Finite Group Databasebeta

← #11792767

Subgroups — #11792767

Order 1152.

813 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64151C211yesmaximal, characteristic
2C22×C3×#216C211yesmaximal, characteristic, complemented
3C2×#63833C211yesmaximal, characteristic
4C2×C3×#836C211yesmaximal, characteristic, complemented
5C2×#63560C211yesmaximal, characteristic, complemented
6#10591073C211yesmaximal, characteristic
7#10589166C211yesmaximal, characteristic, complemented
8#72074—31nomaximal
9#70125—31nomaximal
10C22×C32×Q8C2211yescharacteristic
11C2×#3828C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C2×#3756—21no—
14#64151—22no—
15#64151—22no—
16C2×#3755—21no—
17C2×#3748C2211yescharacteristic
18C2×C3×#216—22no—
19C2×C3×#216—22no—
20C2×C3×#216—22no—
21C2×C3×#208C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24C2×#3719—21no—
25#63833—22no—
26#63833—22no—
27C2×#3717—21no—
28C3×#836—22no—
29#63560—22no—
30#63557—22no—
31C22×C3×Q16—31no—
32C22×#216D311yescharacteristic, complemented
33C2×#942—31no—
34C2×#942—31no—
35C2×#942—31no—
36C2×#942—61no—
37C2×#909—31no—
38C2×C3×#101—31no—
39C2×#836—31no—
40C2×#836D311yescharacteristic, complemented
41C2×#824—31no—
42#5129—31no—
43#5110—31no—
44#4558—31no—
45#4552—31no—
46C2×C32×Q8—21no—
47C2×C32×Q8—21no—
48C2×C32×Q8—21no—
49C2×C32×Q8—21no—
50C2×C32×Q8—21no—
51C2×C32×Q8D412yes—
52C22×C32×C4—21no—
53C22×C32×C4C2311yescharacteristic
54C2×#638—21no—
55#3828—21no—
56#3828—21no—
57#3828—21no—
58#3828—21no—
59C2×C4⋊(C3⋊D3)D411yescharacteristic
60C2×C4×C3⋊D3—21no—
61C2×#634D411yescharacteristic
62C22×C3×Dic3—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6D412yes—
69#3756—24no—
70#3755—24no—
71#3748—21no—
72#3748D412yes—
73C2×C3⋊(C3⋊C8)—21no—
74C3×#216—28no—
75C3×#208—21no—
76C3×#208—21no—
77C3×#208—21no—
78C3×#208D412yes—
79C2×C3×C3⋊C8—21no—
80C2×C3×C3⋊C8—21no—
81C2×C3×C3⋊C8—21no—
82C2×C3×C3⋊C8—21no—
83C2×C3×C3⋊C8D412yes—
84#3719—24no—
85#3717—24no—
86#2020—91noSylow
87C22×C3×Q8—21no—
88C22×C3×Q8—31no—
89C22×C3×Q8C2×D311yescharacteristic
90C22×C3×Q8C2×D311yescharacteristic
91C2×#235—31no—
92C2×#235—31no—
93C2×#235—61no—
94C22×Dic6C2×D311yescharacteristic
95C2×C3×Q16—62no—
96C2×C3×Q16—62no—
97C2×C3×Q16—62no—
98C2×C3×M16—31no—
99C2×C3×M16—31no—
100C22×C3×C8—31no—
101C2×#216—22no—
102C2×#216—22no—
103C2×#216—22no—
104C2×#216—61no—
105C2×#216—61no—
106C2×#216—61no—
107C2×#216—62no—
108#942—62no—
109#942—62no—
110#942—62no—
111#942—64no—
112#942—64no—
113#942—62no—
114#942—62no—
115#942—62no—
116C2×#215—61no—
117C2×#215—61no—
118C2×#215—61no—
119C2×#215—62no—
120C2×#208—31no—
121C2×#208—31no—
122C2×#208—61no—
123C2×#208C2×D311yescharacteristic
124C2×#208C2×D311yescharacteristic
125C22×C3⋊C8C2×D311yescharacteristic
126#909—62no—
127#909—62no—
128C2×Dic12—61no—
129C2×#204—61no—
130C3×#101—62no—
131#836—22no—
132#836—62no—
133#824—62no—
134(C2×D3)⋊C8—62no—
135(C2×D3)⋊C8—62no—
136C23×C32#4111yescharacteristic
137C22×C3⋊D3—21no—
138C32×Q8—22no—
139C32×Q8—22no—
140C32×Q8—41no—
141C32×Q8—41no—
142C2×C32×C4—22no—
143C2×C32×C4—41no—
144C2×C32×C4—41no—
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D411yescharacteristic
151#638—41no—
152#638—41no—
153C2×C3⋊Dic3—21no—
154C4⋊(C3⋊D3)—21no—
155C4⋊(C3⋊D3)—21no—
156C4×C3⋊D3—41no—
157C4×C3⋊D3—41no—
158#634—21no—
159#634—21no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—41no—
162C2×C3×Dic3—41no—
163C3×Dic6—22no—
164C3×Dic6—22no—
165C3×Dic6—41no—
166C3×Dic6—41no—
167C3⋊(C3⋊C8)—22no—
168C3×C3⋊C8—22no—
169C3×C3⋊C8—22no—
170C3×C3⋊C8—41no—
171C2×#137—91no—
172C2×#137—91no—
173C22×Q16—31no—
174C2×#101—31no—
175C2×#101—91no—
176#408—91no—
