Finite Group Databasebeta

← #10692978

Subgroups — #10692978

Order 768.

576 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 109 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#82196C211yesmaximal, characteristic
2#78602C211yesmaximal, characteristic
3C3×#1823C211yesmaximal, characteristic, complemented
4#71553C211yesmaximal, characteristic, complemented
5#69530C211yesmaximal, characteristic
6#67676C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9474—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#929—21no—
11C2×C4×#67C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14#5212C2211yescharacteristic
15#5212C2211yescharacteristic
16#5211—21no—
17C2×#830C2211yescharacteristic
18#5111—22no—
19#4992C412yescomplemented
20#4987—21no—
21#4983C412yescomplemented
22#4972—21no—
23C2×#806—21no—
24#4730—22no—
25#4726—21no—
26#4719—21no—
27C3×#340—22no—
28#4529—22no—
29#4527—21no—
30#2973—31no—
31#2486—31no—
32#1823—31no—
33#1823D311yescharacteristic, complemented
34#1799—31no—
35#1289—31no—
36#1289—31no—
37C23×C3×C4—21no—
38C23×C3×C4C2311yescharacteristic
39C22×#67D411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C22×C3×C8D412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4C2×C412yes—
49C2×C3×C22⋊C4—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217D411yescharacteristic
53#929—42no—
54#929—42no—
55C4×#67—21no—
56C4×#67—21no—
57C4×#67—21no—
58C4×#67—21no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4C2×C411yescharacteristic
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×Dic3⋊C4—22no—
64C2×Dic3⋊C4C2×C411yescharacteristic
65C2×C4×Dic3D411yescharacteristic
66C22×C3⋊C8D412yes—
67C2×ΓL(1,𝔽25)—22no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#95—21no—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95—21no—
74C3⋊#95C2×C412yes—
75#830—21no—
76#830—21no—
77(C2×D3)⋊C8—22no—
78#815—22no—
79Dic3⋊C8—22no—
80#806—42no—
81C2×#123—61no—
82C2×C4×D4—31no—
83C22×C4⋊C4C2×D311yescharacteristic
84#435—62no—
85#435—62no—
86#432—61no—
87C2×#99—61no—
88#407—61no—
89C2×C22⋊C8—31no—
90C2×C22⋊C8C2×D311yescharacteristic
91#398—32no—
92#397—32no—
93#389—61no—
94#381—61no—
95#380—31no—
96#380—31no—
97#380—61no—
98#380C2×D311yescharacteristic
99#342—61no—
100#340—22no—
101#340—62no—
102C24×C3C2×D411yescharacteristic
103C23×D3C22⋊C411yescharacteristic
104C22×C3×C4—22no—
105C22×C3×C4—42no—
106C22×C3×C4—41no—
107C22×C3×C4—41no—
108C22×C3×C4—41no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C2×#67—21no—
120C2×#67—21no—
121C2×#67—21no—
122C2×#67—21no—
123C2×#67—21no—
124C2×#67—21no—
125C22×Dic3—22no—
126C22×Dic3—41no—
127C22×Dic3C22⋊C411yescharacteristic
128C22×Dic3C22⋊C411yescharacteristic
129C22×Dic3C22⋊C411yescharacteristic
130C2×C4×D3—21no—
131C2×C4×D3—21no—
132C2×C4×D3—42no—
133C2×C3×C8—22no—
134C2×C3×C8—42no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—42no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142#217—21no—
143#217—21no—
144#217—41no—
145#217—41no—
146#217—41no—
147#217—41no—
148(C2×D3)⋊C4—21no—
149(C2×D3)⋊C4—21no—
150(C2×D3)⋊C4—44no—
151C4⋊Dic3—42no—
152C4⋊Dic3—42no—
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155Dic3⋊C4—44no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—42no—
160ΓL(1,𝔽25)—82no—
161C22×D4—31no—
162C23×C4—21no—
163C23×C4—31no—
164C23×C4C22×D311yescharacteristic
165C2×M16—62no—
166C22×C8—32no—
167C22×C8#6712yes—
168#123—122no—
169#123—122no—
170C4×D4—61no—
171C4×D4—61no—
172C4×D4—61no—
173C4×D4—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4C4×D312yes—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C42—31no—
192C4⋊C8—62no—
193#99—122no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C22⋊C8—62no—
197C22⋊C8—61no—
198C22⋊C8—61no—
199#97—62no—
200#95—21no—
201#95—32no—
202#95—62no—
203#95—62no—
204#95—61no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3C2×C22⋊C411yescharacteristic
