Finite Group Databasebeta

← #10695675

Subgroups — #10695675

Order 768.

770 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C3×#3187C211yesmaximal, characteristic
2C2×#5050C211yesmaximal, characteristic, complemented
3#82654C211yesmaximal, characteristic
4#71701C211yesmaximal, characteristic, complemented
5#71675C211yesmaximal, characteristic
6#69963C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9956—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×C3×#121C2211yescharacteristic
12C22×#216—21no—
13C2×#936—21no—
14C3×#483—21no—
15C3×#481—21no—
16C3×#465—22no—
17C3×#450—21no—
18C3×#448—21no—
19C2×C3×C22⋊C8C2211yescharacteristic
20#5220—22no—
21#5212C2211yescharacteristic
22C4×#217C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#816—22no—
25#5050—22no—
26#5050—22no—
27#4973—21no—
28C2×#810—21no—
29#4947—22no—
30#4537—22no—
31#4526—22no—
32#3187D311yescharacteristic
33C2×#451—31no—
34#2436—31no—
35#1996—31no—
36#1787—31no—
37#1519—31no—
38#1282—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C22×Dic6—21no—
45C22×Dic6—21no—
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—44no—
50C22×Dic6D412yes—
51C2×C3×SD16—22no—
52C22×C3×C8D412yes—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×#217—24no—
67C2×#217—21no—
68C2×#217D412yes—
69C2×#216—24no—
70C2×#216—44no—
71#936—44no—
72C2×C4⋊Dic3—21no—
73C2×C4×Dic3—21no—
74C22×C3⋊C8—21no—
75C3×#106—22no—
76C3×#103—22no—
77C3×#102—22no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8—21no—
80C3⋊#95—24no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83#830—21no—
84#830D412yes—
85#816—44no—
86#810—24no—
87C23×Q8—31no—
88C22×Q16—61no—
89C2×#124—61no—
90C2×C22⋊Q8C2×D311yescharacteristic
91C2×#121C2×D311yescharacteristic
92#483—21no—
93#481—21no—
94#465—22no—
95#451—62no—
96#451—62no—
97#450—21no—
98#448—21no—
99C2×#103—61no—
100C2×#103—62no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8C2×D311yescharacteristic
103#393—62no—
104#391—61no—
105#386—62no—
106#380—31no—
107C4×C22⋊C4—31no—
108#342—62no—
109#327—62no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—42no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—42no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—42no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×Dic3—22no—
133C22×Dic3—22no—
134C22×Dic3—44no—
135C22×Dic3—41no—
136C22×Dic3—41no—
137C22×Dic3—41no—
138C2×Dic6—24no—
139C2×Dic6—24no—
140C2×Dic6—22no—
141C2×Dic6—22no—
142C2×Dic6—24no—
143C2×Dic6—44no—
144C2×Dic6—44no—
145C2×Dic6—44no—
146C2×Dic6—42no—
147C2×Dic6—42no—
148C2×Dic6—42no—
149C2×Dic6—41no—
150C2×Dic6—41no—
151C3×SD16—82no—
152C2×C3×C8—22no—
153C2×C3×C8—42no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164#217—22no—
165#217—22no—
166#217—48no—
167#217—42no—
168#216—48no—
169C4⋊Dic3—42no—
170C4×Dic3—42no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—42no—
173C22×Q8—32no—
174C22×Q8—61no—
175C22×Q8—61no—
176C22×Q8—61no—
177C22×Q8—61no—
178C22×Q8—61no—
179C22×Q8—124no—
180C22×Q8#6711yescharacteristic
181C22×D4—21no—
182C22×D4#6711yescharacteristic
183C23×C4—61no—
184C23×C4C22×D311yescharacteristic
185C2×Q16—64no—
186C2×Q16—124no—
187C2×SD16—22no—
188C22×C8—61no—
189C22×C8#6712yes—
190#124—124no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195#121—21no—
196#121—21no—
197#121—21no—
198#121—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4#6711yescharacteristic
202C2×C4⋊C4#6711yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—32no—
206C2×C22⋊C4—64no—
207C2×C22⋊C4—61no—
208C2×C42—61no—
209#106—22no—
210#103—22no—
211#103—64no—
212#103—124no—
213#102—22no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216C22⋊C8—32no—
217C22⋊C8—61no—
218#95—64no—
219#95—61no—
220#95—61no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—41no—
226C23×C3—42no—
