Finite Group Databasebeta

← #13147829

Subgroups — #13147829

Order 1728.

742 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#11721716C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic
3#64079⋊C3C211yesmaximal, characteristic, complemented
4#11721134C212yesmaximal, complemented
5#11721133C212yesmaximal, complemented
6#11721125C212yesmaximal
7#11721111C212yesmaximal, complemented
8#11721108C212yesmaximal, complemented
9#11721079C212yesmaximal, complemented
10#10593952—31nomaximal
11#10593857—31nomaximal
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
15C2×#6196C2212yes—
16C42×H(3,𝔽3)C2211yescharacteristic
17C4⋊#1084C2211yescharacteristic
18C4⋊#1084C2211yescharacteristic
19#88540C2212yes—
20#88540C2212yes—
21#3752⋊C3C2212yes—
22#3752⋊C3C2212yes—
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25#3750⋊C3C2212yes—
26#3750⋊C3C2212yes—
27#88450C2212yes—
28#88450C2212yes—
29#88450C2212yes—
30#88449C2212yes—
31#88448C2212yes—
32#88447C2212yes—
33#64084D311yescharacteristic
34#64079D311yescharacteristic, complemented
35C3×#875—31no—
36C3×#870—31no—
37C3×#870—31no—
38C3×#870—31no—
39C3×#870—61no—
40#63899—32no—
41#63897—32no—
42#63895—32no—
43#63895—32no—
44#63893—32no—
45#63884—32no—
46#63861—32no—
47#63858—32no—
48#63848—32no—
49#63848—32no—
50#63840—32no—
51#63839—32no—
52C22×(C3⋊D3)⋊C3C2312yes—
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×#1084C2312yes—
57C2×#1084C2312yes—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×(C3⋊Dic3)⋊C3C2312yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C4⋊((C3⋊D3)⋊C3)—41no—
62#634⋊C3—41no—
63((C3⋊D3)⋊C3)⋊C4—42no—
64#6196—42no—
65#5699—91no—
66#5699—91no—
67#5672—91no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×#634C2×D311yescharacteristic
70C2×C3×D12—31no—
71C2×C3×Dic6—31no—
72C2×#627—32no—
73C2×#627—32no—
74C2×#624—32no—
75C2×D3×Dic3—32no—
76C32×C42—21no—
77C32×C42C2×D311yescharacteristic
78C32×C42C2×D311yescharacteristic
79#3752C2×D312yes—
80#3752C2×D312yes—
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic
83#3750C2×D312yes—
84#3750C2×D312yes—
85C3×(C2×D3)⋊C4—32no—
86C3×(C2×D3)⋊C4—32no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3—61no—
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—32no—
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4—32no—
101C3×Dic3⋊C4—62no—
102C3×Dic3⋊C4—62no—
103#3724—32no—
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107#3722—32no—
108#3722—32no—
109#3722—32no—
110(C2×D3)⋊Dic3—32no—
111(C2×D3)⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113Dic32—32no—
114Dic32—32no—
115C22×H(3,𝔽3)C2411yescharacteristic
116C2×(C3⋊D3)⋊C3—22no—
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3—22no—
119C4×H(3,𝔽3)—21no—
120C4×H(3,𝔽3)—21no—
121C4×H(3,𝔽3)—21no—
122#1084—22no—
123#1084—22no—
124(C3⋊Dic3)⋊C3—22no—
125(C3⋊Dic3)⋊C3—22no—
126(C3⋊Dic3)⋊C3—22no—
127C3×#125—91no—
128C3×#125—91no—
129C3×#124—91no—
130#944—92no—
131#935—92no—
132#898—92no—
133#898—92no—
134#898—92no—
135#898—92no—
136#897—92no—
137#897—92no—
138#896—92no—
139#896—92no—
140#894—92no—
