Finite Group Databasebeta

← #10692501

Subgroups — #10692501

Order 768.

570 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78601C211yesmaximal, characteristic
2#73358C211yesmaximal, characteristic
3#69946C211yesmaximal, characteristic, complemented
4#69530C211yesmaximal, characteristic
5#69510C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9383—31nomaximal, Sylow
9C2×C4×#67C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13C2×C3×#95C2211yescharacteristic
14#5213—21no—
15#5212—21no—
16#5212C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C8×#67—22no—
19C2×#806—21no—
20#67⋊C8—22no—
21D3×C22⋊C8—21no—
22#4718—21no—
23#4677C411yescharacteristic, complemented
24#4673C411yescharacteristic, complemented
25#4668—21no—
26#4659C411yescharacteristic, complemented
27#4656C411yescharacteristic, complemented
28#4653—21no—
29#4527—21no—
30#4467—22no—
31#4452—22no—
32#2972—31no—
33#2564—31no—
34#1799—31no—
35#1520—31no—
36#1520—31no—
37#1294—31no—
38#1294D311yescharacteristic, complemented
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67D411yescharacteristic
42C22×C4×D3D411yescharacteristic
43C22×Dic6C2×C411yescharacteristic
44C22×C3×C8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×#217—21no—
48C2×#217—21no—
49C2×#217D411yescharacteristic
50C4×#67—21no—
51C4×#67—21no—
52C4×#67—21no—
53C4×#67—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4C2×C411yescharacteristic
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3C2×C411yescharacteristic
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4C2×C411yescharacteristic
67C2×C4×Dic3D411yescharacteristic
68C22×C3⋊C8—21no—
69C2×D3×C8—22no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8D412yes—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95C2×C411yescharacteristic
75C3×#95C2×C411yescharacteristic
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#830D412yes—
83(C2×D3)⋊C8—22no—
84Dic3⋊C8—22no—
85C8×Dic3—22no—
86#806—42no—
87C2×C22⋊Q8—31no—
88C2×C4×D4—31no—
89C8×D4—62no—
90C8×D4—62no—
91#431—61no—
92C2×#99—61no—
93C2×C22⋊C8—31no—
94C2×C22⋊C8—61no—
95C2×C22⋊C8C2×D311yescharacteristic
96#400—31no—
97#399—31no—
98#395—31no—
99#393—31no—
100#387—61no—
101#386—61no—
102#385—61no—
103#380—31no—
104#380—61no—
105#380C2×D311yescharacteristic
106C2×#95C2×D311yescharacteristic
107#342—61no—
108#328—62no—
109#328—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3C22⋊C411yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—81no—
119C22×C3×C4—81no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—21no—
127C2×#67—21no—
128C2×#67—21no—
129C2×#67—21no—
130C2×#67SD1612yes—
131C2×#67Q1612yes—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—41no—
135C22×Dic3C22⋊C411yescharacteristic
136C22×Dic3C22⋊C411yescharacteristic
137C22×Dic3C22⋊C411yescharacteristic
138C2×C4×D3—21no—
139C2×C4×D3—21no—
140C2×C4×D3—81no—
141C2×Dic6—21no—
142C2×Dic6—21no—
143C2×Dic6—81no—
144C2×C3×C8—22no—
145C2×C3×C8—81no—
146C3×C4⋊C4—81no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149#217—21no—
150#217—21no—
151#217—41no—
152#217—41no—
153#217—41no—
154#217—41no—
155(C2×D3)⋊C4—21no—
156(C2×D3)⋊C4—21no—
157(C2×D3)⋊C4—42no—
158(C2×D3)⋊C4—81no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—42no—
162C4⋊Dic3—81no—
163C4⋊Dic3—81no—
164Dic3⋊C4—21no—
165Dic3⋊C4—21no—
166Dic3⋊C4—41no—
167Dic3⋊C4—41no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—81no—
172D3×C8—82no—
173C22×Q8—31no—
174C22×D4—31no—
175C23×C4—21no—
176C23×C4—31no—
177C23×C4C22×D311yescharacteristic
178C22×C8—21no—
179C22×C8—62no—
180C22×C8—61no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C22⋊Q8—61no—
184C22⋊Q8—61no—
185C4×D4—61no—
186C4×D4—61no—
187C4×D4—61no—
188C4×D4—61no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—31no—
198C2×C22⋊C4—31no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C42—31no—
204C4⋊C8—62no—
205#99—122no—
206C22⋊C8—21no—
207C22⋊C8—32no—
208C22⋊C8—62no—
209C22⋊C8—61no—
210C22⋊C8D1212yes—
211C4×C8—62no—
212#95—21no—
213#95—21no—
214#95—61no—
215#95—61no—
216#95—61no—
217#95—61no—
218#95—61no—
219#95C4×D311yescharacteristic
220#95C4×D311yescharacteristic
221C23×C3—41no—
222C23×C3—41no—
223C23×C3C2×C22⋊C411yescharacteristic
224C23×C3#12111yescharacteristic
225C23×C3#12111yescharacteristic