208C23×C3#12111yescharacteristic
209C23×C3#12111yescharacteristic
210C23×C3AGL(1,ℤ/8)11yescharacteristic
211C23×C3AGL(1,ℤ/8)11yescharacteristic
212C23×C3AGL(1,ℤ/8)11yescharacteristic
213C23×C3#13711yescharacteristic
214C22×D3—21no—
215C22×D3—21no—
216C22×D3—21no—
217C22×D3—41no—
218C22×D3—41no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—81no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12311yescharacteristic
239#67—41no—
240#67—41no—
241#67—41no—
242#67—41no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255C2×Dic3—81no—
256C4×D3—42no—
257C4×D3—81no—
258C3×C8—42no—
259C3⋊C8—42no—
260C24—31no—
261C24D3×D411yescharacteristic
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—61no—
268C22×C4—22no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—62no—
283C22×C4—122no—
284C22×C4—121no—
285C22×C4C2×C4×D311yescharacteristic
286C22×C4#23511yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289C22×C4#23711yescharacteristic
290C22×C4C2×#6711yescharacteristic
291M16—242no—
292C2×C8—22no—
293C2×C8—42no—
294C2×C8—62no—
295C2×C8—122no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—122no—
304C4⋊C4—122no—
305C4⋊C4—124no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—61no—
309C22⋊C4—61no—
310C22⋊C4—61no—
311C22⋊C4—61no—
312C22⋊C4—124no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C42—61no—
318C42—61no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3#38611yescharacteristic
330C22×C3#41711yescharacteristic
331C22×C3#41811yescharacteristic
332C22×C3#46811yescharacteristic
333C22×C3#46811yescharacteristic
334C22×C3#46911yescharacteristic
335C22×C3#46911yescharacteristic
336C2×D3—42no—
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—41no—
340C2×D3—81no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—82no—
345C3×C4—161no—
346C3×C4#45812yes—
347Dic3—41no—
348Dic3—41no—
349Dic3—41no—
350Dic3—82no—
351Dic3—161no—
352C23—41no—
353C23—41no—
354C23—61no—
355C23—61no—
356C23—61no—
357C23—121no—
358C23—121no—
359C23D3×C22⋊C411yescharacteristic
360C23#88411yescharacteristic
361C23#91111yescharacteristic
362C23#91111yescharacteristic
363C23#91411yescharacteristic
364C23#91511yescharacteristic
365C23#93811yescharacteristic
366D4—121no—
367D4—121no—
368D4—121no—
369D4—121no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—122no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4—241no—
400C2×C4—241no—
401C2×C4#67⋊C411yescharacteristic
402C2×C4#88311yescharacteristic
403C2×C4C4×#6711yescharacteristic
404C2×C4#93711yescharacteristic
405C8—42no—
406C8—122no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3#189011yescharacteristic
414C2×C3#189511yescharacteristic
415C2×C3#190911yescharacteristic
416D3—81no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—122no—
428C22—121no—
429C22—121no—
430C22—121no—
431C22—241no—
432C22#477011yescharacteristic
433C22#477111yescharacteristic
434C22#478011yescharacteristic
435C22#478211yescharacteristic
436C22#495611yescharacteristic
437C22#515911yescharacteristic
438C22#517411yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—82no—
443C4—121no—
444C4—121no—
445C4—121no—
446C4—161no—
447C4—242no—
448C4—481no—
449C4#505312yes—
450C3#947411yescharacteristic, complemented, Sylow
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—241no—
458C2#6953811yescharacteristic
459C2#7020311yescharacteristic
460C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1823C2
socleC22×C3#386
prime radicalC22×#67D4
cosocleC23×C3×C4C23
hypercenter#1823D3
hypocenterC3#9474
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
3. C3×#1823C2semidirect
4. #71553C2semidirect
6. #67676C2semidirect
7. #67625C2semidirect
19. #4992C4semidirect
21. #4983C4semidirect
33. #1823D3semidirect
450. C3#9474semidirect

Maximal subgroups

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

subgroupindex
#821962
#786022
C3×#18232
#715532
#695302
#676762
#676252
#94743