141#894—92no—
142#888—92no—
143#888—92no—
144#879—92no—
145#879—92no—
146#878—92no—
147#878—92no—
148#875—31no—
149#875—91no—
150#870—31no—
151#870—31no—
152#870—31no—
153#870—61no—
154#870—91no—
155C22×C3⋊D3C22×D312yes—
156C22×C3×D3—32no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C3⋊Dic3C22×D312yes—
167C2×C3⋊Dic3C22×D312yes—
168C2×C3⋊Dic3C22×D312yes—
169C4⋊(C3⋊D3)—41no—
170#634—41no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—62no—
180C3×D12—121no—
181C3×Dic6—121no—
182#627—122no—
183#627—122no—
184#624—122no—
185D3×Dic3—122no—
186#563—271noSylow
187C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
188C2×H(3,𝔽3)2−1+411yescharacteristic
189C2×H(3,𝔽3)2−1+411yescharacteristic
190(C3⋊D3)⋊C3—42no—
191C2×C3×Q8—91no—
192C2×C3×D4—91no—
193C22×Dic3—92no—
194C2×D12—31no—
195C2×D12—91no—
196C2×C4×D3—92no—
197C2×C4×D3—92no—
198C2×Dic6—31no—
199C2×Dic6—91no—
200C2×Dic6—92no—
201C2×Dic6—92no—
202C2×Dic6—92no—
203C3×C4⋊C4—92no—
204C3×C4⋊C4—91no—
205C3×C4⋊C4—92no—
206C3×C4⋊C4—91no—
207C3×C4⋊C4—92no—
208C3×C4⋊C4—91no—
209C3×C4⋊C4—92no—
210C3×C4⋊C4—91no—
211C3×C22⋊C4—92no—
212C3×C22⋊C4—92no—
213C3×C42—31no—
214C3×C42—31no—
215C3×C42—61no—
216C3×C42D3211yescharacteristic
217#217—92no—
218#217—92no—
219#217—92no—
220(C2×D3)⋊C4—32no—
221(C2×D3)⋊C4—32no—
222(C2×D3)⋊C4—92no—
223(C2×D3)⋊C4—92no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4—92no—
226(C2×D3)⋊C4—92no—
227(C2×D3)⋊C4—92no—
228(C2×D3)⋊C4—92no—
229(C2×D3)⋊C4—92no—
230C4⋊Dic3—31no—
231C4⋊Dic3—31no—
232C4⋊Dic3—31no—
233C4⋊Dic3—31no—
234C4⋊Dic3—31no—
235C4⋊Dic3—31no—
236C4⋊Dic3—61no—
237C4⋊Dic3—61no—
238C4⋊Dic3—91no—
239C4⋊Dic3—91no—
240C4⋊Dic3—92no—
241C4⋊Dic3—92no—
242Dic3⋊C4—32no—
243Dic3⋊C4—32no—
244Dic3⋊C4—32no—
245Dic3⋊C4—32no—
246Dic3⋊C4—32no—
247Dic3⋊C4—32no—
248Dic3⋊C4—62no—
249Dic3⋊C4—62no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256C4×Dic3—92no—
257C4×Dic3—92no—
258C4×Dic3—92no—
259C22×C32—21no—
260C22×C32C23×D311yescharacteristic
261C22×C32C23×D311yescharacteristic
262C2×C3⋊D3—22no—
263C2×C3⋊D3—22no—
264C2×C3⋊D3—22no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C32×C4—22no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—41no—
276C32×C4—41no—
277C3⋊Dic3—22no—
278C3⋊Dic3—22no—
279C3⋊Dic3—22no—
280C3×Dic3—62no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3—62no—
286C3×Dic3—62no—
287C3×Dic3—122no—
288C3×Dic3—122no—
289C4⋊Q8—272no—
290#126—272no—
291#126—272no—
292#125—91no—
293#125—91no—
294#124—91no—
295#123—272no—
296C22⋊Q8—272no—
297C22⋊Q8—272no—
298H(3,𝔽3)#56311yescharacteristic, complemented, Sylow
299C23×C3—92no—
300C22×D3—32no—
301C22×D3—92no—
302C3×Q8—361no—
303C3×D4—361no—
304C2×C3×C4—31no—
305C2×C3×C4—31no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—31no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—92no—
314C2×C3×C4—92no—
315C2×C3×C4—92no—
316C2×C3×C4—92no—
317C2×C3×C4—92no—
318C2×C3×C4C2×D3211yescharacteristic
319C2×C3×C4C2×D3211yescharacteristic
320C2×C3×C4C2×D3211yescharacteristic
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3—32no—
328C2×Dic3—62no—
