Finite Group Databasebeta

← #10694712

Subgroups — #10694712

Order 768.

510 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 111 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×#5051C211yesmaximal, characteristic
2C3×#1996C211yesmaximal, characteristic
3#71690C211yesmaximal, characteristic
4#69511C211yesmaximal, characteristic
5#69510C211yesmaximal, characteristic
6#69457C211yesmaximal, characteristic
7#69447C211yesmaximal, characteristic
8#9784—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×#216C411yescharacteristic, complemented
11C2×#924C2211yescharacteristic
12C23×C3⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C2×#835C411yescharacteristic, complemented
16#5212C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5051—21no—
19#5051—21no—
20#5051—21no—
21#5051—21no—
22C2×#810C411yescharacteristic, complemented
23C2×#808—21no—
24C2×#807C411yescharacteristic, complemented
25C2×#803—21no—
26C3×#342—22no—
27#4538—22no—
28#4529—21no—
29#4527—22no—
30#4524—21no—
31#4452—22no—
32C2×#467—31no—
33#1996—31no—
34#1996D311yescharacteristic
35#1807—31no—
36#1752—31no—
37#1520—31no—
38#1520—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×Dic6C2×C411yescharacteristic
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4C2×C411yescharacteristic
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4D412yes—
52C2×C3×C42—21no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#216—21no—
56C2×#216—21no—
57C2×#216—42no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3C2×C411yescharacteristic
60#924—21no—
61#924—21no—
62#924—21no—
63#924—21no—
64C2×Dic3⋊C4—21no—
65C22×C3⋊C8—21no—
66C22×C3⋊C8—42no—
67C22×C3⋊C8C2×C411yescharacteristic
68C22×C3⋊C8C2×C411yescharacteristic
69C22×C3⋊C8D411yescharacteristic
70C22×C3⋊C8D411yescharacteristic
71C22×C3⋊C8D411yescharacteristic
72C22×C3⋊C8D411yescharacteristic
73C3×#95—21no—
74C3×#95—21no—
75#835—21no—
76#835—21no—
77C3⋊#95—21no—
78#830—21no—
79#830D412yes—
80#810—21no—
81#810—21no—
82#808—42no—
83#807—21no—
84#807—21no—
85#803—42no—
86C22×Q16—31no—
87C23×C8—31no—
88C2×C22⋊Q8—31no—
89C2×C22⋊Q8C2×D311yescharacteristic
90#467—61no—
91#467—61no—
92#467—61no—
93#467—61no—
94C2×#107—31no—
95C2×#106—61no—
96C2×#103—31no—
97C2×#103—31no—
98C2×C22⋊C8—31no—
99C2×#97—61no—
100#380—31no—
101#380C2×D311yescharacteristic
102C4×C22⋊C4C2×D311yescharacteristic
103#342—22no—
104#342—62no—
105#340—61no—
106#336—61no—
107#328—62no—
108#328—62no—
109C24×C3C2×D411yescharacteristic
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C22×C3×C4—22no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—41no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—41no—
132C2×Dic6—41no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C22⋊C4—22no—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—42no—
144C3×C42—41no—
145C3×C42—41no—
146#217—41no—
147#217—41no—
148#217—42no—
149#216—42no—
150#216—42no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8C22⋊C411yescharacteristic
167C2×C3⋊C8C22⋊C411yescharacteristic
168C2×C3⋊C8C22⋊C411yescharacteristic
169C2×C3⋊C8C22⋊C411yescharacteristic
170C22×Q8—31no—
171C22×Q8C4×D311yescharacteristic
172C23×C4C22×D311yescharacteristic
173C2×Q16—61no—
174C2×Q16—61no—
175C2×Q16—122no—
176C22×C8—31no—
177C22×C8—31no—
178C22×C8—31no—
179C22×C8—31no—
180C22×C8—31no—
181C22×C8—31no—
182C22×C8—61no—
183C22×C8—122no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187C22⋊Q8—21no—
188C22⋊Q8—61no—
189C22⋊Q8—61no—
190C22⋊Q8—61no—
191C22⋊Q8—61no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4C4×D311yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4#6712yes—
203C2×C42—21no—
204#107—61no—
205#107—61no—
206#106—122no—
207#103—61no—
208#103—61no—
209#103—61no—
