescharacteristic
134C2×C3⋊Dic3D1211yescharacteristic
135#634—21no—
136#634—21no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C3×D12—32no—
142C3×D12—32no—
143C3×D12—62no—
144C3×D12—62no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—121no—
152C3×D12—121no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155D3×Dic3—122no—
156D3×Dic3—122no—
157C32×C8—22no—
158C32×C8—21no—
159C32×C8—21no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—61no—
162#463—271noSylow
163C2×H(3,𝔽3)#12111yescharacteristic
164C2×H(3,𝔽3)#13511yescharacteristic
165C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
166#272—42no—
167#272—81no—
168C2×C3×Q8—91no—
169C2×C3×D4—91no—
170C2×C3×D4—91no—
171C22×Dic3—181no—
172C2×D12—31no—
173C2×D12—31no—
174C2×D12—31no—
175C2×D12—31no—
176C2×D12—61no—
177C2×D12—61no—
178C2×C4×D3—181no—
179C2×C4×D3—181no—
180C2×Dic6—31no—
181C2×Dic6—91no—
182C3×D8—182no—
183C2×C3×C8—31no—
184C2×C3×C8—31no—
185C2×C3×C8—61no—
186C2×C3×C8—91no—
187C2×C3×C8D3211yescharacteristic
188C3×C4⋊C4—91no—
189C3×C42—91no—
190#217—181no—
191#217—181no—
192#217—181no—
193#215—182no—
194#214—182no—
195(C2×D3)⋊C4—181no—
196(C2×D3)⋊C4—181no—
197(C2×D3)⋊C4—181no—
198(C2×D3)⋊C4—181no—
199(C2×D3)⋊C4—181no—
200(C2×D3)⋊C4—181no—
201C4⋊Dic3—31no—
202C4⋊Dic3—91no—
203C4×Dic3—31no—
204C4×Dic3—91no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—91no—
207D24—62no—
208D24—62no—
209D24—64no—
210#204—182no—
211C22×C32—21no—
212C22×C32C2×D1211yescharacteristic
213C22×C32D3×D411yescharacteristic
214C2×C3×D3—62no—
215C2×C3×D3—62no—
216C2×C3×D3—122no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C2×C3×D3—122no—
222C2×C3×D3—121no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—241no—
229C2×C3×D3—241no—
230C2×C3×D3—241no—
231C2×C3×D3—241no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4#23511yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4D3×D411yescharacteristic
237C32×C4#23911yescharacteristic
238C3⋊Dic3—21no—
239C3⋊Dic3—21no—
240C3⋊Dic3—41no—
241C3⋊Dic3—41no—
242C3×Dic3—61no—
243C3×Dic3—61no—
244C3×Dic3—121no—
245C3×Dic3—121no—
246C2×SD16—271no—
247C2×D8—91no—
248#124—271no—
249C4×D4—271no—
250C4⋊C8—91no—
251#103—91no—
252#102—271no—
253H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
254C23×C3—181no—
255C23×C3—181no—
256C22×D3—61no—
257C22×D3—61no—
258C22×D3—61no—
259C22×D3—61no—
260C22×D3—121no—
261C22×D3—121no—
262C3×Q8—181no—
263C3×Q8—181no—
264C3×D4—92no—
265C3×D4—181no—
266C3×D4—181no—
267C3×D4—181no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—61no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—181no—
274C2×C3×C4—181no—
275C2×C3×C4C2×D3211yescharacteristic
276C2×Dic3—31no—
277C2×Dic3—31no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—91no—
281C2×Dic3—91no—
282C2×Dic3—181no—
283C2×Dic3—181no—
284C2×Dic3—362no—
285D12—32no—
286D12—32no—
287D12—62no—
288D12—61no—
289D12—61no—
290D12—61no—
291D12—61no—
292D12—61no—
293D12—61no—
294D12—62no—
295D12—121no—
296D12—121no—
297C4×D3—362no—
298C4×D3—362no—
299Dic6—61no—
300Dic6—61no—
301Dic6—181no—
302Dic6—181no—
303C3×C8—21no—
304C3×C8—61no—
305C3×C8—61no—