329C2×Dic3—62no—
330C2×Dic3—92no—
331C2×Dic3—92no—
332C2×Dic3—92no—
333C2×Dic3—362no—
334D12—121no—
335D12—361no—
336C4×D3—362no—
337C4×D3—362no—
338Dic6—121no—
339Dic6—361no—
340Dic6—362no—
341Dic6—362no—
342Dic6—362no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32#100411yescharacteristic
347C2×C32#100711yescharacteristic
348C2×C32#100911yescharacteristic
349C2×C32#101011yescharacteristic
350C2×C32#101011yescharacteristic
351C2×C32#101011yescharacteristic
352C3⋊D3—42no—
353C3×D3—122no—
354C2×Q8—91no—
355C2×Q8—272no—
356C2×D4—91no—
357C22×C4—272no—
358C4⋊C4—91no—
359C4⋊C4—91no—
360C4⋊C4—92no—
361C4⋊C4—92no—
362C4⋊C4—92no—
363C4⋊C4—92no—
364C4⋊C4—91no—
365C4⋊C4—91no—
366C4⋊C4—272no—
367C4⋊C4—272no—
368C22⋊C4—92no—
369C22⋊C4—92no—
370C22⋊C4—272no—
371C22⋊C4—272no—
372C22⋊C4—272no—
373C42—272no—
374C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
375C22×C3—31no—
376C22×C3—31no—
377C22×C3—61no—
378C22×C3—182no—
379C22×C3—182no—
380C22×C3—182no—
381C22×C3C22×D3211yescharacteristic
382C2×D3—62no—
383C2×D3—62no—
384C2×D3—62no—
385C2×D3—182no—
386C2×D3—182no—
387C2×D3—182no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—61no—
392C3×C4—61no—
393C3×C4—61no—
394C3×C4—61no—
395C3×C4—61no—
396C3×C4—61no—
397C3×C4—62no—
398C3×C4—121no—
399C3×C4—121no—
400C3×C4—182no—
401C3×C4—182no—
402C3×C4—182no—
403C3×C4—182no—
404C3×C4—182no—
405Dic3—62no—
406Dic3—62no—
407Dic3—62no—
408Dic3—62no—
409Dic3—62no—
410Dic3—62no—
411Dic3—62no—
412Dic3—122no—
413Dic3—122no—
414Dic3—182no—
415Dic3—182no—
416Dic3—182no—
417C32—21no—
418C32#567211yescharacteristic, complemented
419C32#569911yescharacteristic, complemented
420C23—92no—
421Q8—361no—
422Q8—1082no—
423D4—361no—
424C2×C4—92no—
425C2×C4—92no—
426C2×C4—92no—
427C2×C4—92no—
428C2×C4—92no—
429C2×C4—1082no—
430C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
431C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
432C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—61no—
440C2×C3—61no—
441C2×C3—61no—
442C2×C3—362no—
443C2×C3#6429711yescharacteristic
444C2×C3#6429811yescharacteristic
445C2×C3#6430711yescharacteristic
446D3—122no—
447D3—362no—
448C22—182no—
449C22—182no—
450C22—182no—
451C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—182no—
456C4—182no—
457C4—182no—
458C4—182no—
459C4—182no—
460C3—31no—
461C3—31no—
462C3—61no—
463C3#1059395211yescharacteristic
464C2—362no—
465C2#1172248211yescharacteristic
466C2#1172248311yescharacteristic
467C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#563
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721716C2semidirect
3. #64079⋊C3C2semidirect
4. #11721134C2semidirect
5. #11721133C2semidirect
7. #11721111C2semidirect
8. #11721108C2semidirect
9. #11721079C2semidirect
34. #64079D3semidirect
298. H(3,𝔽3)#563semidirect
418. C32#5672semidirect
419. C32#5699semidirect

Maximal subgroups

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

subgroupindex
#117217162
#64084⋊C32
#64079⋊C32
#117211342
#117211332
#117211252
#117211112
#117211082
#117210792
#105939523
#105938573