177#341—91no—
178C2×C3×Q8—21no—
179C2×C3×Q8—21no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—22no—
191C2×C3×Q8—32no—
192C2×C3×Q8—41no—
193C2×C3×Q8—41no—
194C2×C3×Q8—41no—
195C2×C3×Q8—61no—
196C2×C3×Q8—61no—
197C2×C3×Q8—61no—
198C2×C3×Q8—61no—
199C2×C3×Q8—61no—
200C2×C3×Q8D1212yes—
201C2×C3×Q8#6712yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—41no—
206C22×C3×C4—61no—
207C22×C3×C4C22×D311yescharacteristic
208C22×C3×C4C22×D311yescharacteristic
209C2×#67—61no—
210C2×#67—61no—
211C2×#67—121no—
212C22×Dic3—21no—
213#235—62no—
214#235—62no—
215#235—61no—
216#235—61no—
217#235—61no—
218#235—61no—
219#235—61no—
220#235—61no—
221#235—61no—
222#235—61no—
223#235—121no—
224#235—121no—
225C2×D12—31no—
226C2×D12—31no—
227C2×D12—61no—
228C2×C4×D3—61no—
229C2×C4×D3—61no—
230C2×C4×D3—121no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—31no—
237C2×Dic6—31no—
238C2×Dic6—61no—
239C2×Dic6#6712yes—
240C3×Q16—68no—
241C3×M16—32no—
242C3×M16—61no—
243C3×M16—61no—
244C3×M16—61no—
245C2×C3×C8—32no—
246C2×C3×C8—61no—
247C2×C3×C8—61no—
248C2×C3×C8—61no—
249C2×C3×C8—61no—
250#216—28no—
251#216—64no—
252#216—68no—
253#216—64no—
254#216—64no—
255#215—64no—
256#215—68no—
257#215—64no—
258#215—64no—
259#208—21no—
260#208—21no—
261#208—21no—
262#208—32no—
263#208—32no—
264#208—62no—
265#208—62no—
266#208—61no—
267#208—61no—
268#208#6712yes—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—62no—
274C2×C3⋊C8—61no—
275C2×C3⋊C8—61no—
276C2×C3⋊C8#6712yes—
277Dic12—64no—
278#204—64no—
279C22×C32—22no—
280C22×C32#12111yescharacteristic
281C22×C32#12111yescharacteristic
282C22×C32#12111yescharacteristic
283C2×C3⋊D3—41no—
284C2×C3⋊D3—42no—
285C32×C4—42no—
286C32×C4#12011yescharacteristic
287C32×C4#12011yescharacteristic
288C32×C4#12011yescharacteristic
289C32×C4#12711yescharacteristic
290C3⋊Dic3—41no—
291C3×Dic3—42no—
292C2×#41—91no—
293C22×Q8—31no—
294C22×Q8D3211yescharacteristic
295#137—182no—
296#137—182no—
297#137—182no—
298#137—182no—
299C2×Q16—62no—
300C2×Q16—62no—
301C2×Q16—62no—
302C2×Q16—181no—
303C2×Q16—181no—
304C2×SD16—181no—
305C2×SD16—181no—
306C2×M16—31no—
307C2×M16—31no—
308C2×M16—91no—
309C22×C8—31no—
310#101—62no—
311#101—182no—
312C22⋊C8—182no—
313C23×C3—21no—
314C23×C3#23511yescharacteristic
315C23×C3#23711yescharacteristic
316C22×D3—61no—
317C22×D3—61no—
318C22×D3—121no—
319C3×Q8—24no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8—42no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—41no—
328C3×Q8—41no—
329C3×Q8—41no—
330C3×Q8—41no—
331C3×Q8—62no—
332C3×Q8—62no—
333C3×Q8—121no—
334C3×Q8—121no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—62no—
351C2×C3×C4—121no—
352C2×C3×C4—121no—
353C2×C3×C4C2×D1211yescharacteristic
354C2×C3×C4D3×D411yescharacteristic
355C2×C3×C4D3×D411yescharacteristic
356C2×C3×C4D3×D411yescharacteristic
357C2×C3×C4D3×D411yescharacteristic
358C2×C3×C4D3×D411yescharacteristic
359C2×C3×C4D3×D411yescharacteristic
360C2×C3×C4D3×D411yescharacteristic
361C2×C3×C4C2×#6711yescharacteristic
362C2×C3×C4C2×#6711yescharacteristic
363C2×C3×C4C2×#6711yescharacteristic
364C2×C3×C4C2×#6711yescharacteristic
365#67—122no—
366#67—122no—
367#67—121no—
368#67—121no—
369#67—121no—
370#67—121no—
371C2×Dic3—22no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—121no—
377D12—62no—
378D12—61no—
379D12—61no—
380D12—61no—
381D12—61no—
382D12—121no—
383C4×D3—122no—
384C4×D3—122no—
385C4×D3—121no—
386C4×D3—121no—
387C4×D3—121no—
388C4×D3—121no—
389Dic6—22no—
390Dic6—22no—
391Dic6—41no—
392Dic6—41no—
393Dic6—62no—
394Dic6—61no—
395Dic6—61no—
396Dic6—61no—
397Dic6—61no—
398Dic6—121no—
399C3×C8—62no—