210#103—61no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213#97—122no—
214#95—21no—
215#95—21no—
216#95—61no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3C4×D411yescharacteristic
220C23×C3#12111yescharacteristic
221C23×C3#12711yescharacteristic
222C23×C3#13712yes—
223C23×C3#13712yes—
224C3×Q8—41no—
225C3×Q8—41no—
226C3×Q8—81no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4C2×C22⋊C411yescharacteristic
247C2×C3×C4C4×D411yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12411yescharacteristic
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—81no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255C2×Dic3—81no—
256C2×Dic3—81no—
257Dic6—41no—
258Dic6—41no—
259Dic6—81no—
260C3⋊C8—21no—
261C3⋊C8—21no—
262C3⋊C8—21no—
263C3⋊C8—42no—
264C3⋊C8#9912yes—
265C24C2×#6711yescharacteristic
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—61no—
271C2×Q8—61no—
272C2×Q8—121no—
273C2×Q8—121no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—121no—
283C22×C4C2×C4×D311yescharacteristic
284C22×C4#23511yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4#23711yescharacteristic
289Q16—122no—
290Q16—122no—
291C2×C8—31no—
292C2×C8—31no—
293C2×C8—31no—
294C2×C8—31no—
295C2×C8—62no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8—62no—
299C2×C8—121no—
300C2×C8—121no—
301C2×C8—121no—
302C2×C8—121no—
303C2×C8—121no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—121no—
313C4⋊C4—121no—
314C4⋊C4—121no—
315C4⋊C4—121no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—121no—
322C22⋊C4—121no—
323C22⋊C4—122no—
324C42—41no—
325C42—41no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—42no—
333C22×C3—42no—
334C22×C3#39011yescharacteristic
335C22×C3#44112yes—
336C22×C3#47012yes—
337C22×C3#49711yescharacteristic
338C22×C3#49711yescharacteristic
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—82no—
343C3×C4—81no—
344C3×C4—81no—
345C3×C4—161no—
346C3×C4C2×#9911yescharacteristic
347C3×C4#41011yescharacteristic
348Dic3—81no—
349Dic3—81no—
350Dic3—161no—
351C23—41no—
352C23—41no—
353C23C4×#6711yescharacteristic
354C23#93911yescharacteristic
355C23#94011yescharacteristic
356C23#94212yes—
357C23#95112yes—
358Q8—41no—
359Q8—41no—
360Q8—81no—
361Q8—121no—
362Q8—121no—
363Q8—241no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4—241no—
388C2×C4—241no—
389C2×C4—241no—
390C2×C4D3×C22⋊C411yescharacteristic
391C2×C4#67⋊C411yescharacteristic
392C2×C4#88411yescharacteristic
393C2×C4#88511yescharacteristic
394C8—32no—
395C8—61no—
396C8—61no—
397C8—61no—
398C8—122no—
399C2×C3—21no—
400C2×C3—22no—
401C2×C3—21no—
402C2×C3—22no—
403C2×C3#196411yescharacteristic
404C2×C3#197111yescharacteristic
405C2×C3#197611yescharacteristic
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—42no—
413C22—42no—
414C22#495711yescharacteristic
415C22#503012yes—
416C22#505212yes—
417C22#506011yescharacteristic
418C22#508611yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—82no—
423C4—81no—
424C4—81no—
425C4—161no—
426C4—241no—
427C4—241no—
428C4—481no—
429C4#474011yescharacteristic
430C4D3×#9911yescharacteristic
431C3#978411yescharacteristic, complemented, Sylow
432C2—22no—
433C2—21no—
434C2—21no—
435C2—22no—
436C2#6967811yescharacteristic
437C2#7003011yescharacteristic
438C2#7004211yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1996C2
socleC22×C3#390
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1996D3
hypocenterC3#9784
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
10. C22×#216C4semidirect
15. C2×#835C4semidirect
22. C2×#810C4semidirect
24. C2×#807C4semidirect
431. C3#9784semidirect

Maximal subgroups

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

subgroupindex
C2×#50512
C3×#19962
#716902
#695112
#695102
#694572
#694472
#97843