227C23×C3#12011yescharacteristic
228C23×C3#12111yescharacteristic
229C23×C3#12711yescharacteristic
230C23×C3#13512yes—
231C23×C3#13712yes—
232C3×Q8—42no—
233C3×Q8—42no—
234C3×D4—42no—
235C3×D4—81no—
236C3×D4—82no—
237C3×D4—81no—
238C3×D4—81no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—81no—
250C2×C3×C4—81no—
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×Dic3—24no—
256C2×Dic3—44no—
257C2×Dic3—44no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—81no—
264C2×Dic3—81no—
265C2×Dic3—81no—
266Dic6—24no—
267Dic6—24no—
268Dic6—42no—
269Dic6—42no—
270Dic6—44no—
271Dic6—44no—
272C3×C8—42no—
273C3⋊C8—42no—
274C24—21no—
275C24D3×D411yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—42no—
279C2×Q8—64no—
280C2×Q8—64no—
281C2×Q8—64no—
282C2×Q8—62no—
283C2×Q8—62no—
284C2×Q8—122no—
285C2×Q8—122no—
286C2×Q8—122no—
287C2×Q8—124no—
288C2×Q8—124no—
289C2×Q8—124no—
290C2×Q8—121no—
291C2×Q8—121no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—124no—
306C22×C4—121no—
307C22×C4—121no—
308C22×C4—121no—
309C22×C4D3×D411yescharacteristic
310C22×C4D3×D411yescharacteristic
311C22×C4#23711yescharacteristic
312C22×C4C2×#6711yescharacteristic
313C22×C4C2×#6711yescharacteristic
314C22×C4C2×#6711yescharacteristic
315Q16—128no—
316SD16—82no—
317C2×C8—22no—
318C2×C8—42no—
319C2×C8—62no—
320C2×C8—122no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—122no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C22⋊C4—128no—
335C22⋊C4—122no—
336C42—122no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—84no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3#39211yescharacteristic
350C22×C3#45212yes—
351C22×C3#46412yes—
352C22×C3#49511yescharacteristic
353C22×C3#49711yescharacteristic
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—81no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4B(4,𝔽2)12yes—
361Dic3—44no—
362Dic3—82no—
363Dic3—161no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—41no—
370C23#91512yes—
371C23#91612yes—
372C23#93711yescharacteristic
373C23#93911yescharacteristic
374C23#94011yescharacteristic
375Q8—42no—
376Q8—42no—
377Q8—64no—
378Q8—64no—
379Q8—122no—
380Q8—122no—
381Q8—124no—
382Q8—124no—
383D4—42no—
384D4—81no—
385D4—82no—
386D4—81no—
387D4—81no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—64no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—122no—
402C2×C4—124no—
403C2×C4—124no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4—241no—
409C2×C4—241no—
410C2×C4—241no—
411C2×C4#93711yescharacteristic
412C2×C4#93811yescharacteristic
413C2×C4#93911yescharacteristic
414C2×C4#95311yescharacteristic
415C8—42no—
416C8—122no—
417C2×C3—22no—
418C2×C3—21no—
419C2×C3—22no—
420C2×C3—21no—
421C2×C3—161no—
422C2×C3#200911yescharacteristic
423C2×C3#201511yescharacteristic
424C2×C3#202111yescharacteristic
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—42no—
432C22—42no—
433C22—84no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22#516612yes—
438C22#516711yescharacteristic
439C22#516811yescharacteristic
440C22#517312yes—
441C22#522311yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—124no—
449C4—242no—
450C4—481no—
451C4#524412yes—
452C3#995611yescharacteristic, complemented, Sylow
453C2—22no—
454C2—21no—
455C2—22no—
456C2—21no—
457C2—161no—
458C2#7185111yescharacteristic
459C2#7205811yescharacteristic
460C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3187C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#3187D3
hypocenterC3#9956
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
2. C2×#5050C2semidirect
4. #71701C2semidirect
6. #69963C2semidirect
7. #67677C2semidirect
452. C3#9956semidirect

Maximal subgroups

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

subgroupindex
C3×#31872
C2×#50502
#826542
#717012
#716752
#699632
#676772
#99563