Finite Group Databasebeta

← #13139037

Subgroups — #13139037

Order 1728.

629 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 87 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×#88649C211yesmaximal, characteristic
2C2×C8×#272C211yesmaximal, characteristic, complemented
3#11721089C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic, complemented
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10590954—31nomaximal
9#10590690—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#6200—21no—
14C2×((C3⋊D3)⋊C3)⋊C4—21no—
15#88649—22no—
16#88649C414yescomplemented
17C2×C8×H(3,𝔽3)C2211yescharacteristic
18C2×#6180C2211yescharacteristic
19C8×#272—22no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
22#88451—21no—
23#88447—21no—
24C2×#3799—31no—
25C2×#3795—31no—
26C2×C3×D3×C8—31no—
27C2×C3×D3×C8—31no—
28C2×C3×D3×C8—61no—
29#63841—31no—
30#63841—31no—
31#63648D311yescharacteristic, complemented
32#63643D311yescharacteristic, complemented
33C3×#821—31no—
34C3×#816—31no—
35#63567—31no—
36#63566—31no—
37#63564—31no—
38#63563—31no—
39C22×#272D411yescharacteristic
40C22×(C3⋊D3)⋊C3—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084D411yescharacteristic
43C4×#272C2×C412yes—
44C4×#272D412yes—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)C2×C412yes—
50#634⋊C3—21no—
51#634⋊C3C2×C412yes—
52#6200—24no—
53((C3⋊D3)⋊C3)⋊C4—24no—
54C8×H(3,𝔽3)—21no—
55#6180—21no—
56#5113—91no—
57#4804—91no—
58#4772—91no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×C4×D3—31no—
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—31no—
66C2×#626—61no—
67C2×#625—61no—
68C2×#624—61no—
69C2×D3×Dic3—61no—
70#3799—34no—
71#3799—62no—
72#3795—34no—
73#3795—62no—
74C2×C32×C8—21no—
75C2×C32×C8C2×D311yescharacteristic
76C2×C32×C8C2×D311yescharacteristic
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—61no—
84C3×D3×C8—62no—
85C3×D3×C8—62no—
86C3×D3×C8—64no—
87(C2×D3)⋊Dic3—61no—
88(C2×D3)⋊Dic3—61no—
89Dic32—61no—
90Dic32—61no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×#272—21no—
93C2×#272—21no—
94C2×#272C22⋊C412yes—
95C2×(C3⋊D3)⋊C3—22no—
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3—41no—
98C4×H(3,𝔽3)C22×C411yescharacteristic
99C4×H(3,𝔽3)C2×D411yescharacteristic
100#1084C22⋊C412yes—
101(C3⋊Dic3)⋊C3—22no—
102(C3⋊Dic3)⋊C3—41no—
103(C3⋊Dic3)⋊C3—41no—
104C2×#239—91no—
105C2×#237—91no—
106C2×#235—91no—
107C22×C3×C8—91no—
108#926—91no—
109C2×D3×C8—31no—
110C2×D3×C8—31no—
111C2×D3×C8—61no—
112#892—91no—
113#892—91no—
114C3×#103—91no—
115C3×#102—91no—
116#835—91no—
117#832—91no—
118#821—31no—
119#821—91no—
120#821—91no—
121#816—31no—
122#816—91no—
123#816—91no—
124#810—91no—
125#809—91no—
126C22×C3⋊D3—21no—
127C22×C3×D3—31no—
128C22×C3×D3—31no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C2×C32×C4—21no—
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)C4×D312yes—
139#634—21no—
140#634C4×D312yes—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C3×D12—32no—
148C3×D12—61no—
149C3×C4×D3—32no—
150C3×C4×D3—32no—
151C3×C4×D3—32no—
152C3×C4×D3—32no—
153C3×C4×D3—62no—
154C3×C4×D3—62no—
155C3×Dic6—32no—