400C3×C8—62no—
401C3×C8—121no—
402C3⋊C8—22no—
403C3⋊C8—22no—
404C3⋊C8—41no—
405C3⋊C8—64no—
406C3⋊C8—62no—
407C3⋊C8—62no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32#39211yescharacteristic
411C2×C32#47312yes—
412C3⋊D3—81no—
413#41—181no—
414#41—181no—
415#41—181no—
416#41—181no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—32no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—91no—
429C2×Q8#62612yes—
430C2×D4—91no—
431C2×D4—181no—
432C22×C4—21no—
433C22×C4—61no—
434C22×C4—181no—
435C22×C4C2×D3211yescharacteristic
436Q16—68no—
437Q16—184no—
438Q16—184no—
439SD16—184no—
440SD16—184no—
441M16—32no—
442M16—61no—
443M16—61no—
444M16—61no—
445M16—92no—
446M16—181no—
447C2×C8—32no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—181no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—42no—
459C22×C3#88411yescharacteristic
460C22×C3#88411yescharacteristic
461C22×C3#93011yescharacteristic
462C22×C3#93811yescharacteristic
463C22×C3#93911yescharacteristic
464C22×C3#93911yescharacteristic
465C2×D3—122no—
466C2×D3—122no—
467C2×D3—121no—
468C2×D3—121no—
469C2×D3—122no—
470C2×D3—242no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—44no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—122no—
479C3×C4#88211yescharacteristic
480C3×C4#88211yescharacteristic
481C3×C4#93711yescharacteristic
482C3×C4#93711yescharacteristic
483C3×C4#93711yescharacteristic
484C3×C4#94011yescharacteristic
485C3×C4#94011yescharacteristic
486C3×C4#95311yescharacteristic
487Dic3—42no—
488Dic3—122no—
489Dic3—121no—
490Dic3—121no—
491C32#202011yescharacteristic, complemented, Sylow
492C23—181no—
493C23#380411yescharacteristic
494Q8—22no—
495Q8—22no—
496Q8—41no—
497Q8—41no—
498Q8—62no—
499Q8—62no—
500Q8—121no—
501Q8—121no—
502Q8—181no—
503Q8—181no—
504D4—181no—
505D4—181no—
506D4—361no—
507D4—361no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—62no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—181no—
515C2×C4—361no—
516C2×C4—361no—
517C2×C4C2×#62611yescharacteristic
518C2×C4D3×#6711yescharacteristic
519C2×C4D3×#6711yescharacteristic
520C2×C4D3×#6711yescharacteristic
521C2×C4#381111yescharacteristic
522C2×C4#381111yescharacteristic
523C8—62no—
524C8—62no—
525C8—121no—
526C8—182no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—22no—
532C2×C3—21no—
533C2×C3—41no—
534C2×C3—41no—
535C2×C3#475512yes—
536C2×C3#496011yescharacteristic
537C2×C3#519212yes—
538C2×C3#522311yescharacteristic
539D3—242no—
540D3—241no—
541D3—241no—
542C22—22no—
543C22—362no—
544C22—361no—
545C22#6395811yescharacteristic
546C22#6397111yescharacteristic
547C22#6397811yescharacteristic
548C4—42no—
549C4—122no—
550C4—361no—
551C4#6397711yescharacteristic
552C4#6397911yescharacteristic
553C4#6398111yescharacteristic
554C4#6398311yescharacteristic
555C3—21no—
556C3#7012511yescharacteristic, complemented
557C3#7207411yescharacteristic, complemented
558C2—21no—
559C2—21no—
560C2—721no—
561C2#1059154312yes—
562C2#1059172411yescharacteristic
subgroupG/H
centerC22#63958
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC22×C32#121
prime radicalC2×#3828C22
hypercenterC22×Q8D32
hypocenterC32#2020
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C22×C3×#216C2semidirect
4. C2×C3×#836C2semidirect
5. C2×#63560C2semidirect
7. #10589166C2semidirect
32. C22×#216D3semidirect
40. C2×#836D3semidirect
491. C32#2020semidirect
556. C3#70125semidirect
557. C3#72074semidirect

Maximal subgroups

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

subgroupindex
C2×#641512
C22×C3×#2162
C2×#638332
C2×C3×#8362
C2×#635602
#105910732
#105891662
#720743
#701253