306C3×C8—62no—
307C3×C8—181no—
308C3⋊C8—61no—
309C3⋊C8—181no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32#88411yescharacteristic
314C2×C32#89511yescharacteristic
315C2×C32#90411yescharacteristic
316C2×C32#90811yescharacteristic
317C2×C32#91411yescharacteristic
318C2×C32#91911yescharacteristic
319C3×D3—122no—
320C3×D3—122no—
321C3×D3—242no—
322C3×D3—242no—
323C3×D3—241no—
324C3×D3—241no—
325C2×Q8—91no—
326C2×D4—91no—
327C2×D4—91no—
328C22×C4—541no—
329SD16—542no—
330D8—182no—
331C2×C8—91no—
332C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C4⋊C4—91no—
334C22⋊C4—541no—
335C22⋊C4—541no—
336C22⋊C4—541no—
337C42—91no—
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—61no—
341C22×C3—182no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3—361no—
345C22×C3—361no—
346C22×C3—361no—
347C22×C3D3×D1211yescharacteristic
348C2×D3—62no—
349C2×D3—62no—
350C2×D3—122no—
351C2×D3—121no—
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—121no—
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—121no—
361C2×D3—122no—
362C2×D3—241no—
363C2×D3—241no—
364C2×D3—241no—
365C2×D3—241no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—61no—
371C3×C4—61no—
372C3×C4—181no—
373C3×C4—181no—
374C3×C4—361no—
375C3×C4—361no—
376C3×C4#379911yescharacteristic
377C3×C4#381111yescharacteristic
378Dic3—61no—
379Dic3—61no—
380Dic3—121no—
381Dic3—121no—
382Dic3—181no—
383Dic3—181no—
384Dic3—361no—
385Dic3—361no—
386C32—21no—
387C32#481911yescharacteristic, complemented
388C32#484411yescharacteristic, complemented
389C23—181no—
390C23—181no—
391Q8—181no—
392Q8—181no—
393D4—92no—
394D4—181no—
395D4—181no—
396D4—181no—
397C2×C4—91no—
398C2×C4—91no—
399C2×C4—181no—
400C2×C4—181no—
401C2×C4—1082no—
402C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
403C8—21no—
404C8—181no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—61no—
412C2×C3—61no—
413C2×C3—61no—
414C2×C3—362no—
415C2×C3—721no—
416C2×C3#6379411yescharacteristic
417C2×C3#6380811yescharacteristic
418C2×C3#6391011yescharacteristic
419D3—122no—
420D3—122no—
421D3—242no—
422D3—241no—
423D3—241no—
424D3—242no—
425C22—182no—
426C22—361no—
427C22—361no—
428C22—361no—
429C22—361no—
430C22—361no—
431C22#8865411yescharacteristic
432C4—181no—
433C4—181no—
434C4—361no—
435C4—361no—
436C4#8865211yescharacteristic
437C4#8867711yescharacteristic
438C3—31no—
439C3—31no—
440C3—61no—
441C3#1059077011yescharacteristic
442C2—362no—
443C2—721no—
444C2#1172084811yescharacteristic
445C2#1172085011yescharacteristic
446C2#1172115711yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×#88529C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#463
LCSG ▷ C2×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. #11721149C2semidirect
3. #11721114C2semidirect
4. C2×#88437C2semidirect
5. #63643⋊C3C2semidirect
6. #63641⋊C3C2semidirect
7. #11720324C2semidirect
32. #63643D3semidirect
33. #63641D3semidirect
253. H(3,𝔽3)#463semidirect
387. C32#4819semidirect
388. C32#4844semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
#117211492
#117211142
C2×#884372
#63643⋊C32
#63641⋊C32
#117203242
#105907553
#105907513