226C23×C3C2×SD1611yescharacteristic
227C23×C3C2×Q1611yescharacteristic
228C23×C3#13511yescharacteristic
229C23×C3#13511yescharacteristic
230C22×D3—21no—
231C22×D3—21no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3#10312yes—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—82no—
248C2×C3×C4—82no—
249C2×C3×C4—81no—
250C2×C3×C4—81no—
251C2×C3×C4—81no—
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12311yescharacteristic
256#67—22no—
257#67—22no—
258#67—22no—
259#67—22no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—21no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—81no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275C2×Dic3#10312yes—
276C4×D3—42no—
277C4×D3—81no—
278Dic6—81no—
279Dic6—81no—
280C3×C8—42no—
281C3⋊C8—42no—
282C24—31no—
283C24D3×D411yescharacteristic
284C2×Q8—61no—
285C2×Q8—61no—
286C2×Q8—241no—
287C2×D4—32no—
288C2×D4—32no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—81no—
307C22×C4—81no—
308C22×C4—121no—
309C22×C4—241no—
310C22×C4C2×C4×D311yescharacteristic
311C22×C4C2×D1211yescharacteristic
312C22×C4#23511yescharacteristic
313C22×C4D3×D411yescharacteristic
314C22×C4D3×D411yescharacteristic
315C22×C4#23911yescharacteristic
316C2×C8—22no—
317C2×C8—62no—
318C2×C8—81no—
319C2×C8—242no—
320C2×C8—241no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—81no—
326C4⋊C4—122no—
327C4⋊C4—121no—
328C4⋊C4—121no—
329C4⋊C4—241no—
330C4⋊C4—241no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—122no—
338C22⋊C4—121no—
339C22⋊C4—121no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C22⋊C4—241no—
343C42—61no—
344C42—61no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3#38611yescharacteristic
356C22×C3C2×#10311yescharacteristic
357C22×C3#41611yescharacteristic
358C22×C3#46511yescharacteristic
359C22×C3#46511yescharacteristic
360C22×C3#46711yescharacteristic
361C22×C3#46711yescharacteristic
362C2×D3—22no—
363C2×D3—41no—
364C2×D3—41no—
365C2×D3—42no—
366C2×D3—81no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—42no—
372C3×C4—81no—
373C3×C4—161no—
374Dic3—22no—
375Dic3—41no—
376Dic3—41no—
377Dic3—81no—
378Dic3—81no—
379Dic3—161no—
380C23—32no—
381C23—41no—
382C23—41no—
383C23—61no—
384C23—61no—
385C23—121no—
386C23—121no—
387C23D3×C22⋊C411yescharacteristic
388C23#88411yescharacteristic
389C23#89511yescharacteristic
390C23#91211yescharacteristic
391C23D3×SD1611yescharacteristic
392C23#91611yescharacteristic
393C23D3×Q1611yescharacteristic
394Q8—241no—
395Q8—241no—
396D4—62no—
397D4—62no—
398D4—62no—
399D4—62no—
400C2×C4—22no—
401C2×C4—32no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—122no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—241no—
432C2×C4—241no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4C4×D1211yescharacteristic
436C2×C4#88211yescharacteristic
437C2×C4#89311yescharacteristic
438C2×C4#89411yescharacteristic
439C8—42no—
440C8—122no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3#188411yescharacteristic
448C2×C3#189211yescharacteristic
449C2×C3#191311yescharacteristic
450D3—42no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—62no—
462C22—122no—
463C22—121no—
464C22—121no—
465C22—241no—
466C22#466511yescharacteristic
467C22#477211yescharacteristic
468C22#477811yescharacteristic
469C22D3×#10311yescharacteristic
470C22#481311yescharacteristic
471C22#486411yescharacteristic
472C22#488711yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—42no—
478C4—62no—
479C4—81no—
480C4—121no—
481C4—121no—
482C4—161no—
483C4—241no—
484C4—241no—
485C4—481no—
486C3#938311yescharacteristic, complemented, Sylow
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—122no—
494C2#6877711yescharacteristic
495C2#6888211yescharacteristic
496C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC22×C3#386
prime radicalC22×#67D4
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9383
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
3. #69946C2semidirect
5. #69510C2semidirect
6. C3×#1294C2semidirect
7. #67612C2semidirect
23. #4677C4semidirect
24. #4673C4semidirect
26. #4659C4semidirect
27. #4656C4semidirect
38. #1294D3semidirect
486. C3#9383semidirect

Maximal subgroups

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

subgroupindex
#786012
#733582
#699462
#695302
#695102
C3×#12942
#676122
#93833