156C3×Dic6—61no—
157#626—64no—
158#625—64no—
159#624—64no—
160D3×Dic3—64no—
161C32×C8—22no—
162C32×C8—21no—
163C32×C8—21no—
164C3×C3⋊C8—62no—
165C3×C3⋊C8—61no—
166C3×C3⋊C8—61no—
167#416—271noSylow
168C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
169C2×H(3,𝔽3)#13511yescharacteristic
170C2×H(3,𝔽3)#13511yescharacteristic
171#272—22no—
172(C3⋊D3)⋊C3—42no—
173C2×C3×Q8—91no—
174C2×C3×D4—91no—
175C22×C3×C4—91no—
176C2×#67—181no—
177C2×#67—181no—
178C22×Dic3—181no—
179#239—94no—
180#239—182no—
181#237—94no—
182#237—182no—
183#235—94no—
184#235—182no—
185C2×D12—31no—
186C2×D12—91no—
187C2×D12—181no—
188C2×C4×D3—31no—
189C2×C4×D3—31no—
190C2×C4×D3—61no—
191C2×C4×D3—181no—
192C2×C4×D3—181no—
193C2×Dic6—31no—
194C2×Dic6—91no—
195C2×C3×C8—31no—
196C2×C3×C8—31no—
197C2×C3×C8—61no—
198C2×C3×C8—91no—
199C2×C3×C8—182no—
200C2×C3×C8D3211yescharacteristic
201C3×C4⋊C4—91no—
202C3×C4⋊C4—91no—
203#217—181no—
204(C2×D3)⋊C4—181no—
205(C2×D3)⋊C4—181no—
206C4⋊Dic3—31no—
207C4⋊Dic3—31no—
208C4⋊Dic3—91no—
209C4⋊Dic3—91no—
210C4×Dic3—181no—
211C4×Dic3—181no—
212C4×Dic3—181no—
213C2×C3⋊C8—31no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—61no—
216D3×C8—62no—
217D3×C8—62no—
218D3×C8—64no—
219C22×C32—21no—
220C22×C32D3×D411yescharacteristic
221C22×C32D3×D411yescharacteristic
222C2×C3⋊D3—22no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—41no—
225C2×C3×D3—32no—
226C2×C3×D3—32no—
227C2×C3×D3—62no—
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C2×C3×D3—61no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C2×C3×D3—62no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4C2×C4×D311yescharacteristic
240C32×C4C2×C4×D311yescharacteristic
241C32×C4D3×D411yescharacteristic
242C32×C4D3×D411yescharacteristic
243C3⋊Dic3—22no—
244C3⋊Dic3—41no—
245C3⋊Dic3—41no—
246C3×Dic3—32no—
247C3×Dic3—32no—
248C3×Dic3—62no—
249C3×Dic3—62no—
250C3×Dic3—121no—
251C3×Dic3—121no—
252C2×#41—271no—
253C22×C8—91no—
254#117—271no—
255#103—91no—
256#103—271no—
257#102—91no—
258#102—271no—
259H(3,𝔽3)#41611yescharacteristic, complemented, Sylow
260C23×C3—91no—
261C23×C3—181no—
262C22×D3—31no—
263C22×D3—31no—
264C22×D3—61no—
265C22×D3—61no—
266C22×D3—181no—
267C3×Q8—92no—
268C3×Q8—181no—
269C3×D4—92no—
270C3×D4—181no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—61no—
274C2×C3×C4—91no—
275C2×C3×C4—92no—
276C2×C3×C4—92no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4C2×D3211yescharacteristic
281#67—184no—
282#67—184no—
283C2×Dic3—31no—
284C2×Dic3—31no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—181no—
290C2×Dic3—181no—
291C2×Dic3—181no—
292C2×Dic3—184no—
293D12—32no—
294D12—61no—
295D12—92no—
296D12—181no—
297D12—184no—
298C4×D3—32no—
299C4×D3—32no—
300C4×D3—32no—
301C4×D3—32no—
302C4×D3—62no—
303C4×D3—62no—
304C4×D3—184no—
305C4×D3—184no—
306Dic6—32no—
307Dic6—61no—
308Dic6—92no—
309Dic6—181no—
310C3×C8—21no—
311C3×C8—61no—
312C3×C8—61no—
313C3×C8—62no—
314C3×C8—181no—
315C3⋊C8—61no—
316C3⋊C8—61no—
317C3⋊C8—62no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32D3×C22⋊C411yescharacteristic
322C2×C32D3×C22⋊C411yescharacteristic
323C2×C32#91211yescharacteristic
324C2×C32#91611yescharacteristic
325C2×C32#91611yescharacteristic
326C2×C32#91911yescharacteristic
327C3⋊D3—42no—
328C3×D3—64no—
329C3×D3—62no—
330C3×D3—62no—
331C3×D3—122no—
332#41—274no—
333#41—542no—
334C2×Q8—91no—
335C2×D4—91no—
336C2×D4—541no—
337C22×C4—91no—
338C22×C4—541no—
339C2×C8—91no—
340C2×C8—182no—
341C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C4⋊C4—91no—
343C4⋊C4—91no—
344C22⋊C4—541no—
345C42—541no—
346C22×C3—31no—
347C22×C3—31no—
348C22×C3—61no—
349C22×C3—92no—
350C22×C3—182no—
351C22×C3—181no—
352C22×C3—181no—
353C22×C3—361no—
354C22×C3—361no—
355C22×C3#380211yescharacteristic
356C2×D3—32no—
357C2×D3—32no—
358C2×D3—62no—
359C2×D3—61no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—61no—
363C2×D3—62no—
364C2×D3—62no—
365C2×D3—62no—
366C2×D3—121no—
367C2×D3—121no—
368C2×D3—182no—
369C2×D3—361no—
370C2×D3—361no—
371C3×C4—31no—
372C3×C4—31no—
373C3×C4—31no—
374C3×C4—31no—
375C3×C4—61no—
376C3×C4—61no—
377C3×C4—92no—
378C3×C4—182no—
379C3×C4—361no—
380C3×C4—361no—
381C3×C4C4×D3211yescharacteristic
382C3×C4#381111yescharacteristic
383Dic3—32no—
384Dic3—32no—
385Dic3—62no—
386Dic3—62no—
387Dic3—121no—
388Dic3—121no—
389Dic3—182no—
390Dic3—361no—
391Dic3—361no—
392C32—21no—
393C32#477211yescharacteristic, complemented
394C32#480411yescharacteristic, complemented
395C23—91no—
396C23—181no—
397Q8—92no—
398Q8—181no—
399D4—92no—
400D4—181no—
401D4—544no—
402C2×C4—91no—
403C2×C4—92no—
404C2×C4—92no—
405C2×C4—181no—
406C2×C4—181no—
407C2×C4—181no—
408C2×C4—544no—
409C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
410C8—21no—
411C8—181no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—31no—
416C2×C3—31no—
417C2×C3—31no—
418C2×C3—61no—
419C2×C3—61no—
420C2×C3—61no—
421C2×C3—182no—
422C2×C3—362no—
423C2×C3#6380911yescharacteristic
424C2×C3#6381011yescharacteristic
425C2×C3#6392111yescharacteristic
426D3—62no—
427D3—62no—
428D3—64no—
429D3—122no—
430D3—362no—
431C22—92no—
432C22—181no—
433C22—181no—
434C22—182no—
435C22—361no—
436C22—361no—
437C22#8865511yescharacteristic
438C4—92no—
439C4—182no—
440C4—361no—
441C4—361no—
442C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
443C4#8867711yescharacteristic
444C3—31no—
445C3—31no—
446C3—61no—
447C3#1059095611yescharacteristic
448C2—182no—
449C2—362no—
450C2#1172084311yescharacteristic
451C2#1172084411yescharacteristic
452C2#1172116111yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×#88649C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#416
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×C8×#272C2semidirect
3. #11721089C2semidirect
4. #63648⋊C3C2semidirect
5. #63643⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720313C2semidirect
16. #88649C4semidirect
31. #63648D3semidirect
32. #63643D3semidirect
259. H(3,𝔽3)#416semidirect
393. C32#4772semidirect
394. C32#4804semidirect

Maximal subgroups

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

subgroupindex
C2×#886492
C2×C8×#2722
#117210892
#63648⋊C32
#63643⋊C32
#117203182
#117203132
#105909543
#105906903